Princeton University

Tue 14 Jul 2009, 2:00pm
Diff. Geom, Math. Phys., PDE Seminar
MATX 1118

Second boundary value problem for special Lagrangian submanifolds

MATX 1118
Tue 14 Jul 2009, 2:00pm3:00pm
Abstract
Given any two uniformly convex regions in Euclidean space, we show that there exists a unique diffeomorphism between them, such that the graph of the diffeomorphism is a special Lagrangian submanifold in the product space. This is joint work with Simon Brendle.
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UBC

Wed 16 Sep 2009, 4:00pm
Diff. Geom, Math. Phys., PDE Seminar
MATH 103

Dimensional reduction of the meanfield dynamics of bosons in strongly anisotropic harmonic potentials

MATH 103
Wed 16 Sep 2009, 4:00pm5:00pm
Abstract
I discuss recent results on the spatial dimensional reduction of the effective meanfield dynamics of manybody bosonic systems in strongly anisotropic harmonic potentials. In particular, the dynamics in the limit of strong anisotropy is effectively described by the nonlinear Hartree equation that is restricted to a submanifold of the original configuration space. Time permitting, I will discuss open problems regrading the meanfield dynamics of manybody constraint quantum systems.
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UBC and Nancy 1

Tue 22 Sep 2009, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

Controllability results for a fishlike swimming body

WMAX 110
Tue 22 Sep 2009, 3:30pm4:30pm
Abstract
We study the controllability of a shape changing body immersed in a perfect fluid. The shape changes are prescribed as functions of time and satisfy constraints ensuring that they are due to the work of body's internal forces only. The net locomotion of the body results from the exchange of momentum between the shape changes and the fluid. We consider the control problem that associates to any given shape changes the trajectory of the body in the fluid and we will show how this nonstandard control problem can be solved within the framework of geometric control theory.
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U. Virginia

Tue 29 Sep 2009, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

On an isoperimetric inequality for a Schroedinger operator depending on the curvature of a loop

WMAX 110
Tue 29 Sep 2009, 3:30pm4:30am
Abstract
Let \gamma be a smooth closed curve of length 2\pi in R^3, and let \kappa(s) be its curvature regarded as a function of arc length s. We associate with this curve the onedimensional Schroedinger operator H_\gamma = d^2/ds^2 + \kappa^2(s) acting on the space of square integrable 2\piperiodic functions. A natural conjecture is that the lowest eigenvalue e_0(\gamma) of H_\gamma is bounded below by 1 for any \gamma (this value is assumed when \gamma is a circle). We study a family of curves which includes the circle and for which e_0(\gamma)=1 as well, and show that the curves in this family are local minimizers; i.e., e_0(\gamma) does not decrease under small perturbations. A connection between the inequality and a dynamical elastica will be described. The conjecture remains open.
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Northwestern University

Tue 6 Oct 2009, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar / Probability Seminar
WMAX 110

Volume Growth, Brownian motion, and Conservation of the heat kernel on a Riemannian manifold

WMAX 110
Tue 6 Oct 2009, 3:30pm4:30pm
Abstract
The minimal heat kernel on a Riemannian manifold is conservative if it integrates to 1. If this is the case, the manifold is said to be stochastically complete. Since the heat kernel is the transition density function of Brownian motion, a manifold is stochastically complete if and only if Brownian motion does not explode. This interpretation opens a way for investigating conservation of the heat kernel by probability theory. To find a proper geometric condition for heat kernel conservation is an old geometric problem. The first result in this direction was due to S. T. Yau, who proved that a Riemannian manifold is stochastically complete if its Ricci curvature is bounded from below by a constant. However, it has been known for quite some time that the heat kernel conservation property is intimately related to the volume growth of a Riemannian manifold. We study this problem by looking at the more refined question of how fast Brownian motion escapes to infinity, for the existence of a deterministic upper bound for the escaping rate implies heat kernel conservation. We show how the Neumann heat kernel, time reversal of reflecting Brownian motion, and volumes of geodesic balls all come together in this problem and give an elegant and often sharp upper bound of the escaping rate solely in terms of the volume growth function without any extra geometric restriction besides geodesic completeness. The talk should be interesting and accessible to differential geometers, people in partial differential equations (pdeers), and probabilists.
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UBC

Tue 13 Oct 2009, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

On the best constant in the MoserOnofriAubin inequality

WMAX 110
Tue 13 Oct 2009, 3:30pm4:30pm
Abstract
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UBC

Tue 27 Oct 2009, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar

Inverse problems via variational methods

Tue 27 Oct 2009, 3:30pm4:30pm
Abstract
We present a general variational method, involving selfdual variational calculus, for recovering nonlinearities from prescribed solutions for certain types of PDEs which are not necessarily of EulerLagrange type, including parabolic equations. The approach can also be used for optimal control problems. The topological aspects involved, for the space of selfdual Lagrangians, and the space of maximal monotone vector fields on a reflexive Bancah space will be discussed.
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Princeton University

Tue 3 Nov 2009, 3:00pm
Diff. Geom, Math. Phys., PDE Seminar

Mass critical generalized KdV equation

Tue 3 Nov 2009, 3:00pm4:00pm
Abstract
I will discuss the scattering problem of masscritical generalized KdV equation. We will see if the scattering of gKdV fails, then a minimal mass blowup solution exist on the condition that scattering of masscritical 1D NLS is true. We use concentration compactness argument in addition to an observation that a certain modulated, rescaled version of NLS solution is approximately gKdV solution for highly oscillatory profile.
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U. of Washington

Mon 9 Nov 2009, 2:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
MATX 1118

Applications of Optimal Transport I

MATX 1118
Mon 9 Nov 2009, 2:30pm3:30pm
Abstract
In this first talk we'll give an advertisement for the rigorous and formal tools of optimal transportation, highlighting their contribution to diffusion equations, simple proofs of Sobolev and isoperimetric inequalities, generalizing the Ricciboundedbelow condition beyond smooth manifolds, and geometrically reinterpreting the Schroedinger equation. We'll then learn about two ideas at the center of these applications: 1) that probability measures can be formally seen as a Riemannian manifold (F. Otto '01) and 2) certain entropy functionals are convex in this geometry (R. M cCann '94). We'll fill out the hour by reviewing the formal Riemannian structure (local geometry) and rigorous aspects of (global) Wasserstein distance.
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U. of Washington

Tue 10 Nov 2009, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (at PIMS) (Notice the title change)

Applications of Optimal Transport II

WMAX 110 (at PIMS) (Notice the title change)
Tue 10 Nov 2009, 3:30pm4:30pm
Abstract
In this second talk we'll see how commonly studied PDEs like the heat equation, nonlinear diffusion, thin film equation, and Schroedinger equation can be formally seen as geometric evolutions in the Riemannian geometry of probability measures. The work on Schroedinger equation is due to MaxK. von Renesse ('09).
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U. of Victoria

Thu 12 Nov 2009, 3:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS minisymposium in PDE); time changed

Traffic Flow and Traffic Jams: From Kinetic Theory to Functional Differential Equations

WMAX 110 (PIMS minisymposium in PDE); time changed
Thu 12 Nov 2009, 3:00pm4:00pm
Abstract
I will speak on certain kinetic and macroscopic models of traffic flow. After a review of the concept of a fundamental diagram the highdensity regime will be considered, and the emergence of macroscopic models with nolocalities will be discussed. Numerical evidence (and real traffic data) suggest that travelling "braking" waves form and propagate in response to trigger events. A traveling wave ansatz for solutions of the macroscopic models leads to an unusual functional differential equation, for which preliminary studies will be shown.
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University of Washington

Fri 13 Nov 2009, 1:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110: PIMS minisymposium in PDE

Fourth order diffusion with geometric link to second order diffusion

WMAX 110: PIMS minisymposium in PDE
Fri 13 Nov 2009, 1:00pm2:00pm
Abstract
We describe a fourth order family generalizing the linearmobility thin film equation on R^n. In joint work with R. McCann we derive formally sharp converence rates to selfsimilarity, using a link to DenzlerMcCann's analysis of a second order diffusion. We then show (joint with Matthes, McCann, Savare) that a certain range of nonlinearity allows the obtaining of rigorous results for the fourthorder evolution in 1 dimension.
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UCLA

Fri 13 Nov 2009, 2:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
MATX 1100: PIMS minisymposium in PDE

Dynamics of Kinematic Aggregation Patterns

MATX 1100: PIMS minisymposium in PDE
Fri 13 Nov 2009, 2:00pm3:00pm
Abstract
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UBC

Tue 17 Nov 2009, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 PIMS

General Hardy inequalities with improvements and applications

WMAX 110 PIMS
Tue 17 Nov 2009, 3:30pm4:30pm
Abstract
We derive a general Hardy inequality and show most Hardy inequalities can be seen as special cases of this inequality. In addition we characterize the improvements of this inequality and (time permitting) we show an application of this inequality to the regularity of stable solutions to a nonvariational equation.
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UBC

Tue 24 Nov 2009, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (at PIMS)

Nonnegatively crosscurved transportation costs

WMAX 110 (at PIMS)
Tue 24 Nov 2009, 3:30pm4:30pm
Abstract
The theory of optimal transport is concerned with phenomena arising when one matches two mass distributions in a most economic way, minimizing transportation cost of moving mass from one location to another. We consider an optimal transportation problem with costs satisfying certain type of degenerate curvature condition. This condition is a slightly stronger but still degenerate version of the MaTrudinger Wang condition for regularity of optimal transport maps. We explain a continuity result of optimal maps with rough data on local and global domains. If time permits, we will also explain a connection to Principal Agent problem in microeconomics. These reflect joint work in progress with Alessio Figalli and Robert McCann.
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UBC

Tue 1 Dec 2009, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (at PIMS)

Small solutions of Nonlinear Schrodinger Equations with Many Bound States

WMAX 110 (at PIMS)
Tue 1 Dec 2009, 3:30pm4:30pm
Abstract
Consider a nonlinear Schr\"{o}dinger equation in $\mathbb{R}^3$ with a shortrange potential. The linear Hamiltonian is assumed to have three or more eigenvalues satisfying some resonance conditions. We study the asymptotic behavior at time infinity of solutions with small initial data in $H^1 \cap L^1(\mathbb{R}^3)$. The results include the case that all of the eigenvalues are simple and also the case that the second eigenvalues are degenerate. These are joint works with Stephen Gustafson, Kenji Nakanishi and TaiPeng Tsai.
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UBC

Tue 19 Jan 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

Singularities and asymptotics for some dynamics of maps into the sphere

WMAX 110
Tue 19 Jan 2010, 3:30pm4:30pm
Abstract
I will describe some background and recent results on singularity formation (and nonformation) for some simple, physical, and popular geometric PDE describing dynamics of maps into spheres  the heatflow, wave map, and Schroedinger map  in the energycritical 2D case. I'll try to keep it simple and accessible by illustrating the methods on a symmetric reduction of the heatflow, leading to a single scalar PDE.
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Stanford University

Tue 26 Jan 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX110

The Inverse Calderon Problem for Schoedinger Operator on Riemann Surfaces

WMAX110
Tue 26 Jan 2010, 3:30pm4:30am
Abstract
We show that on a smooth compact Riemann surface with boundary (M_0, g) the DirichlettoNeumann map of the Schr\"odinger operator \Delta_g + V determines uniquely the potential V. This seemingly analytical problem turns out to have connections with ideas in symplectic geometry and differential topology. We will discuss how these geometrical features arise and the techniques we use to treat them.
This is joint work with Colin Guillarmou of CNRS Nice.
The speaker is partially supported by NSF Grant No. DMS0807502 during this work.
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University of Chicago

Thu 28 Jan 2010, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX110

Traveling Fronts in Combustible Media

WMAX110
Thu 28 Jan 2010, 3:30pm4:30am
Abstract
Traveling fronts are special solutions of reactiondiffusion equations which model phenomena such as propagation of species in an environment or spreading of flames in combustible media. In this talk we will address questions of existence, uniqueness, and stability of traveling fronts in general inhomogeneous media. We will show that in certain circumstances a unique front exists and it is a global attractor of the corresponding parabolic evolution, thus describing long time dynamics for very general solutions of the PDE. In contrast to this, we will also present examples of media where no traveling front solutions exist.
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University of Toronto

Thu 4 Feb 2010, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar / Probability Seminar
WMAX 110

Wellposedness of stochastic PDEs

WMAX 110
Thu 4 Feb 2010, 3:30pm4:30pm
Abstract
In this talk, we first discuss the second iteration argument introduced by Bourgain to establish LWP of KdV with measures as initial data. Then, we establish LWP of the stochastic KdV (SKdV) with additive spacetime white noise by estimating the stochastic convolution via Ito calculus and showing its continuity via the factorization method. Next, we discuss
wellposedness of SKdV with multiplicative noise in $L^2$. In order to treat the nonzero mean case, we derive a coupled system of a SDE and a SPDE.
Lastly, as a toy model to study KPZ equation and stochastic Burgers equation, we study stochastic KdVBurgers equation (SKdVB). We discuss how Fourier analytic technique can be applied to show LWP. If time permits, we discuss how one can obtain global wellposedness of these equations via (1) analogue of conservation laws, (2) Applying Bourgain's argument for invariant measures (for deterministic PDEs) to SPDEs.
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University of Chicago

Tue 9 Feb 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX110

cancelled

WMAX110
Tue 9 Feb 2010, 3:30pm4:30pm
Abstract
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University of Iowa

Tue 2 Mar 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

Threshold solutions in critical nonlinear Schrodinger equations

WMAX 110
Tue 2 Mar 2010, 3:30pm4:30pm
Abstract
I will explain recent joint work with Xiaoyi Zhang on threshold solutions to critical nonlinear Schrodinger equations. These results are analogues of Liouvilletype theorems in the dispersive setting. I will cover mainly the masscritical case. Time permitting the energycritical case will also be discussed.
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UBC

Tue 9 Mar 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX110

The first eigenvalue of the DirichlettoNeumann map, conformal geometry, and minimal surfaces

WMAX110
Tue 9 Mar 2010, 3:30pm4:30pm
Abstract
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Columbia University

Tue 16 Mar 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

Optimal conditions for the extension of the mean curvature flow

WMAX 110
Tue 16 Mar 2010, 3:30pm4:30pm
Abstract
In this talk, we will discuss several optimal (global) conditions for the existence of a smooth solution to the mean curvature flow. Our focus will be on quantities involving only the mean curvature. We will also discuss several applications of a local curvature estimate which is a parabolic analogue of ChoiSchoen estimate for minimal submanifolds. This is joint work with Natasa Sesum.
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UBC

Tue 30 Mar 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

Regularity of the extremal solution in fourth order problems on general domains

WMAX 110
Tue 30 Mar 2010, 3:30pm4:30am
Abstract
I will discuss recent results concerning the regularity of the extremal solution associated with fourth order nonlinear eigenvalue problems on general domains. We show that the extremal solution is bounded under various assumptions on the nonlinearity and/or the space dimension. This is a joint work with Pierpaolo Esposito and Nassif Ghoussoub.
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Louisiana State University

Tue 6 Apr 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

Nonlinear singular operators and measure data quasilinear Riccati type equations with nonstandard growth

WMAX 110
Tue 6 Apr 2010, 3:30pm4:30pm
Abstract
We establish explicit criteria of solvability for the quasilinear Riccati type equation $\Delta_p u =\nabla u^q + \omega$ in a bounded $\mathcal{C}^1$ domain $\Omega\subset\mathbb{R}^n$, $n\geq 2$. Here $\Delta_p$, $p>1$, is the $p$Laplacian, $q$ is critical $q=p$, or super critical $q>p$, and the datum $\omega$ is a measure. Our existence criteria are given in the form of potential theoretic or geometric (capacitary) estimates that are sharp when $\omega$ is compactly supported in the ground domain $\Omega$. A key in our approach to this problem is capacitary inequalities for certain nonlinear singular operators arising from the $p$Laplacian.
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Johns Hopkins U.

Thu 15 Apr 2010, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 216

Closed geodesics and Alexandrov spaces

WMAX 216
Thu 15 Apr 2010, 3:30pm4:30pm
Abstract
In this talk we will present our recent work on ‘’Closed geodesics in Alexandrov spaces of curvature bounded from above’’. This is an extension of Colding and Minicozzi’s widthsweepout construction of closed geodesics on closed Riemannian manifold to the Alexandrov setting, which provides a generalized version of the BirkhoffLyusternik theorem on the existence of nontrivial closed geodesics. We will explain how the widthsweepout construction works and discuss some future work in this direction.
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McMaster University

Tue 27 Apr 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX110

Symmetrybreaking bifurcation in the GrossPitaevskii equation with a doublewell potential

WMAX110
Tue 27 Apr 2010, 3:30pm4:30pm
Abstract
We classify bifurcations of the asymmetric states from a family of symmetric states in the focusing (attractive) GrossPitaevskii equation with a symmetric doublewell potential. Depending on the shape of the potential, both supercritical and subcritical pitchfork bifurcations may occur. We also consider the limit of large energies and show that the asymmetric states always exist near a nondegenerate extremum of the symmetric potential. These states are stable (unstable) in the case of subcritical nonlinearity if the extremum is a minimum (a maximum). All states are unstable for large energy in the case of supercritical nonlinearity. This is a joint work with E. Kirr and P. Kevrekidis.
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Université ParisDauphine

Wed 30 Jun 2010, 3:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
PIMS WMAX 110

Branched transport problems and elliptic approximation

PIMS WMAX 110
Wed 30 Jun 2010, 3:00pm4:00pm
Abstract
The branched transport problem is the minimization of a concave functional on vecror measures with prescribed divergence. The only admissible measures are those concentrated on 1rectifiable sets and the energy is the integral of a power $\theta^\alpha$ of their multiplicity $\theta$. I'll present an approximation by Gammaconvergence, through elliptic functionals defined on more regular functions : the idea is minimizing fucntionals such as $\frac 1 \varepsilon \int v^\alpha + \varepsilon Dv^2$ under constraints on the divergence of the $H^1$ function $v$. Obviously the exponents on the $\varepsilon$ and on the power of $v$ are to be changed if the result wants to be true. This approximation result recalls those of ModicaMortola for the perimeter functional, where a doublewell potential $W$, minimal on $0$ and $1$, is considered, and the energies $\frac 1 \varepsilon \int W(v)+ \varepsilon Dv^2$ converge to the perimeter of the interface between $\{v=0\}$ and $\{v=0\}$. Here the doublewell is replaced with a concave power, so that there is a sort of doublewell at $0$ and $\infty$. In ths case as well, the energy at the limit concentrates on a lower dimensional structure. Besides the link with the theory of elliptic approximations, the interest of this convergence lies in its applications for numerics. Actually, we built (in collaboration with E. Oudet, Chambéry) a quite efficient method, which allows to find reasonable local minima of the limit problem, avoiding the NP complications of the usual combinatorial approaches. The Steiner problem of minimal connection may be approached in this way as well.
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National Taiwan University

Thu 5 Aug 2010, 10:30am
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 216 (PIMS) "Note the room change"

Revisiting an idea of Brezis and Nirenberg

WMAX 216 (PIMS) "Note the room change"
Thu 5 Aug 2010, 10:30am11:30am
Abstract
Usually, a nonlinear functional involving the Sobolev exponent does not satisfy the PalaisSmale condition. However, Brezis and Nirenberg showed that under a threshold , the minimax value is in fact a critical value. In this talk, we should extend this idea to the equation involving with the Hardy singular potential.
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UCSD

Mon 13 Sep 2010, 4:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
MATH 203 (< Room changed)

Contracting exceptional divisors by the KahlerRicci flow

MATH 203 (< Room changed)
Mon 13 Sep 2010, 4:00pm5:00pm
Abstract
We give a criterion under which a solution g(t) of the KahlerRicci flow contracts exceptional divisors on a compact manifold and can be uniquely continued on a new manifold. This is a joint work with Jian Song.
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U. Oregon

Mon 20 Sep 2010, 4:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
MATX 1100

The complex Mongeampere equation on compact Kahler manifolds

MATX 1100
Mon 20 Sep 2010, 4:00pm5:00pm
Abstract
We consider the complex MongeAmp\`ere equation on a compact K\"ahler manifold $(M, g)$ when the right hand side $F$ has rather weak regularity. In particular we prove that estimate of second order and the gradient estimate hold when $F$ is in $W^{1, p_0}$ for any $p_0>2n$. As an application, we show that there exists a classical solution in $W^{3, p_0}$ for the complex MongeAmp\`ere equation when $F$ is in $W^{1, p_0}$.
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UBC

Tue 28 Sep 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 218 (note the schedule change)

Blowup of the cubic focusing nonlinear Schrodinger equation in dimension two with vortex soliton profile

WMAX 218 (note the schedule change)
Tue 28 Sep 2010, 3:30pm4:30pm
Abstract
Vortex solitons are standing wave solutions with complex phase that is an (integer) multiple of the angular polar coordinate. This multiple we call the 'spin', and indexes a family of solutions with increasing L2 norm. In the case of no spin, Merle and Raphael have shown that there exists a range of data that blowup with the Townes profile (the regular soliton) and whose H1 norm grows at a precise 'loglog' rate. We prove that in the case of spin 1, there is comparable data that blows up with the vortex profile and the loglog rate. The case of spin 2 and 3 will be discussed. This is joint work with Gideon Simpson (Toronto)
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University of Victoria

Tue 5 Oct 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

Illposedness of the 3DNavierStokes equation and related topics

WMAX 110
Tue 5 Oct 2010, 3:30pm4:30pm
Abstract
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UBC

Tue 19 Oct 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

Asymptotics of small exterior NavierStokes flows with nondecaying boundary data

WMAX 110
Tue 19 Oct 2010, 3:30pm4:30am
Abstract
We prove the unique existence of solutions of the 3D incompressible NavierStokes equations in an exterior domain with small nondecaying boundary data, for all $t \in R$ or $t \in (0,\infty)$. In the case $t \in (0,\infty)$ it is coupled with a small initial data in weak $L^{3}$. If the boundary data is timeperiodic, the spatial asymptotics of the timeentire solution is given by a Landau solution which is the same for all time. If the boundary data is timeperiodic and the initial data is asymptotically discretely selfsimilar, the solution is asymptotically the sum of a timeperiodic vector field and a forward discretely selfsimilar vector field as time goes to infinity. This is a joint work with Kyungkuen Kang and Hideyuki Miura.
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Korea Institute for Advanced Study and Stanford

Mon 25 Oct 2010, 4:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
MATX 1100

First eigenvalue of the Laplacian on minimal surfaces in $\mathbb S^3$

MATX 1100
Mon 25 Oct 2010, 4:00pm5:00pm
Abstract
Yau conjectured that the first eigenvalue of the Laplacian on compact embedded minimal surfaces in $\mathbb S^3$ should be equal to 2. We prove that Yau's conjecture is true for all minimal surfaces that are known to exist so far: the minimal surfaces constructed by Lawson, by KarcherPinkallSterling, and by KapouleasYang. (Joint work with M. Soret)
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McMaster U.

Thu 4 Nov 2010, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

On the size of the Navier  Stokes singular set

WMAX 110
Thu 4 Nov 2010, 3:30pm4:30am
Abstract
We consider the situation in which a weak solution of the NavierStokes equations fails to be continuous in the strong L^2 topology at some singular time t=T. We identify a closed set S_T in space on which the L^2 norm concentrates at this time T. The famous Caffarelli, Kohn Nirenberg theorem on partial regularity gives an upper bound on the Hausdorff dimension of this set. We study microlocal properties of the Fourier transform of the solution in the cotangent bundle T*(R^3) above this set. Our main result is a lower bound on the L^2 concentration set. Namely, that L^2 concentration can only occur on subsets of T*(R^3) which are sufficiently large. An element of the proof is a new global estimate on weak solutions of the NavierStokes equations which
have sufficiently smooth initial data.
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UBC

Tue 9 Nov 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

An inverse function theorem for differentiable maps between Fr\'{e}chet spaces

WMAX 110
Tue 9 Nov 2010, 3:30pm4:30pm
Abstract
I state and prove an inverse function theorem between Fr\'{e}chet spaces, which does not require that the function to be inverted is C^{2}, or even C^{1}, or even Fr\'{e}chetdifferentiable.
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University of Washington

Thu 18 Nov 2010, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

Boundary rigidity, lens rigidity and travel time tomography

WMAX 110
Thu 18 Nov 2010, 3:30pm4:30pm
Abstract
The boundary rigidity problem consists in determining the Riemannian metric of a compact Riemannian manifold with boundary by measuring the lengths of geodesics joining points of the boundary. The lens rigidity problem consists in determining the Riemannian metric of a compact Riemannian manifold with boundary by measuring the scattering relation or lens relation: We know the point of exit and direction of exit of a geodesic if we know its point of entrance and direction of entrance.
These two problems arise in travel time tomography in which one attempts to determine the index of refraction of a medium by measuring the travel times of waves going through the medium.
We will survey what is known about this problem and some recent results.
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UBC

Tue 23 Nov 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

Multisoliton solutions for the supercritical gKdV equations

WMAX 110
Tue 23 Nov 2010, 3:30pm4:30pm
Abstract
We consider the problem of existence and uniqueness of multisoliton solutions for the LÂ²supercritical generalized Kortewegde Vries equation. We recall that a multisoliton is a solution which behaves as a sum of N solitons in large time. After a survey of existing results in the subcritical and critical cases, and also in the 1soliton case, we will state the theorem of existence and uniqueness of an Nparameter family of Nsolitons in the supercritical case. Finally, we will sketch a proof of the classification part of this theorem.
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MIT

Thu 6 Jan 2011, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

A variational Characterization of the catenoid

WMAX 110
Thu 6 Jan 2011, 3:30pm4:30pm
Abstract
We show that the catenoid is the unique surface of least area (suitably understood) within a geometrically natural class of minimal surfaces. The proof relies on a techniques involving the Weierstrass representation used by Osserman and Schiffer to show the sharp isoperimetric inequality for minimal annuli. An alternate approach that avoids the Weierstrass representation will also be discussed. This latter approach depends on a conjectural sharp eigenvalue estimate for a geometric operater and has interesting connections with spectral theory. This is joint work with J. Bernstein
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Brown University

Thu 20 Jan 2011, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

Global existence for the energy critical Schrodinger equation in different spaces

WMAX 110 (PIMS)
Thu 20 Jan 2011, 3:30pm4:30pm
Abstract
We will prove that solutions to the defocusing energycritical Schrodinger equations are global in the hyperbolic space H^3. The relevance of the energycritical case is that in this case, one needs to understand how to take into account the scaling limits of the equation. In particular, one needs to see how to connect solutions to the corresponding equation on a Euclidian space to solutions of the original equation which concentrate as they evolve. To try and understand the influence of the geometry, we will also look at some results on other spaces in the other directions (the volume of balls grows slowly as the radius goes to infinity). This is a joint work with A. Ionescu and G. Staffilani.
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University of Iowa

Thu 27 Jan 2011, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

On the Skyrme model in quantum field theory

WMAX 110 (PIMS)
Thu 27 Jan 2011, 3:30pm4:30pm
Abstract
The Skyrme model is one of the important nonlinear sigma models in quantum field theory. In this talk I will report some recent progress on he dynamics of Skyrmions, focusing on the (3+1) spacetime case.
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University of Nice

Tue 1 Feb 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

The Heat Flow as gradient flow

WMAX 110
Tue 1 Feb 2011, 3:30pm4:30pm
Abstract
Aim of the talk is to make a survey on some recent results concerning analysis over spaces with Ricci curvature bounded from below. I will show that the heat flow in such setting can be equivalently built either as gradient flow of the natural Dirichlet energy in L^2 or as gradient flow if the relative entropy in the Wasserstein space. I will also show how such identification can lead to interesting analytic and geometric insights on the structures of the spaces themselves. From a collaboration with L.Ambrosio and G.Savare'.
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Mon 7 Feb 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar


Mon 7 Feb 2011, 3:30pm4:30pm
Abstract
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University of Alabama at Birmingham

Thu 10 Feb 2011, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
PIMS

Zerovelocity LiebRobinson bounds in the disordered xyspin chain

PIMS
Thu 10 Feb 2011, 3:30pm4:30pm
Abstract
The well understood phenomenon of Anderson localization says (in its dynamical formulation) that adding random fluctuations to the potential of a Schrodinger operator will lead to the absence of wave transport for the solution of the timedependent Schrodinger equation. Several years ago it was argued by Burrell and Osborne that a corresponding phenomenon should hold in quantum spin systems. As an example they used the xyspin chain to show on the physical level of rigor that the introduction of disorder will lead to zerovelocity LiebRobinson bounds. We will show
how recent results on Anderson localization can be used to make this result rigorous and, in fact, to improve on the conclusions reached by Burrell and Osborne.
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Rice University

Tue 1 Mar 2011, 3:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (Note the time change to 3:00)

Bernstein's Theorem for the twovalued minimal surface equation.

WMAX 110 (Note the time change to 3:00)
Tue 1 Mar 2011, 3:00pm4:00pm
Abstract
We explore the question of whether there are nontrivial solutions to the twovalued minimal surface (2MSE) equation defined over the punctured plane. The 2MSE is a nonuniformly elliptic PDE, degenerate at the origin, originally introduced by N.Wickramasekera and L.Simon to produce examples of stable branched minimal immersions.
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University of Tokyo

Tue 1 Mar 2011, 4:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (Note the time change to 4:00)

Fefferman's program in conformal geometry and the singularities of the Green functions of the conformal powers of the Laplacian

WMAX 110 (Note the time change to 4:00)
Tue 1 Mar 2011, 4:00pm5:00pm
Abstract
Motivated by the analysis of the singularity of the Bergman kernel on strictly pseudoconvex complex domains, Fefferman launched in the late 70s the program of determining all local biholomorphic invariants of a strictly pseudoconvex complex domain. This program has since been extended to other "parabolic" geometries such as conformal geometry. After a review of Fefferman's program, we shall explain how to compute explicitly the logarithmic singularities of the Green kernels of the conformal powers of the Laplacian, including the Yamabe and Paneitz operators. As applications we obtain a new characterization of locally conformally flat manifolds and a spectraltheoretic characterization of the conformal class of the round sphere.
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University of Texas at Austin

Fri 8 Apr 2011, 1:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
MATX 1118 (Note the special location and special time)

Regularity for the parabolic obstacle problem with fractional Laplacian

MATX 1118 (Note the special location and special time)
Fri 8 Apr 2011, 1:00pm2:00pm
Abstract
In recent years, there has been an increasing interest in studying constrained variational problems with a fractional diffusion. One of the motivations comes from mathematical finance: jumpdiffusion processes where incorporated by Merton into the theory of option evaluation to introduce discontinuous paths in the dynamics of the stock's prices, in contrast with the classical lognormal diffusion model of Black and Scholes. These models allow to take into account large price changes, and they have become increasingly popular for modeling market fluctuations, both for risk management and option pricing purposes.
In a joint paper with Luis Caffarelli we study the parabolic version of the fractional obstacle problem, i.e. where the elliptic part of the operator is given (at least at the leading order) by a fractional laplacian. We prove optimal spatial regularity and almost optimal time regularity of the solution, recovering in particular the optimal regularity for the stationary case. To obtain this result, we crucially exploit the fact that the solution coincides with the obstacle at the initial time, which corresponds to the fact that (for the backward operator) the stock's price coincides with the payoff at the final time.
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University of Wisconsin at Madison

Wed 8 Jun 2011, 11:00am
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110, PIMS

Rigidity and stability of Einstein metrics for quadratic curvature functionals

WMAX 110, PIMS
Wed 8 Jun 2011, 11:00am12:00pm
Abstract
ABSTRACT: I will discuss rigidity (existence or nonexistence of
infinitesimal deformations) and stability (strict local minimization)
properties of Einstein metrics for quadratic curvature functionals on
Riemannian manifolds. This is joint work with Matt Gursky.
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Sungkyunkwan University, Korea

Mon 25 Jul 2011, 2:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX110 (PIMS) note the room change

On the blowup problem for the Euler equations and the Liouville type results for the fluid equations

WMAX110 (PIMS) note the room change
Mon 25 Jul 2011, 2:00pm3:00pm
Abstract
In the first part of the talk we discuss some new observations on the blowup problem in the 3D Euler equations. We consider the scenarios of the selfsimilar blowups and the axisymmetric blowup. For the selfsimilar blowup we prove a Liouville type theorem for the selfsimilar Euler equations. For the axisymmetric case we show that some uniformity condition for the pressure is not consistent with the global regularity. In the second part we present Liouville type theorems for the steady NavierStokes equations for both of the incompressible and the compressible cases. In the time dependent case we prove that some pressure integrals have definite sign unless the solution is trivial.
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Academy of Finland and University of Sydney

Thu 8 Sep 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS) (Schedule and location subject to change)

The AharonovBohm effect and the Calderon problem for connection Laplacians

WMAX 110 (PIMS) (Schedule and location subject to change)
Thu 8 Sep 2011, 3:30pm4:30pm
Abstract
The AharonovBohm eﬀect is a quantum mechanical phenomenon where electrons passing through a region of vanishing magnetic ﬁeld gets scattered due to topological eﬀects. It turns
out that this phenomenon is closely related to the cohomology of forms with integer coeﬃcients. We study this relationship from the point of view of the Calder´n problem and see that it can be captured in how Cauchy data of the connection laplacian determines uniquely the holonomy representation of the connection.
The work was partially supported by Finnish Academy of Science and by NSF Grant No.DMS0807502.
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UBC

Tue 13 Sep 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

A Selfdual Polar Factorization for Vector Fields

WMAX 110 (PIMS)
Tue 13 Sep 2011, 3:30pm4:30pm
Abstract
We show that any nondegenerate vector field u in L^{\infty}(\Omega, \R^N), where \Omega is a bounded domain in \R^N, can be written as {equation} \hbox{u(x)= \nabla_1 H(S(x), x) for a.e. x \in \Omega}, {equation} where S is a measure preserving point transformation on \Omega such that S^2=I a.e (an involution), and H: \R^N \times \R^N \to \R is a globally Lipschitz antisymmetric convexconcave Hamiltonian. Moreover, u is a monotone map if and only if S can be taken to be the identity, which suggests that our result is a selfdual version of Brenier's polar decomposition for the vector field u as u(x)=\nabla \phi (S(x)), where \phi is convex and S is a measure preserving transformation. We also describe how our polar decomposition can be reformulated as a selfdual mass transport problem.
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UBC

Tue 20 Sep 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

Backward uniqueness for the heat equation in cones

WMAX 110 (PIMS)
Tue 20 Sep 2011, 3:30pm4:30pm
Abstract
I will talk about the backward uniqueness of the heat equation in unbounded domains. It is known that a bounded solution of the heat equation in a halfspace which becomes zero at some time must be identically zero, even though no assumptions are made on the boundary values of the solutions. In a recent example, Luis Escauriaza showed that this statement fails if the halfspace is replaced by cones with opening angle smaller than 90 degrees. In a joint work with Vladimir Sverak we show the result remains true for cones with opening angle larger than 110 degrees. Our proof covers heat equations having lowerorder terms with bounded measurable coefficients.
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U. ParisSud, Orsay

Mon 26 Sep 2011, 4:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar / Probability Seminar
MATX 1100

The Langevin process and the trace formula

MATX 1100
Mon 26 Sep 2011, 4:00pm5:00pm
Abstract
I will explain the probabilistic interpretation of the hypoelliptic Laplacian L_b . To L_b, one can associate the diﬀusion on the manifold X that is a solution of the diﬀerential equation b^2 x'' = −x' + w'. For b = 0, we get x' = w', the equation of Brownian motion, and for b = +∞, we obtain the equation of geodesics x'' = 0. I will explain the rigorous results one can derive on the corresponding heat kernels via the Malliavin calculus. These will include uniform Gaussian decay of the hypoelliptic heat kernel over a symmetric space.
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U. ParisSud, Orsay

Tue 27 Sep 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

Orbital integrals and the hypoelliptic Laplacian

WMAX 110 (PIMS)
Tue 27 Sep 2011, 3:30pm5:00pm
Abstract
Third talk in the series. If G is a reductive Lie group with Lie algebra g, orbital integrals are key ingredient in Selberg’s trace formula. I will explain how one can think of the evaluation of orbital integrals as the computation of a Lefschetz trace. Using in particular the Dirac operator of Kostant, the standard Casimir operator of X = G/K is deformed to a hypoelliptic operator L_b acting on the total space of a canonically ﬂat vector bundle on X, that contains TX as a subbundle. The symbol of this hypoelliptic operator is exactly the one described in the previous talks. When descending the situation to a locally symmetric space, the spectrum of the original Casimir remains rigidly embedded in the spectrum of the hypoelliptic deformation. Making b → +∞ gives an explicit evaluation of semisimple orbital integrals.
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UBC

Tue 4 Oct 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

Regularity for the optimal transport problem with Euclidean distance squared cost on the embedded sphere

WMAX 110 (PIMS)
Tue 4 Oct 2011, 3:30pm4:30pm
Abstract
We consider regularity for Monge solutions to the optimal transport problem when the initial and target measures are supported on the embedded sphere, and the cost function is the Euclidean distance squared. Gangbo and McCann have shown that when the initial and target measures are supported on boundaries of strictly convex domains in $\mathbb{R}^n$, there is a unique Kantorovich solution, but it can fail to be a Monge solution. By using PDE methods, in the case when we are dealing with the sphere with measures absolutely continuous with respect to surface measure, we present a condition on the densities of the measures to ensure that the solution given by Gangbo and McCann is indeed a Monge solution, and obtain higher regularity as well. This talk is based on joint work with Micah Warren.
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Cornell University

Tue 11 Oct 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

Harnack Inequalities, Heat Kernel Estimates and the Ricci flow

WMAX 110 (PIMS)
Tue 11 Oct 2011, 3:30pm4:30pm
Abstract
In this talk, we will discuss about LiYauHamilton type differential Harnack inequalities, heat kernel estimates and their applications to study type I ancient solutions of the Ricci flow. Some of this is joint work with Q. S. Zhang.
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UBC

Tue 18 Oct 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

Uniqueness of the compactly supported weak solutions of the relativistic VlasovDarwin system

WMAX 110 (PIMS)
Tue 18 Oct 2011, 3:30pm4:30pm
Abstract
The relativistic VlasovDarwin (RVD) system is a kinetic model that describes the evolution of a collisionless plasma whose particles interact through the selfinduced electromagnetic ﬁeld. In contrast with the VlasovMaxwell system, the particle interaction is assumed to be a loworder relativistic correction (i.e., the Darwin approximation) to the full Maxwell case. A consequence of this assumption is that instead of the less tractable hyperbolic Maxwell equations, the resulting system has elliptic features even though there is a fully coupled magnetic ﬁeld. We use optimal transportation techniques to show uniqueness of the compactly supported weak solutions of the RVD system. Our proof extends the method used by Loeper in [J. Math. Pures Appl., 86 (2006), pp. 6879 ] to obtain uniqueness results for the VlasovPoisson system. This is a joint work with Martial Agueh.
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Universite Libre de Bruxelles

Tue 25 Oct 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
PIMS (WMAX 110)

A rough guide to reduction methods for strongly coupled elliptic systems

PIMS (WMAX 110)
Tue 25 Oct 2011, 3:30pm4:30pm
Abstract
In this talk, I will first recall the notions of superlinearity and subcriticality for strongly coupled elliptic systems. I will present various functional frameworks and their limitations. I will then discuss two reduction methods that allow to get rid of the indefiniteness of the energy functional. These reductions to a single equation are powerful to treat basic questions for superlinear systems. For instance, I will discuss the notion of ground states, in bounded domains and in R^N, show how to get the information on the symmetry and the sign of the ground states through the definition of a convenient Nehari manifold or constrained minimization problem. I will also discuss the classical question of existence of infinitely many critical points of perturbed indefinite symmetric functionals and how one of the reduction method allow to use the notion of Morse index. Finally, I will show how these reduction methods can help in proving partial symmetry and symmetry breaking. As a paradigm, I will illustrate the ideas on the LaneEmden system with Hénon weights.
References : B.Ramos ANIHP 2009  B.dos Santos JDE 2010  B.Ramosdos Santos Trans. AMS 2012 & preprint.
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UBC

Tue 1 Nov 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
PIMS (WMAX 110) (Notice the date change)

Liouvilletype theorems for some elliptic equations and systems

PIMS (WMAX 110) (Notice the date change)
Tue 1 Nov 2011, 3:30pm4:30pm
Abstract
In this talk, we consider the problem of nonexistence of solutions for some basic elliptic equations and systems with weights. Starting with HenonLaneEmden system, we present a Liouvilletype theorem for bounded solutions in dimension N=3 as well as the full HenonLaneEmden conjecture in higher dimensions. Since systems are normally much more complicated than equations, in higher dimensions we back to single equations (both second order and fourth order) to prove such theorems under some additional assumptions on solutions.
Also, during the talk we will see many open problems.
This work has been done under supervision of N. Ghoussoub.
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UBC

Tue 15 Nov 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (Schedule and time tentative)

Free boundary problem for embedded minimal surfaces

WMAX 110 (Schedule and time tentative)
Tue 15 Nov 2011, 3:30pm4:30pm
Abstract
For any smooth compact Riemannian 3manifold with boundary, we prove that there always exists a smooth, embedded minimal surface with (possibly empty) free boundary. We also obtain a priori upper bound on the genus of such minimal surfaces in terms of the topology of the ambient compact 3manifold. An interesting note is that no convexity assumption on the boundary is required. In this talk, we will describe the minmax construction for the free boundary problem, and then we will sketch a proof of the existence part of the theory.
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Sogang University and UBC

Tue 17 Jan 2012, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

Mathematical analysis of the stationary motion of an incompressible viscous fluid

WMAX 110
Tue 17 Jan 2012, 3:30pm4:30pm
Abstract
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University of Chicago

Tue 31 Jan 2012, 2:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
MATX 1102 The time and location are changed!

Partial regularity for fully nonlinear elliptic PDE.

MATX 1102 The time and location are changed!
Tue 31 Jan 2012, 2:00pm3:00pm
Abstract
We prove that solutions to a fully nonlinear elliptic equation F(D^2u)=0 are classical outside a set of dimension at most nepsilon, where n is the dimension and epsilon is a small constant depending on the ellipticity bounds of F and dimension. We do not make any convexity assumption on the equation F, but we assume that it is differentiable. We will also discuss the relationship of the partial regularity result with the question of unique continuation of solutions. This is a joint work with Scott Armstrong and Charles Smart.
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National Chiao Tung University, Taiwan

Tue 14 Feb 2012, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar / Mathematical Biology Seminar
WMAX 110 (PIMS) (PDEMath Biology Joint seminar)

Asymptotic Limit in a Cell Differentiation Model

WMAX 110 (PIMS) (PDEMath Biology Joint seminar)
Tue 14 Feb 2012, 3:30pm4:30pm
Abstract
T cells of the immune system, upon maturation, differentiate into either Th1 or Th2 cells that have different functions. The decision to which cell type to differentiate depends on the concentrations of transcription factors Tbet (x_1) and GATA3 (x_2). These factors are translated by the mRNA whose levels of expression, y_1 and y_2, depend, respectively, on x_1 and x_2 in a nonlinear nonlocal way. The population density of T cells, \phi(t,x_1,x_2, y_1, y_2), satisfies a hyperbolic conservation law with coefficients depending nonlinearly and nonlocally on (t, x_1,x_2, y_1, y_2), while the x_i, y_i satisfy a system of ordinary differential equations. We study the long time behavior of \phi and show, under some conditions on the parameters of the system of differential equations, that the gene expressions in the Tcell population aggregate at one, two or four points, which connect to various cell differentiation scenarios.
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Yonsei University

Thu 16 Feb 2012, 2:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ANGU 235 (It is in *Sauder School*, the second floor)

Harnack inequality for second order elliptic operators on Riemannian manifolds.

ANGU 235 (It is in *Sauder School*, the second floor)
Thu 16 Feb 2012, 2:00pm3:00pm
Abstract
In this talk, I will give a survey on Harnack inequalities for solutions of secondorder elliptic equations on Riemannian manifolds.
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The Chinese University of Hong Kong

Thu 1 Mar 2012, 3:45pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

Specialized Seminar: Mathematical Analysis of Localized Patterns in ReactionDiffusion Systems

WMAX 110 (PIMS)
Thu 1 Mar 2012, 3:45pm4:45pm
Abstract
Abstract: I will describe some mathematical issues related to the analysis of localized patterns (spikes and interfaces) in reaction diffusion systems. For spikes, the analysis of the spectrum of various classes of nonlocal eigenvalue problems (NLEP) is essential. I will discuss some new and interesting NLEPs arising in cross diffusion systems and crime models. For interfaces, a nonlocal geometric problem involving mean curvature and Newtonian potential is derived and analyzed. Our goal is to give mathematically rigorous proofs of existence and stability of various classes of patterns that have been observed in experiments and simulations in the physics literature. Our further goal is to predict the existence of some new patterns which have not yet been found in experiments. (Joint works with T. Kolokolnikov, X. Ren, M. Ward, and M. Winter.)
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UBC

Tue 6 Mar 2012, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

Local dynamics near unstable branches of NLS solitons

WMAX 110 (PIMS)
Tue 6 Mar 2012, 3:30pm4:30pm
Abstract
TBA
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Brown University

Tue 13 Mar 2012, 2:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

Elliptic equations in convex wedges with irregular coefficients

WMAX 110 (PIMS)
Tue 13 Mar 2012, 2:00pm3:00pm
Abstract
I will present a recent result on the $W^2_p$solvability of elliptic equations in convex wedge domains or in convex polygonal domains with discontinuous coefficients. A corresponding result for parabolic equations in polyhedrons with timeirregular coefficients will also be discussed.
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UCLA

Thu 29 Mar 2012, 2:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
Cancelled

Liquid drops sliding down an inclined plane.

Cancelled
Thu 29 Mar 2012, 2:00pm3:00pm
Abstract
We investigate a onedimensional model describing the motion of liquid drops sliding down an inclined plane (the socalled quasistatic approximation model). We prove existence and uniqueness of a solution and investigate its long time behavior for both homogeneous and inhomogeneous medium (i.e. constant and nonconstant contact angle). This is joint work with Antoine Mellet (U.Maryland).
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Institut de Mathématiques de Toulouse, Université Paul Sabatier

Tue 3 Apr 2012, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110

Excited MultiSolitons for a Nonlinear Schrödinger Equation

WMAX 110
Tue 3 Apr 2012, 3:30pm4:30pm
Abstract
We consider a nonlinear Schrödinger equation with a general nonlinearity. In space dimension 2 or higher, this equation admits solitons (standing/traveling waves) with a fixed profile which is not a ground state. These types of profiles are called excited states. Due to instability, excited solitons are singular objects for the dynamics of NLS. Nevertheless, we will show in this talk how to exhibit solutions of NLS behaving in large time like a sum of excited solitons with high relative speeds.
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University of Toronto

Wed 22 Aug 2012, 3:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

[PIMS distinguished lecture] Optimal transportation with capacity constraints

WMAX 110 (PIMS)
Wed 22 Aug 2012, 3:00pm4:00pm
Abstract
The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a given cost function. Here we consider a variation of this problem by imposing an upper bound constraining the joint measures, namely: among all joint measures with fixed marginals and dominated by a fixed measure, find the optimal one. After computing illustrative examples, we given conditions guaranteeing uniqueness of the optimizer and initiate a study of its properties. Based on a preprint arXived with Jonathan Korman.
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University of Toronto

Fri 24 Aug 2012, 3:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

[PIMS distinguished lecture] Multisector matching with cognitive and social skills: a stylized model for education, work and marriage

WMAX 110 (PIMS)
Fri 24 Aug 2012, 3:00pm4:00pm
Abstract
Economists are interested in studying who matches with whom (and why) in the educational, labour, and marriage sectors. With Aloysius Siow, Xianwen Shi, and Ronald Wolthoff, we propose a toy model for this process, which is based on the assumption that production in any sector requires completion of two complementary tasks. Individuals are assumed to have both social and cognitive skills, which can be modified through education, and which determine what they choose to specialize in and with whom they choose to partner.
Our model predicts variable, endogenous, manytoone matching. Given a fixed initial distribution of characteristics, the steady state equilibrium of this model is the solution to an (infinite dimensional) linear program, for which we develop a duality theory which exhibits a phase transition depending on the number of students who can be mentored. If this number is two or more, then a continuous distributions of skills leads to formation of a pyramid in the education market with a few gurus having unbounded wage gradients. One preprint is on the arXiv; a sequel is in progress.
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University of Sussex

Thu 6 Sep 2012, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

Blowup of critical Besov norms at a NavierStokes singularity

ESB 2012 (in the new PIMS building)
Thu 6 Sep 2012, 3:30pm4:30pm
Abstract
In this talk we describe a generalization of the result of EscauriazaSereginSverak on blowup of the L^3 norm at a NavierStokes singularity by establishing the blowup of any weaker critical Besov norm with finite third index as well. Following previous joint works with C. Kenig and with I. Gallagher and F. Planchon respectively, we use the "dispersivetype" method of concentration compactness and critical elements developed by C. Kenig and F. Merle. Joint work with I. Gallagher and F. Planchon
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UBC

Tue 18 Sep 2012, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

NonUniqueness Phenomena for the 3D Euler Equations

ESB 2012 (in the new PIMS building)
Tue 18 Sep 2012, 3:30pm4:30pm
Abstract
Since the famous work of V. Scheffer about 20 years ago, it has been known that the Cauchy problem for the incompressible Euler equations has nonunique weak solutions. Recently, De Lellis and Szekelyhidi demonstrated that this phenomenon can be viewed as an instance of the socalled hprinciple, thereby providing a shorter and more general proof of the nonuniqueness. In this talk I will briefly review their method and then present some subsequent results, including global existence and nonuniqueness for 3D Euler, the approximation of measurevalued solutions by weak ones, and nonuniqueness for shear flow initial data.
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UBC

Tue 25 Sep 2012, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

Regularity for solutions of non local parabolic equations

ESB 2012 (in the new PIMS building)
Tue 25 Sep 2012, 3:30pm4:30pm
Abstract
We study the regularity of solutions of parabolic equations of the form
u_t  Iu = f,
where I is a fully non linear non local operator. We prove C^\alpha regularity in space and time and, under different assumptions on the kernels,
C^{1,\alpha}; in space for translation invariant equations. The proofs rely on a weak parabolic ABP and the classic ideas of K. Tso and L. Wang. Our results remain uniform as \sigma goes to 2 allowing us to recover most of the regularity results of the local case.
This is a joint work with H ector Chang Lara.
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UBC

Tue 2 Oct 2012, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

On the degeneracy of optimal transportation

ESB 2012 (in the new PIMS building)
Tue 2 Oct 2012, 3:30pm4:30pm
Abstract
It is well known that an upper and lower bound on the MongeAmp{\`e}re measure of a convex function u implies this function must actually be strictly convex. A lesser known result, also by Caffarelli, states that if the MongeAmp{\`e}re of u has only a lower bound, the contact set between u and a supporting affine function must have affine dimension strictly less than n/2. By means of a careful geometric construction involving the subdifferential, we give an alternative proof of Caffarelli's result, and extend the result to optimal transportation problems with cost functions satisfying the weak MaTrudingerWang condition. This talk is based on a joint work with YoungHeon Kim.
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UBC

Tue 9 Oct 2012, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

Forward Discrete SelfSimilar Solutions of the NavierStokes Equations

ESB 2012 (in the new PIMS building)
Tue 9 Oct 2012, 3:30pm4:30pm
Abstract
Extending the work of Jia and Sverak on selfsimilar solutions of the NavierStokes equations, we show the existence of large, forward, discrete selfsimilar solutions.
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UBC

Tue 16 Oct 2012, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building) TBA

Absolutely continuous spectrum for random Schrödinger operators on treestrips of finite conetype.

ESB 2012 (in the new PIMS building) TBA
Tue 16 Oct 2012, 3:30pm4:30pm
Abstract
One of the biggest challenges in the field of random Schrödinger operators is to prove the existence of absolutely continuous spectrum for the Anderson model for small disorder in dimensions greater equal to 3. So far, the existence of absolutely continuous spectrum is only known for models on infinitedimensional tree structures. The first proof, done by Abel Klein for a regular tree, dates back to 1994.
Recent developments considered trees of finite cone type and cross products of trees with finite graphs, so called treestrips. I will present a proof for the existence of absolutely continuous spectrum for models on treestrips of finite cone type. The proof uses a version of the Implicit Function Theorem in Banach spaces which are constructed by a supersymmetric formalism using Grassmann variables.
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CNRS and Université Joseph Fourier

Tue 23 Oct 2012, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

Nondifferentiability locus of distance functions and Federer's curvature measures

ESB 2012 (in the new PIMS building)
Tue 23 Oct 2012, 3:30pm4:30pm
Abstract
I will present an upper bound on the (d1)volume and covering numbers of a filtration of the nondifferentiability locus of the distance function of a compact set in R^d. A consequence of this upper bound is that the projection function to a compact subset K depends in a Hoelder way on the compact set, in the L^1 sense. This in turn implies that Federer's curvature measure of a compact set with positive reach can be reliably estimated from a Hausdorff approximation of this set, regardless of any regularity assumption on the approximation.
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Université Paul Sabatier, Toulouse, France

Tue 20 Nov 2012, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

On analytical properties of Alexandrov spaces

ESB 2012 (in the new PIMS building)
Tue 20 Nov 2012, 3:30pm4:30pm
Abstract
In this talk, I will discuss some analytical aspects in the study of a finite dimensional Alexandrov space. Loosely speaking, the question I will consider is: to what extent does an Alexandrov space resemble a Riemannian manifold? In the first part of the talk, I will recall the background of Alexandrov's theory of metric spaces with curvature bounded from below, including results on the topology of these spaces.
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University of Alberta

Tue 11 Dec 2012, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 4127 (PIMS video conference room)

Multimarginal optimal transport and multiagent matching problems: uniqueness and structure of solutions

ESB 4127 (PIMS video conference room)
Tue 11 Dec 2012, 3:30pm4:30pm
Abstract
I will discuss uniqueness and Monge solution results for multimarginal optimal transportation problems with a certain class of cost functions; this class arises naturally in multiagent matching problems in economics. This result generalizes a seminal result of Gangbo and \'Swi\c{e}ch on multimarginal problems. I will also discuss some related observations about multimarginal optimal transport on Riemannian manifolds.
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UBC

Tue 8 Jan 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

mLiouville theorems for elliptic PDEs

ESB 2012 (in the new PIMS building)
Tue 8 Jan 2013, 3:30pm4:30pm
Abstract
De Giorgi in 1978 conjectured that bounded and monotone solutions of the AllenCahn equation must be onedimensional up to dimension eight. This conjecture is known to be true for N=<3 and with extra (natural) assumptions for 4=<N=<8. We state a counterpart of the above conjecture for gradient systems introducing the concept of monotonicity for systems. Then, we prove this conjecture for dimensions up to three and applying a geometric Poincare inequality for stable solutions we show that the gradients of various components of the solutions are parallel.
On the other hand, we ask under what conditions we can prove solutions of a PDE are mdimensional for 0=<m=<N1. This leads us to define the concept of “mLiouville theorem” for PDEs. The motivation to this definition is the Liouville theorem (or 0Liouville theorem) that we have seen in elementary analysis stating that bounded harmonic functions on the whole space must be constant (0dimensional).
This is the main part of my dissertation under the supervision of Nassif Ghoussoub.
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UBC

Tue 15 Jan 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

A Bernstein theorem for the Willmore equation

ESB 2012 (in the new PIMS building)
Tue 15 Jan 2013, 3:30pm4:30pm
Abstract
A classical theorem in minimal surface theory says that any entire minimal graph in R^3 is a plane. We ask the same question for the Willmore equation which is of 4th order. We prove that an entire Willmore graph is a plane if its Willmore functional is finite (i.e. if the mean curvature of the graph is square integrable). This is joint work with Tobias Lamm.
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UBC

Tue 22 Jan 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

Decoupling DeGiorgi's systems via multimarginal mass transport

ESB 2012 (in the new PIMS building)
Tue 22 Jan 2013, 3:30pm4:30pm
Abstract
We expose and exploit a surprising relationship between elliptic gradient systems of PDEs and a multimarginal MongeKantorovich optimal transport problem. We show that the notion of an "orientable" elliptic system (FazlyGhoussoub) conjectured to imply that stable solutions are essentially 1dimensional, is equivalent to the definition of a "compatible" cost function (CarlierPass), known to imply uniqueness and structural results for optimal measures to certain MongeKantorovich problems. We use this equivalence to show that solutions to these elliptic PDEs, with appropriate monotonicity properties, are related to optimal measures in the MongeKantorovich problem. We also prove a decoupling result for solutions to elliptic PDEs and show that under the orientability condition, the decoupling has additional properties, due to the connection to optimal transport.
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University du Sud Toulon Var, visiting McGill

Tue 29 Jan 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

Infrared (and ultraviolet) aspects of a model of QFT on a static space time

ESB 2012 (in the new PIMS building)
Tue 29 Jan 2013, 3:30pm4:30pm
Abstract
We consider the Nelson model with variable coeffcients, which can be seen as a model describing a particle interacting with a scalar field on a static space time. We consider the problem of the existence of the ground state, showing that it depends on the decay rate of the coeffcients at infinity. We also show that it is possible to remove the ultraviolet cutoff, as it is in the flat case. We'll explain some open conjecture. (joint work with C.Gérard, F.Hiroshima, A.Suzuki)
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Georgia Tech

Tue 5 Feb 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

Small BGK waves and Landau damping

ESB 2012 (in the new PIMS building)
Tue 5 Feb 2013, 3:30pm4:30pm
Abstract
In this talk, we discuss the Landau damping  the asymptotic stability of the linearly stable homogeneous states of the VlasovPossion system. It has been proved that solutions to the system linearized at stable homogeneous states decay algebraically in time. In such a Hamiltonian system, this decay is not caused by any dissipation. The nonlinear asymptotic stability is open until recently when Mouhot and Villani proved it of solutions in the Gevery class. The problem in Sobolev spaces remains open. We show that the nonlinear damping does not happen in Sobolev space with too low regularity by constructing BKG waves  traveling waves  arbitrarily close to stable homogeneous states. In the contrary, in Sobolev spaces with higher regularity, we show that there are no invariant structures  including BGK waves  near any stable homogeneous states and thus the same obstacle for the damping as in the rough Sobolev spaces does not appear. Similar results have also been proved for the Euler equation near Couette flow. These are joint works with Zhiwu Lin.
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University of Tennessee, Knoxville

Tue 12 Feb 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

Geometric Inequalities for Hypersurfaces

ESB 2012 (in the new PIMS building)
Tue 12 Feb 2013, 3:30pm4:30pm
Abstract
I will begin this talk by recalling the classic inequalities of AlexandrovFenchel and PolyaSzego for convex surfaces of 3dimensional Euclidean space.
Then, I will present my joint work with Freire, which generalizes the inequalities with rigidity to both a larger class of hypersurfaces and to arbitrary dimensions. I will conclude by mentioning some applications of the results, including a masscapacity inequality for black holes.
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Universite de Toulouse

Tue 12 Mar 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

Blow up dynamics for the 1corotational energy critical harmonic heat flow

ESB 2012 (in the new PIMS building)
Tue 12 Mar 2013, 3:30pm4:30pm
Abstract
After a short presentation of the equation of harmonic heat flow and corotational solutions, I am presenting a result of finite time blowup dynamics. We will have to see how similar results for wave maps and Schrödinger map allow us to conjecture the instability of this regime in the general case. Finally, I will give a strategy of the proof.
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New York University

Mon 18 Mar 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB Rm 4127, (PIMS video conference room). Note the date, time and place change.

Scattering for nonlinear dispersive equations in the presence of a potential

ESB Rm 4127, (PIMS video conference room). Note the date, time and place change.
Mon 18 Mar 2013, 3:30pm4:30pm
Abstract
Questions related to the asymptotic behavior of nonlinear dispersive equations in the presence of a potential term are of great interest both for mathematical and physical reasons. Our main concern will be equations with lowdegree nonlinearities, namely below the Strauss exponent threshold, for which classical energy and decay methods fail to suffice. For this, we we use the spectral theory of the operator H=\Delta+V to develop a spacetime resonance analysis adapted to the inhomogeneous setting. A key ingredient in this setup is the development of a sufficiently comprehensive multilinear harmonic analysis in the context of the corresponding distorted Fourier transform. This turns out to exhibit several intriguing differences in comparison to the unperturbed Euclidean setting (no matter how small V is). As a first application, we treat the case of a quadratic nonlinear Schrodinger equation on \R^3.
This is joint work with Pierre Germain and Samuel Walsh (Courant Institute, NYU).
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Georgia Tech

Tue 19 Mar 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

The Stability of Cylindrical Pendant Drops

ESB 2012 (in the new PIMS building)
Tue 19 Mar 2013, 3:30pm4:30pm
Abstract
In 1980 Henry Wente considered the variational stability of rotationally symmetric pendant drops and obtained a number of results for various problems.
We consider one version of one of those problems for cylindrical pendant drops trapped between parallel planes. The analysis is different in various ways, and leads to results of a different nature. Most notably, Wente's rotationally symmetric pendant drops form stable families which terminate at a maximum volume. We find stable families which terminate at a maximum volume, but are followed by (distinct disconnected) families of stable drops. As a result, we may have "large" stable pendant drops which become unstable and "drip" when the volume is decreased.
We will attempt to explain these results using the simpler zero gravity case as a model.
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UCLA

Tue 26 Mar 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)

Quasistatic evolution in randomly perforated media

ESB 2012 (in the new PIMS building)
Tue 26 Mar 2013, 3:30pm4:30pm
Abstract
We consider a quasistatic free boundary problem (the HeleShaw problem) in a randomly perforated domain with zero Neumann boundary conditions. A homogenization limit is obtained as the characteristic scale of the domain goes to zero. Specifically, we prove that the solutions as well as their free boundaries converge uniformly to those corresponding to a homogeneous and anisotropic HeleShaw problem set in $\mathbb{R}^d$. This is joint work with Nestor Guillen.
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Yonsei University, Korea

Tue 23 Apr 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in PIMS building)

Green's function for secondorder elliptic and parabolic systems with boundary conditions.

ESB 2012 (in PIMS building)
Tue 23 Apr 2013, 3:30pm4:30pm
Abstract
In this talk, I will describe construction and estimates for Green's function for elliptic and parabolic systems of second order in divergence form subject to various boundary conditions.
Here, we assume minimal regularity assumptions on the coefficients and domains.
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Ecole Polytechnique, France

Tue 28 May 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
Earth Sciences Bldg ESB, Room 2012 (PIMS building)

Diffusion of knots and magnetic relaxation

Earth Sciences Bldg ESB, Room 2012 (PIMS building)
Tue 28 May 2013, 3:30pm4:30pm
Abstract
Motivated by seeking stationary solutions to the Euler equations with prescribed vortex topology, H.K. Moffatt has described in the 80s a diffusion process, called "magnetic relaxation", for 3D divergencefree vector fields that (formally) preserves the knot structure of their integral lines. (See also the book by V.I. Arnold and B. Khesin.)
The magnetic relaxation equation is a highly degenerate parabolic PDE which admits as equilibrium points all stationary solutions of the Euler equations. Combining ideas from P.L. Lions for the Euler equations and AmbrosioGigliSavar\'e for the scalar heat equation, we provide a concept of "dissipative solutions" that enforces first the "weakstrong" uniqueness principle in any space dimensions and, second, the existence of global solutions at least in two space dimensions.
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University of Sydney

Tue 11 Jun 2013, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB 4127 (at PIMS)

Some systems of nonlinear elliptic partial differential equations in condensate problems.

ESB 4127 (at PIMS)
Tue 11 Jun 2013, 3:30pm4:30pm
Abstract
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Mon 8 Jul 2013, 8:00am
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
Earth Sciences Building Rm 2012 2207 Main Mall

Analysis and Partial Differential Equations

Earth Sciences Building Rm 2012 2207 Main Mall
Mon 8 Jul 2013, 8:00am6:00pm
Abstract
Schedule to be posted in the conference webpage:
http://www.pims.math.ca/scientificevent/ghoussoub
This conference brings together worldrenowned researchers in areas of mathematical analysis and PDE such as optimal transportation, the calculus of variations, convex analysis, elliptic systems, and geometric analysis, which are grounded in applications to the natural and social sciences while generating exciting new directions for mathematical research. Its primary aims are to survey the stateoftheart in these interrelated fields, expand the connections between them, identify key future directions, and encourage a new generation of scientists to advance this fundamental area of mathematics.
The conference begins Monday 8th morning and ends Friday 12th evening.
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Univeristy of Montreal and CRM

Tue 27 Aug 2013, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (PIMS)

QUANTUM WIRES, ORTHOGONAL POLYNOMIALS AND DIOPHANTINE APPROXIMATION

ESB 2012 (PIMS)
Tue 27 Aug 2013, 3:30pm4:30pm
Abstract
An important problem in Quantum Information is the transfer of states with high fidelity between locations. The devices performing this function are referred to as quantum wires. Spin chains can in principle be used to construct such wires. I shall discuss the design of spin chains that realize perfect and almost perfect transfer, that is that transport a state from one end of the chain to the other with probability one or almost one over some time.
Orthogonal Polynomial Theory and elements of Diophantine approximation will be called upon.
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University of Cape Town

Thu 5 Sep 2013, 1:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
MATX 1102

Existence and nonlinear stability of stationary states for the semirelativistic SchroedingerPoisson system

MATX 1102
Thu 5 Sep 2013, 1:00pm2:00pm
Abstract
We study the stationary states of the semirelativistic SchroedingerPoisson system in the repulsive (plasma physics) Coulomb case. In particular, we establish the existence and the nonlinear stability of a wide class of stationary states by means of the energyCasimir method. Moreover, we establish global wellposedness results for the semirelativistic SchroedingerPoisson system in appropriate functional spaces.
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UBC

Tue 17 Sep 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Relaxation of Variational Problems for OrientationPreserving Maps

ESB 2012
Tue 17 Sep 2013, 3:30pm4:30pm
Abstract
It is wellknown that variational problems may fail to have a classical minimiser if the integrand is not convex. In the 1930s, L. C. Young suggested a relaxation of such problems, where the minimising map is allowed to be measurevalued. In physical applications (e.g. elasticity theory), one often looks at variational problems for gradients of vector fields. A crucial problem in the context of relaxation is to characterise those measurevalued maps that arise as limits of a sequence of gradients. While this was achieved by D. Kinderlehrer and P. Pedregal about 20 years ago, the question remained open whether a similar characterisation could be found under the additional constraint that the gradients have positive determinant, i.e. the underlying maps be orientationpreserving. I will present such a characterisation, recently obtained in joint work with K. Koumatos (Oxford) and F. Rindler (Warwick).
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UBC

Tue 24 Sep 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Symmetric MongeKantorovich problems and polar decompositions of vector fields

ESB 2012
Tue 24 Sep 2013, 3:30pm4:30am
Abstract
For any given integer N larger than 2, we show that every bounded measurable vector field is Ncyclically monotone up to a measure preserving Ninvolution. The proof involves the solution of a multidimensional symmetric MongeKantorovich problem, which we first study in the case of a general cost function on a product domain. The proof exploits a remarkable duality between measure preserving transformations that are Ninvolutions and those Hamiltonians that are Ncyclically antisymmetric.
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Montpellier / PIMSUBC

Tue 1 Oct 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Asymptotically harmonic manifolds of nonpositive curvature

ESB 2012
Tue 1 Oct 2013, 3:30pm4:30pm
Abstract
Harmonic manifolds are those Riemannian manifolds whose harmonic functions satisfy the meanvalue property, or equivalently, whose spheres have constant mean curvature. F. Ledrappier introduced an asymptotic version of harmonicity which was mainly studied in the cocompact and homogeneous cases. In this talk, I will review some classical facts on harmonic manifolds and prove some new results on asymptotically harmonic manifolds, including a characterization in term of the volume form . This is a joint work with Andrea Sambusetti.
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U. Lorraine /PIMSUBC

Tue 8 Oct 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Compactness and stability of some nonlinear elliptic equations: glueing of a peak on a static profile

ESB 2012
Tue 8 Oct 2013, 3:30pm4:30pm
Abstract
In this talk, I will review a few issues and results on compactness of equations of scalar curvature type. In particular, I will focus on the difficulty of the degeneracy of the kernel of the solutions to such equations. This is joint work with Jérôme Vétois (Nice).
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George Washington U.

Tue 15 Oct 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Double bubble and coreshell solutions in an inhibitory ternary system

ESB 2012
Tue 15 Oct 2013, 3:30pm4:30pm
Abstract
We consider a inhibitory ternary system of three constituents, a model motivated by the triblock copolymer theory. The free energy of the system consists of two parts: the interfacial energy coming from the boundaries separating the three constituents, and the longer range interaction energy that functions as an inhibitor to limit micro domain growth. One solution of this system, found by Lu Xie in her PhD thesis, is a coreshell pattern where the first constituent forms the core, the second forms the shell, and the third fills the back ground. Another solution is shown in a joint work with Juncheng Wei: there is a perturbed double bubble that exists as a stable solution of the system. Each bubble is occupied by one constituent. The third constituent fills the complement of the double bubble. This solution has two triple junction points.
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UBC

Tue 29 Oct 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

On Fractional Minimal Surfaces

ESB 2012
Tue 29 Oct 2013, 3:30pm4:30pm
Abstract
We consider fractional minimal surfaces introduced by Caffarelli, Roquejoffre and Savin (2009). Up to now the only examples of fractional minimal surfaces are hyperplanes. In this talk, we first prove the existence of the analog of fractional Lawson's minimal cones and establish their stability/instability in low dimensions. In particular we find that there are stable fractional minimal cones in dimension 7, in contrast with the case of classical minimal surfaces. Then we prove the existence of fractional catenoids and fractional CostaHoffmanMeeks surfaces. Interestingly the interaction of planes in fractional minimal surfaces is governed by an nonlinear elliptic equation with negative power which arises in the study of MEMS. (Joint work with J. Davila and M. del Pino.)
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Trondheim

Tue 5 Nov 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Homeomorphisms between aperiodic tiling spaces

ESB 2012
Tue 5 Nov 2013, 3:30pm4:30pm
Abstract
In this talk, I will give an introduction to aperiodic tilings. Usually, one studies a topological dynamical system associated to these tilings rather than one specific tiling (this is the analogue to studying a subshift rather that one single word in symbolic dynamics).
It is a natural question to ask what happens to the underlying tilings when there is a homeomorphism between tiling spaces.
The result I will present is the following: whenever two tiling spaces are homeomorphic, the complexity function is preserved up to some multiplicative constants and rescaling.
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University of Marseille

Tue 12 Nov 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

The fractional Yamabe problem

ESB 2012
Tue 12 Nov 2013, 3:30pm4:30pm
Abstract
A great amount of work has been dedicated in the last years to understand
problems with integral diffusion for elliptic, parabolic or hyperbolic
equations and systems. In this talk, I will describe a new Yamabe problem
based on conformally covariant elliptic operators of fractional order. I
will describe some new results on existence of metrics for the regular and
singular problems.
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UBC

Tue 19 Nov 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

On the norm inflation of Incompressible Euler in borderline spaces

ESB 2012
Tue 19 Nov 2013, 3:30pm4:30pm
Abstract
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Universite de Lorraine

Tue 21 Jan 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

HardySobolev equations and related inequalities on compact Riemannian manifolds

ESB 2012
Tue 21 Jan 2014, 3:30pm4:30pm
Abstract
Let (M,g) be a compact Riemannian Manifold without boundry of dimension n \geq 3, x_0 \in M, and s \in (0,2). We let \crit: = \frac{2(ns)}{n2} be the critical HardySobolev exponent. I investigate the influence of geometry on the existence of positive distributional solutions u \in C^0(M) for the critical equation
\Delta_g u+a(x) u=\frac{u^{\crit1}}{d_g(x,x_0)^s} \;\; \hbox{ in} \ M.
Via a minimization method, I prove existence in dimension n\geq 4 when the potential a is sufficiently below the scalar curvature at x_0. In dimension n=3, using a global argument, i prove existence when the mass of the linear operator \Delta_g + a is positive at x_0. On the other hand, by using a Blowup around x_0, i prove that the sharp constant of the related HardySobolev inequalities on (M,g), which is equal to the one of the Euclidean HardySobolev inequalities, is achieved for all compact Riemannian Manifold of dimension n \geq 3 with or without boundary.
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Universidad de Chile

Tue 28 Jan 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Entire stable solutions of a 4rth order elliptic equation and a monotonicity formula

ESB 2012
Tue 28 Jan 2014, 3:30pm4:30pm
Abstract
We consider the nonlinear fourthorder problem \Delta^2 u=u^{p1}u\ \ \mbox{in} \ \mathbb R^n, where p>1 and n\ge1. We give a complete classification of stable and finite Morse index solutions in the full exponent range. We also compute an upper bound of the Hausdorff dimension of the singular set of extremal solutions. A key tool is a new monotonicity formula.
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Basque Center for Applied Mathematics

Tue 4 Feb 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Flow control in the presence of shocks

ESB 2012
Tue 4 Feb 2014, 3:30pm4:30pm
Abstract
In this talk we present some joint work in collaboration with C. Castro (UPM, Madrid), R. Lecaros (CMM Chile) and F. Palacios (Stanford) on flow control.
We address a classical optimal control problem of inverse design, aiming to identify the initial source leading to a desired final configuration.
First, in the onedimensional case, we explain why classical strategies, based on linearization methods, fail, because of the lack of regularity of solutions. We then introduce an alternating descent method that exploits the generalized gradients that take into account the sensitivity of the smooth arcs of the solutions but also of shock locations.
We compare the performance of the method with classical purely discrete strategies through various numerical experiments.
We also address the multidimensional case and point towards perspectives of future development.
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University of Toronto

Thu 13 Feb 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 4133 at PIMS

Big frequency cascades in the nonlinear Schrödinger evolution

ESB 4133 at PIMS
Thu 13 Feb 2014, 3:30pm4:30pm
Abstract
I will outline a construction of an exotic solution of the nonlinear Schrödinger equation that exhibits a big frequency cascade. Recent advances related to this construction and some open questions will be surveyed.
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Oregon State University

Tue 25 Feb 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

How to lift positive Ricci curvature

ESB 2012
Tue 25 Feb 2014, 3:30pm4:30pm
Abstract
We show how to lift positive Ricci and almost nonnegative curvatures from an orbit space M/G to the corresponding Gmanifold, M. We apply the results to get new examples of Riemannian manifolds that satisfy both curvature conditions simultaneously. This is joint work with Fred Wilhelm.
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University of Victoria

Tue 4 Mar 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB2012 (PIMS)

Optimal Transportbased Model and Algorithms for Particle Image Velocimetry

ESB2012 (PIMS)
Tue 4 Mar 2014, 3:30pm4:30pm
Abstract
Particle Image Velocimetry (PIV) is a technique using successive laser images of particles immersed in a fluid to measure the velocity field of the fluid flow. Traditionally, crosscorrelation is employed to extract the field from each pair of recorded images. This talk will introduce a new approach based on Optimal Transport (OT) to approximate the velocity field. More specifically, we consider the solution of the L2 OT problem with initial and final densities given by successive images of tracers. We will first present a model for this situation and investigate the behaviour of the OT map with respect to the model's key parameters. Then, we will present some algorithms and numerical results applying this theory to synthetic and real examples. This is joint work with B.Khouider and M.Agueh.
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Montpellier / PIMSUBC

Tue 11 Mar 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Spectral positivity on surfaces

ESB 2012
Tue 11 Mar 2014, 3:30pm4:30pm
Abstract
We shall see how the positivity of some Schr\"odinger operator on a surface gives information on its topology and its conformal type. The potent of the operators considered here involve the curvature of the surface and appear naturally in the study of minimal and constant mean curvature surfaces. It is a joint work with Pierre B\'erard.
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University of Oregon

Fri 21 Mar 2014, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (PIMS building) note the time and location change

Complex MongeAmpere equation on Kahler manifolds

ESB 2012 (PIMS building) note the time and location change
Fri 21 Mar 2014, 3:30pm4:30pm
Abstract
Complex MongeAmpere (CMA) equation is of fundamental importance in Kahler geometry. We will discuss regularity results for two versions of complex MongeAmpere equation which are extensively studied in Kahler geometry. The first is the classical CMA equation solved by S.T. Yau in 1970s to prove the Calabi conjecture. The second is a homogenous complex MongeAmpere, which is known as a geodesic equation of the space of Kahler metrics.
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UBC

Tue 25 Mar 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Complex analytic, onefrequency cocycles

ESB 2012
Tue 25 Mar 2014, 3:30pm4:30pm
Abstract
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University of Warwick

Tue 15 Apr 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Directional oscillations, concentrations, and compensated compactness via microlocal compactness forms

ESB 2012
Tue 15 Apr 2014, 3:30pm4:30pm
Abstract
Microlocal compactness forms (MCFs) are a new tool to study oscillations and concentrations in L^pbounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending both the theory of (generalized) Young measures and the theory of Hmeasures. Since in L^pspaces oscillations and concentrations precisely discriminate between weak and strong compactness, MCFs allow to quantify the difference between these two notions of compactness. The definition involves a Fourier variable, whereby also differential constraints on the functions in the sequence can be investigated easily. Furthermore, pointwise restrictions are reflected in the MCF as well, paving the way for applications to Tartar's framework of compensated compactness; consequently, we establish a new weaktostrong compactness theorem in a "geometric" way. Moreover, the hierarchy of oscillations with regard to slow and fast scales can be investigated as well since this information is also is reflected in the generated MCF.
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UC Berkeley

Mon 12 May 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (PIMS building)

Hölder continuous Euler flows with compact support in spacetime [Joint with P. Isett (MIT) ]

ESB 2012 (PIMS building)
Mon 12 May 2014, 3:30pm4:30pm
Abstract
In this talk, we will describe a construction of compactly supported solutions to the threedimensional incompressible Euler equations on R \times R^3 with Hölder regularity 1/5 \epsilon in space and time. This work extends the earlier works of De LellisSzékelyhidi, BuckmasterDe LellisSzékelyhidi and Isett on construction of Hölder continuous dissipative Euler flows to the nonperiodic setting. Our key technical innovation is a simple method for finding a compactly supported symmetric 2tensor with a prescribed divergence, which obeys useful bounds.
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Korea Institute for Advanced Study (KIAS)

Tue 13 May 2014, 11:00am
Diff. Geom, Math. Phys., PDE Seminar
ESB 4127

The cPlateau problem for surfaces in space.

ESB 4127
Tue 13 May 2014, 11:00am12:00pm
Abstract
The cPlateau problem for surfaces in space asks, given c>0 and \gamma a closed curve in space, whether we can find M_{c} a smooth orientable surfacewithboundary, with \partial M_{c} = \sigma_{c}+\gamma where \sigma_{c} is a finite union of closed curves disjoint from \gamma, minimizing cisoperimetric mass \mathbf{M}^{c}(M) := \text{area}(M)+c \cdot \text{length}(\partial M)^{2} amongst all M smooth orientable surfaceswithboundary, with \partial M = \sigma+\gamma where \sigma is a finite union of closed curves disjoint from \gamma. In this talk we give several regularity results for solutions to the cPlateau problem, formulated in the more general setting of integer tworectifiable currents.
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University of Hagen

Tue 13 May 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 4127

Scaling of R\'enyi entanglement entropies of the free Fermigas ground state

ESB 4127
Tue 13 May 2014, 3:30pm4:30pm
Abstract
In the remarkable paper [Phys. Rev. Lett. 96, 100503 (2006)], Dimitri Gioev and Israel Klich conjectured an explicit formula for the leading asymptotic growth of the spatially bipartite vonNeumann entanglement entropy of noninteracting fermions in multidimensional Euclidean space at zero temperature. Based on recent progress by one of us (A.V.S.) in semiclassical functional calculus for pseudodifferential operators with discontinuous symbols, we provide here a complete proof of that formula and of its generalization to R\'enyi entropies of all orders \alpha>0. The special case \alpha=1/2 is also known under the name logarithmic negativity and often considered to be a particularly useful quantification of entanglement. These formulas, exhibiting a ``logarithmically enhanced area law'', have been used already in many publications.
This is joint work with Hajo Leschke and Alexander V. Sobolev which will be published in Phys. Rev. Lett.
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Université d'AixMarseille

Tue 20 May 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Rearrangement inequalities and applications to elliptic eigenvalue problems

ESB 2012
Tue 20 May 2014, 3:30pm4:30pm
Abstract
The talk will be concerned with various optimization results for the principal eigenvalues of general secondorder elliptic operators in divergence form with Dirichlet boundary condition in bounded domains of R^n. To each operator in a given domain, one can associate a radially symmetric operator in the ball with the same Lebesgue measure, with a smaller principal eigenvalue. The constraints on the coefficients are of integral, pointwise or geometric types. In particular, we generalize the RayleighFaberKrahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new symmetrization technique, different from the Schwarz symmetrization. This talk is based on a joint work with N. Nadirashvili and E. Russ.
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UBC

Tue 27 May 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
MATX 1118 (Math Annex) NOTE: special location

Monge Ampere functionals and the second boundary value problem

MATX 1118 (Math Annex) NOTE: special location
Tue 27 May 2014, 3:30pm4:30pm
Abstract
I will discuss a family of fourth order PDE's and their corresponding second boundary value problem on a bounded strictly convex domain. Associated MongeAmp\`ere functionals will be discussed as well. Special cases here include the equation for prescribed affine mean curvature of a graph, and also Abreu's equation for prescribed scalar curvature of certain toric varieties. The talk is based on joint work with Ben Weinkove.
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Seoul National University, Korea.

Thu 12 Jun 2014, 3:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
MATX 1118 [Notice the time and location change!]

Regularity estimates for nonlinear elliptic and parabolic problems

MATX 1118 [Notice the time and location change!]
Thu 12 Jun 2014, 3:00pm4:00pm
Abstract
In this talk I will present some recent improvements in regularity estimates for the gradient of solutions to nonlinear elliptic and parabolic equations with nonstandard growth in irregular domains, In particular, when the nonhomogeneous terms belong to various function spaces including weighted Lebesgue spaces, Orlicz spaces and variable exponent spaces.
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EHESS, France

Tue 9 Sep 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

The effect of a line with fast diffusion on biological invasions

ESB 2012
Tue 9 Sep 2014, 3:30pm4:30pm
Abstract
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UBC

Tue 16 Sep 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Singularities in Lagrangian Mean Curvature Flow

ESB 2012
Tue 16 Sep 2014, 3:30pm4:30pm
Abstract
Lagrangian Mean Curvature Flow (LMCF) is a geometric flow, aiming to deform a Lagrangian immersion to a minimal one. To understand the flow, it is important to understand the formation of singularity in LMCF. In this talk, I will introduce the concept of a selfshrinker (a local model for singularity), how it is formed in LMCF, and give some examples of Lagrangian selfshrinkers. Then I will discuss a recent work with Jingyi Chen concerning the space of all compact Lagrangian selfshrinkers in \mathbb C^2.
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UBC

Tue 23 Sep 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

General Hardytype inequalities on manifolds.

ESB 2012
Tue 23 Sep 2014, 3:30pm4:30pm
Abstract
Given a general secondorder, elliptic operator P on a general domain, we discuss the question of finding an "optimal", or "asymptotically optimal", Hardy inequality for P. Such an inequality can be considered as a gneralized spectral gap inequality of P. If time allows, we will also consider the $L^p$ case.
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University of Minnesota

Tue 30 Sep 2014, 2:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB 4127

Geometry of shrinking Ricci solitons

ESB 4127
Tue 30 Sep 2014, 2:30pm3:30pm
Abstract
This talk concerns the geometry of shrinking Ricci solitons, a class of selfsimilar solutions to the Ricci flows. We plan to provide some general background results and explain a recent work with Ovidiu Munteanu on the curvature estimates of four dimensional solitons.
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Universite de CergyPontoise

Tue 30 Sep 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Stationary Kirchhoff systems in closed manifolds

ESB 2012
Tue 30 Sep 2014, 3:30pm4:30pm
Abstract
We investigate various issues for stationary Kirchhoff systems in closed manifolds, such as the questions of existence, nonexistence and compactness of solutions to the equations.
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Université ParisSud

Thu 2 Oct 2014, 4:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB 4133 (PIMS lounge)

Large Time behavior for the cubic Szego évolution

ESB 4133 (PIMS lounge)
Thu 2 Oct 2014, 4:00pm5:00pm
Abstract
The cubic Szegö equation is an Hamiltonian evolution on periodic functions with nonnegative Fourier modes, arising as a normal form for the large time behavior of a nonlinear wave equation on the circle. It defines a flow on every Sobolev space with enough regularity. In this talk, I will give the main arguments for the proof of the following theorem. The trajectories of the cubic Szegö equation are almost periodic in the Sobolev energy space, but
are generically unbounded in every more regular Sobolev space.This is a joint work with Sandrine Grellier and Zaher Hani.
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UBC

Tue 7 Oct 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Analytical properties for the NavierStokes equations and applications

ESB 2012
Tue 7 Oct 2014, 3:30pm4:30pm
Abstract
Strong solutions to the 3D NavierStokes equations are known to exist locallyintime and are real analytic. Providing lower bounds for their analyticity radius is important as this length scale plays an important role in turbulent phenomenologies and can be used to establish blowup criteria. In this talk we discuss one approach to estimating analyticity radii and a related conditional regularity criteria.
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Toulouse

Tue 14 Oct 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

On the blowup speed for modified critical nonlinear Schrödinger equations

ESB 2012
Tue 14 Oct 2014, 3:30pm4:30pm
Abstract
So far, only two blowup regimes have been studied for NLS equations: the pseudoconformal regime, where the blowup speed is like $t ^{1}$ and the loglog regime where the blowup speed is like $t^{1/2}$ with a loglog correction.
In this talk, we consider the nonlinear Schrodinger with a double power nonlinearity where one of the power is L2 critical and the other one is L2subcritical. We construct a minimal mass blowing up solution whose blowup speed is neither the loglog speed nor the pseudoconformal speed, but is of the type $t^{s}$ with $s$ varying between $1/2$ and $1$ depending on the subcritical power. This is based on a joint work with Yvan Martel and Pierre Raphael.
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UBC

Tue 21 Oct 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Traveling fronts to reaction diffusion equations with fractional Laplacian

ESB 2012
Tue 21 Oct 2014, 3:30pm4:30pm
Abstract
We show the nonexistence of traveling fronts in the combustion model with fractional Laplacian (\Delta)^s when s\in(0,1/2]. Our method can be used to give a direct and simple proof of the nonexistence of traveling fronts for the usual FisherKPP nonlinearity. Also we prove the existence and nonexistence of traveling waves solutions for different ranges of the fractional power s for the generalized FisherKPP type model.
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Tue 28 Oct 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Traveling waves involving fractional Laplacians

ESB 2012
Tue 28 Oct 2014, 3:30pm4:30pm
Abstract
In this talk, we will discuss the existence of the traveling wave solution for the AllenCahn equation involving the fractional Laplacians. Based on the existence of the standing waves for the balanced AllenCahn equation, we will use the continuity method to obtain the existence of the traveling waves for unbalanced AllenCahn equation. The key ingredient is the the bound of the traveling speed in terms of the potential. Some open questions will be discussed.
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Kyoto University

Tue 4 Nov 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Global dynamics of nonlinear dispersive equations above the ground state energy

ESB 2012
Tue 4 Nov 2014, 3:30pm4:30pm
Abstract
This is a survey on the joint works with Wilhelm Schlag,
Joachim Krieger and Tristan Roy. We classify global behavior of all
solutions with energy up to slightly more than the ground state for
the nonlinear KleinGordon, Schrodinger, and wave equations. The
dynamics include scatteing (to 0), blowup, and scatttering to
solitons. The solutions scattering to solitons form threshold
hypersurfaces in the energy space, giving a complete classification
under the energy constraint. It also describes how a solution can
disperse in the past and blow up in the future.
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UBC

Tue 18 Nov 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

On Nontopological Solutions of the rank 2 ChernSimons System

ESB 2012
Tue 18 Nov 2014, 3:30pm4:30pm
Abstract
In this talk, I will talk about the ChernSimons equation arising from the study of physics of high critical temperature superconductivity. A longstanding open problem is the existence of nontopological solutions. We proved the existence of nontopological solutions for the rank 2 ChernSimons system. This is joint work with Professor Changshou Lin and Juncheng Wei
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Pont. Cat. Univ. Chile

Tue 25 Nov 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Nondegeneracy of nonradial nodal solutions to Yamabe problem

ESB 2012
Tue 25 Nov 2014, 3:30pm4:30pm
Abstract
We prove the existence of a sequence of nondegenerate, in the sense of DuyckaertsKenigMerle, nodal nonradial solutions to the critical Yamabe problem or stationary energycritical wave equation.
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University of Oregon

Fri 28 Nov 2014, 2:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB 4133 (PIMS lounge)

Geometric flow on almost Hermitian manifolds towards a symplectic structure

ESB 4133 (PIMS lounge)
Fri 28 Nov 2014, 2:00pm3:00pm
Abstract
We propose geometric flows to study the existence of a symplectic structure on an almost Hermitian manifold. We prove the shorttime existence and uniqueness, and show some examples.
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Brown University

Tue 13 Jan 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Recent gluing constructions in Differential Geometry

ESB 2012
Tue 13 Jan 2015, 3:30pm4:30pm
Abstract
I will first discuss doubling and desingularization constructions for minimal surfaces and applications on closed minimal surfaces in the round spheres, free boundary minimal surfaces in the unit ball, and selfshrinkers for the Mean Curvature flow. In the final part of the talk I will discuss my collaboration with Simon Brendle on constructions for Einstein metrics on fourmanifolds and related geometric objects.
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Stanford University

Tue 3 Mar 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

On the topology and index of minimal surfaces

ESB 2012
Tue 3 Mar 2015, 3:30pm4:30pm
Abstract
We show that for an immersed twosided minimal surface in R^3, there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface in R^3 of index 2, as conjectured by Choe. Moreover, we show that the index of an immersed twosided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface. (This is joint work with Otis Chodosh)
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Korea Institute for Advanced Study

Tue 24 Mar 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

New characterizations of the catenoid and helicoid

ESB 2012
Tue 24 Mar 2015, 3:30pm4:30pm
Abstract
Bernstein and Breiner found a characterization of the catenoid that the area of a minimal annulus in a slab is bigger than that of the maximally stable catenoid in the same slab. We give a simpler proof of their theorem and extend the theorem to some minimal surfaces with genus (joint work with Benoit Daniel). New characterizations of the helicoid recently proved by Eunjoo Lee will be also presented.
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University of Utah

Wed 13 May 2015, 3:10pm
Diff. Geom, Math. Phys., PDE Seminar / Probability Seminar
MATH 203

Fluctuations for polymer models in intermediate disorder

MATH 203
Wed 13 May 2015, 3:10pm4:00pm
Abstract
Directed polymer models are finitetemperature versions of first and lastpassage percolation on the lattice. In 1+1 dimensions, the freeenergy of the directed polymer is conjecturally in the TracyWidom universality class at all finite temperatures. However, this has only been proven for a small class of polymers  the socalled solvable models that include Seppalainen's gamma polymers and the O'ConnellYor semidiscrete polymer  with special sets of shapes and edgeweight distributions. We present some new results towards the universality conjecture in the intermediate disorder scaling regime.
This is joint work with Jeremy Quastel.
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National Taiwan University

Tue 15 Sep 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Solutions to PoissonNernstPlanck type systems with crossdiffusion terms

ESB 2012
Tue 15 Sep 2015, 3:30pm4:30pm
Abstract
The PoissonNernstPlanck (PNP) system is a wellknown model of ion transport with many applications in biology, engineering and physics. Crossdiffusion terms may describe the exclusion of steric effects. In this lecture, I shall introduce cross diffusion terms from the LennardJones potential and show the analytical results as follows:
1. Stability of 1D boundary layer solutions to original PoissonNernstPlanck (PNP) systems
2. Multiple solutions of PNP systems with steric effects
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UBC Math

Tue 22 Sep 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Unnormalized conical KahlerRicci flow

ESB 2012
Tue 22 Sep 2015, 3:30pm4:30pm
Abstract
Conical Kahler metrics have become an interesting topic in Kahler geometry, and played an important role in the solution of YauTianDonaldson conjecture. In this talk, we make use of approximation method of GuenanciaPaun to extend TianZhang's maximal existence result of KahlerRicci flow to conic case. Finally if possible, we can talk a little about C^{2,\alpha}estimate for conical KahlerRicci flow based on Tian's master thesis.
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Fernuniversitat Hagen, Germany

Tue 29 Sep 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Area law for the entanglement entropy of the free Fermi gas at nonzero temperature

ESB 2012
Tue 29 Sep 2015, 3:30pm4:30pm
Abstract
The leading asymptotic largescale behavior of the spatially bipartite entanglement entropy (EE) of the free Fermi gas at temperature T=0 is by now well understood. Here, we present and discuss the first rigorous results for the corresponding EE of thermal equilibrium states at T>0. The leading largescale term of this thermal EE turns out to be twice the first finitesize correction to the infinitevolume thermal entropy (density). However, it is given by a rather complicated integral derived from semiclassical trace formulas and differs, at least at high temperature, from simpler expressions previously obtained by arguments based on a conformal field theory. In the zerotemperature limit, the leading largescale term of the thermal EE considerably simplifies and displays a \ln(1/T)singularity which one may identify with the known logarithmic correction at T=0 to the socalled arealaw scaling. This is joint work with Hajo Leschke and Alexander Sobolev.
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UBC

Tue 6 Oct 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

On type II singularity formulation of harmonic map flows

ESB 2012
Tue 6 Oct 2015, 3:30pm4:30pm
Abstract
I will consider the following classical harmonic map flow from a general twodimensional domain D to S^2:
u_t=\Delta u +\nabla u^2 u, u: D \to S^2
We develop a parabolic gluing method to construct finite time blowup solutions of Type II in general domains. We show that type II blowup solutions with blowup rate
(Tt)/\log^2 (Tt)
is stable and generic in arbitrary domains (without any symmetry). I will also discuss the construction of multiple blowups, reverse bubbling, bubbling trees, bubbling at infinity. As a byproduct we can perform new geometric surgeries. (Joint work with Manuel del Pino and Juan Davila.)
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Politechic University of Catalonia

Tue 13 Oct 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

On singular solutions for the fractional Yamabe problem

ESB 2012
Tue 13 Oct 2015, 3:30pm4:30pm
Abstract
Abstract: We construct some ODE solutions for the fractional Yamabe problem in conformal geometry. The fractional curvature, a generalization of the usual scalar curvature, is defined from the conformal fractional Laplacian, which is a nonlocal operator constructed on the conformal infinity of a conformally compact Einstein manifold.
These ODE solutions are a generalization of the usual Delaunay and, in particular, solve the fractional Yamabe problem
$$ (\Delta)^\gamma u= c_{n, {\gamma}}u^{\frac{n+2\gamma}{n2\gamma}}, u>0 \ \mbox{in} \ \r^n \backslash \{0\},$$
with an isolated singularity at the origin.
This is a fractional order ODE for which new tools need to be developed. The key of the proof is the computation of the fractional Laplacian in polar coordinates.
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University of Tennessee, Knoxville

Tue 27 Oct 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Gradient estimates and global existence of smooth solutions to a crossdiffusion system

ESB 2012
Tue 27 Oct 2015, 3:30pm4:30pm
Abstract
We investigate the global time existence of smooth solutions for the ShigesadaKawasakiTeramoto system of crossdiffusion equations of two competing species in population dynamics. If there are selfdiffusion in one species and no crossdiffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension. We obtain this result by deriving global W^{1,p}estimates of Calder\'{o}nZygmund type for a class of nonlinear reactiondiffusion equations with selfdiffusion. These estimates are achieved by employing CaffarelliPeral perturbation technique together with a new twoparameter scaling argument.
The talk is based on the joint work with L. Hoang (Texas Tech) and T. Nguyen (U. of Akron).
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UBC Math

Tue 17 Nov 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Forward discretely selfsimilar solutions of the NavierStokes equations

ESB 2012
Tue 17 Nov 2015, 3:30pm4:30pm
Abstract
For any discretely selfsimilar, incompressible initial data which is arbitrarily large in weak L^3, we construct a forward discretely selfsimilar solution to the 3D NavierStokes equations in the whole space. This also gives a third construction of selfsimilar solutions for any 1homogeneous initial data in weak L^3, improving those in by JiaSverak and KorobkovTsai for H\"older continuous data. Our method is based on a new, explicit a priori bound for the Leray equations. This is a joint work with TaiPeng Tsai.
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UBC Math

Tue 24 Nov 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Resonance Eigenvalues and bound states of the Nonlinear Schroedinger Equation

ESB 2012
Tue 24 Nov 2015, 3:30pm4:30pm
Abstract
There are many interesting questions concerning the Nonlinear Schroedinger Equation such as the existence and stability of solitary wave solutions as well as the long time behaviour of solutions. These problems are made more complicated by the presence of a resonance eigenvalue. Such occurrences are special cases which serve to worsen time decay estimates and complicate resolvent expansions. We will talk about some particular perturbation results whose treatment is nonstandard since a relevant linear operator has a resonance.
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University of Connecticut/University of TexasSan Antonio

Tue 1 Dec 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB2012

Saddle Solutions of AllenCahn Equation on the Plane.

ESB2012
Tue 1 Dec 2015, 3:30pm4:30pm
Abstract
AllenCahn equation arises in the mathematical study of phase transition. Despite it's seemingly simple appearance, It has displayed very rich structure of solutions and involved with deep mathematics. In this talk, I will discuss the existence, symmetry and classification of saddle solutions of AllenCahn equation on the plane. In particular, I will describe the variational characterization of these solutions as a mountain pass solutions.
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University of Toronto

Tue 5 Jan 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB2012

Vortex filaments in the Euler equation

ESB2012
Tue 5 Jan 2016, 3:30pm4:30pm
Abstract
Abstract: Classical fluid dynamics arguments suggest that in certain
limits, the evolution of thin vortex filaments in an ideal incompressible
fluid should roughly be governed by an equation called the binormal
curvature flow. However, these classical arguments rely on assumptions
that are so unrealistic that it would be hard even to extract from them a
precise conjecture that admits any realistic possibility of a proof. We
present a different approach to this question that yields a reasonable
formulation of a conjecture and strong supporting evidence, and that
clarifies the very substantial obstacles to a full proof. Parts of the
talk are based on joint work with Didier Smets and with Christian Seis
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UBC Math

Tue 12 Jan 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar

The singular mass of a domain and critical dimensions associated to the HardySchrodinger operator

Tue 12 Jan 2016, 3:30pm4:30pm
Abstract
I consider two different approaches for breaking scale invariance and restoring compactness for borderline variational problems involving the HardySchrodinger operator \Delta \frac{\gamma}{x^2} on a domain containing the singularity 0, either in its interior or on its boundary. One consists of adding a linear perturbation, another exploits the geometry of the domain. I discuss the role of various ``positive singular mass theorems" that help account for the critical dimensions below which these approaches fail. This is a joint project with Frederic Robert.
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University of Oregon

Tue 2 Feb 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Computing the Hodge Laplacian on 1forms of a manifold using random samples

ESB 2012
Tue 2 Feb 2016, 3:30pm4:30pm
Abstract
Let M be a submanifold of Euclidean space, and let X be a subset of N points, randomly sampled. Belkin and Niyogi showed that one can recover the Laplacian on functions on M as N gets large, by integrating the heat kernel. More recently, Singer and Wu use Principle Component Analysis to construct connection matrices between approximate tangent spaces for nearby points in X. This allows them to construct a rough Laplacian on 1forms. Together with Ache, we show that by iterating the Laplace operator of Belkin and Niyogi, a la Bakry and Emery, and appealing to the Bochner formula, we can reconstruct the Ricci curvature on the approximate tangent spaces. Combining our work with the work of Singer and Wu, we are able to approximate the Hodge Laplacian on 1forms.
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Johns Hopkins University

Tue 15 Mar 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

KAM theory for whiskered tori in Hamiltonian PDEs with applications to illposed ones

ESB 2012
Tue 15 Mar 2016, 3:30pm4:30pm
Abstract
We develop a KAM theory for tori with hyperbolic directions for PDEs coming mainly from fluid dynamics. One of the features of these PDEs is that they are strongly illposed. However, our method allows to construct specific quasiperiodic solutions. The format of the KAM theorem is a posteriori in a sense I will make precise and this allows to use several perturbative expansions to compute approximate solutions.
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University of Edinburgh

Tue 29 Mar 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

On the well posedness of the periodic fourth order Schrodinger equation in negative Sobolev spaces

ESB 2012
Tue 29 Mar 2016, 3:30pm4:30pm
Abstract
We will discuss the Cauchy problem for the cubic fourth order nonlinear Schrodinger equation (4NLS) on the circle. We first prove nonexistence of solutions to (4NLS) for initial data lying strictly in negative Sobolev spaces, by using the short time Fourier restriction norm method. Then, we focus on the wellposedness issue of the renoramilzed 4NLS (so called the Wick ordered W4NLS). In particular, by performing normal form reductions infinite many times, we prove wellposedness of (W4NLS) in negative Sobolev spaces. This talk is based on a joint work with Tadahiro Oh (University of Edinburgh).
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Oklahoma State University

Thu 7 Apr 2016, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB 4127

The twodimensional Boussinesq equations with partial dissipation

ESB 4127
Thu 7 Apr 2016, 3:30pm4:30pm
Abstract
The Boussinesq equations concerned here model geophysical flows such
as atmospheric fronts and ocean circulations. In addition, they play an
important role in the study of RayleighBenard convection. Mathematically
the 2D Boussinesq equations serve as a lowerdimensional model of the 3D
hydrodynamics equations. In fact, the 2D Boussinesq equations retain some
key features of the 3D Euler and the NavierStokes equations such as the
vortex stretching mechanism. The global regularity problem on the 2D
Boussinesq equations with partial or fractional dissipation has attracted
considerable attention in the last few years. This talk presents recent
developments in this direction. In particular, we detail the global regularity
result on the 2D Boussinesq equations with vertical dissipation as
well as some recent work for the 2D Boussinesq equations with general
critical dissipation.
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Karlsruhe Institute of Technology (KIT)

Tue 12 Apr 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB2012

Limits of alphaharmonic maps

ESB2012
Tue 12 Apr 2016, 3:30pm4:30pm
Abstract
I will discuss a recent joint work with A. Malchiodi (Pisa) and M. Micallef (Warwick) in which we show that not every harmonic map can be approximated by a sequence of alphaharmonic maps.
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University of Toronto Mississauga

Tue 6 Sep 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Solvability of some integrodifferential equations with anomalous diffusion

ESB 2012
Tue 6 Sep 2016, 3:30pm4:30pm
Abstract
The work deals with the existence of solutions of an integrodifferential equation in the case of the anomalous diffusion with the Laplace operator in a fractional power. The proof of existence of solutions relies on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains are used.
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UBC

Tue 13 Sep 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
EBS 2012

On some functional and geometric inequalities

EBS 2012
Tue 13 Sep 2016, 3:30pm4:30pm
Abstract
In this talk, we will discuss the TrudingerMoser and CaffarelliKohnNirenberg inequalities in the settings where the classical Schwarz rearrangement cannot be used. We will then talk about some approaches to study the maximizers for these problems. This is joint work with Guozhen Lu.
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University of Minnesota

Tue 20 Sep 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

On the Cauchy problem for vortex rings

ESB 2012
Tue 20 Sep 2016, 3:30pm4:30pm
Abstract
We consider the initialvalue problem for the 3d NavierStokes equation when the initial vorticity is supported on a circle. Such initial datum is in certain function spaces where perturbation theory works for small data, but not for large data, even for short times, and there are good reasons to believe that this is not just a technicality. We prove global existence and uniqueness for large data in the class of axisymmetric solutions. The main tools are Nashtype estimates and certain monotone quantities. Uniqueness in the class of solutions which are not necessarily axisymmetric remains a difficult open problem, which we plan to discuss briefly. Joint work with Thierry Gallay.
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UBC

Tue 27 Sep 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Gibbs' measure and almost sure global wellposedness for one dimensional periodic fractional Schrodinger equation

ESB 2012
Tue 27 Sep 2016, 3:30pm4:20pm
Abstract
In this talk we will present recent local and global wellposedness results on the one dimensional periodic fractional Schrodinger equation. We will also talk about construction of Gibbs' measures on certain Sobolev spaces and how we can prove almost sure global wellposedness using this construction.
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University of California at Irvine

Tue 4 Oct 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

On the first eigenvalue estimate for subLaplacian and Kohn Laplacian and Rigidity Theorems on pseudoHermitian CR manifolds

ESB 2012
Tue 4 Oct 2016, 3:30pm4:30pm
Abstract
In this talk, I will present a CRversion of LichnerowiczObata type theorem in a closed pseudoHermitian CR manifolds. It includes the lower bound estimates for the first positive eigenvalue for the both subLaplacian and Kohn Laplacian. I will also provide Obata type theorem associated to the subLaplacian and Kohn Laplacian on a closed pseudoHermitian manifold. As an application, we give some rigidity theorem when lower bound of eigenvalue is achieved. This is based on a joint work with X. Wang and a joint work with Duong N. Son and Wang. I will also talk about some ongoing work in this topic.
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Harvard University, ENS, France

Thu 6 Oct 2016, 2:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB 4127

Fluid limits of systems of particles

ESB 4127
Thu 6 Oct 2016, 2:00pm3:00pm
Abstract
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UBC

Tue 11 Oct 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Refined Asymptotics of the Teichm\"uller harmonic map flow

ESB 2012
Tue 11 Oct 2016, 3:30pm4:30pm
Abstract
The Teichm\"uller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to evolve. It was introduced by M. Rupflin and P. Topping in 2012. The objective of the flow is to find branched minimal immersions on a given surface. We will give some background on the flow and then describe some recent work. In particular we show that if the flow exists for all times then in a certain sense the maps (sub)converge to a collection of branched minimal immersions with no loss of energy (even when allowing for degeneration of the metric at infinity). We also construct an example of a smooth flow where the image of the limit maps is disconnected. This is joint work with M. Rupflin and P. Topping.
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UBC

Tue 18 Oct 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Higher order elliptic problems with critical sobolev growth on a compact riemannian manifold: Best constants and existence

ESB 2012
Tue 18 Oct 2016, 3:30pm4:30pm
Abstract
We investigate the existence of solutions to a nonlinear elliptic problem involving the critical Sobolev exponent for a Polyharmomic operator on a Riemannian manifold M. We first show that the best constant of the Sobolev embedding on a manifold can be chosen as close as one wants to the Euclidean one, and as a consequence derive the existence of minimizers when the energy functional goes below a quantified threshold. Next, higher energy solutions are obtained by Coron's topological method, provided that the minimizing solution does not exist and the manifold satisfies a certain topological assumption. To perform the topological argument, we obtain a decomposition of PalaisSmale sequences as a sum of bubbles and adapt Lions's concentrationcompactness lemma.
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McGill University

Tue 25 Oct 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB2012

Curvature flows and the isoperimetric problems in geometry

ESB2012
Tue 25 Oct 2016, 3:30pm4:30pm
Abstract
Abstract: We will discuss two types of curvature flows designed to prove isoperimetric type inequalities. The first one is a mean curvature type flow, it was introduced in a previous joint work with Junfang Li in space forms. In a recent joint paper with Junfang Li and MuTao Wang, we consider a similar normalized hypersurface flow in the more general ambient setting of warped product spaces with general base. Under a natural necessary condition, the flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. Under another condition with is related to the notion of “photon sphere” in general relativity, we establish the regularity and convergence of the flow, thereby solve the isoperimetric problem in warped product spaces. In a similar spirit, I will discuss a inverse mean curvature type flow in hyperbolic space to deal with AlexandrovFenchel type isoperimetric inequalities.
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University of Oregon

Tue 1 Nov 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Minimal hypersurfaces with free boundary and positive scalar curvature

ESB 2012
Tue 1 Nov 2016, 3:30pm4:30pm
Abstract
There is a wellknown technique due to SchoenYau from the late 70s which uses (stable) minimal hypersurfaces to find topological implications of a (closed) manifold's ability to support positive scalar curvature metrics. In this talk, we describe a version of this technique for manifolds with boundary and discuss how it can be used to study bordisms of positive scalar curvature metrics.
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Stanford University

Tue 8 Nov 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

The moduli space of 2convex embedded spheres

ESB 2012
Tue 8 Nov 2016, 3:30pm4:30pm
Abstract
The space of smoothly embedded nspheres in R^{n+1} is the quotient space M_{n}:=Emb(S^{n},R^{n+1})/Diff(S^{n}). In 1959 Smale proved that M_{1} is contractible and conjectured that M_{2} is contractible as well, a fact that was proved by Hatcher in 1983.
While it is known that not all M_{n} are contractible, for n\get 3 no single homotopy group of M_{n} is known. Even knowing whether the M_{n} are path connected or not would be extremely interesting. For instance, if M_{3} is not path connected, the 4d smooth Poincare conjecture can not hold true.
In this talk, I will first explain how mean curvature flow can assist in studying the topology of geometric relatives of M_{n}.
I will first illustrate how the theory of 1d mean curvature flow (aka curve shortening flow) yields a very simple proof of Smale's theorem about the contractibility of M_{1}.
I will then describe a recent joint work with Reto Buzzno and Robert Haslhofer, utilizing mean curvature flow with surgery to prove that the space of 2convex embedded spheres is path connected.
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University of Oregon

Thu 24 Nov 2016, 2:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB 4127

The Calabi flow with rough initial data (note special time & room)

ESB 4127
Thu 24 Nov 2016, 2:00pm3:00pm
Abstract
The Calabi flow is a fourth order nonlinear parabolic flow, introduced by Calabi in 1980s, and it aims to find Kahler metrics with constant scalar curvature (or more generally extremal Kahler metrics). We prove that the Calabi flow can have a unique smooth short time solution with continuous initial metric. As a byproduct, we prove some elementary but new Schauder type estimates for biharmonic heat equation on compact manifolds. This is a joint work with Yu Zeng (University of Rochester). Our result partially answers a problem proposed by Xiuxiong Chen.
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Universidade Federal de Juiz de Fora

Tue 29 Nov 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Hénon Problem in Hyperbolic Space

ESB 2012
Tue 29 Nov 2016, 3:30pm4:30pm
Abstract
We deal with a class of the semilinear elliptic equations of the Hénontype in hyperbolic space. The problem involves a logarithm weight in the Poincaré ball model, bringing singularities on the boundary. Considering radial functions, a compact Sobolev embedding result is proved, which extends a former Ni result made for a unit ball in R^N. Combining this compactness embedding with the Mountain Pass Theorem, an existence result is established.
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University of Alberta

Thu 8 Dec 2016, 4:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB Room 4127 (PIMS Videoconferencing Room )

Mulitto onedimensional optimal transport

ESB Room 4127 (PIMS Videoconferencing Room )
Thu 8 Dec 2016, 4:00pm5:00pm
Abstract
I will discuss joint work with PierreAndre Chiappori and Robert McCann on the MongeKantorovich problem of transporting a probability measure on \mathbb{R}^n to another on the real line. We introduce a nestededness criterion relating the cost to the marginals, under which it is possible to solve this problem uniquely (and essentially explicitly), by constructing an optimal map one level set at a time. I plan to discuss examples for which the nestedness condition holds, as well as some for which it fails; some of these examples arise from a matching problem in economics which originally motivated our work. If time permits, I will also briefly discuss how level set dynamics can be used to develop a local regularity theory in the nested case
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Mon 9 Jan 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012


ESB 2012
Mon 9 Jan 2017, 3:30pm4:30pm
Abstract
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Columbia University

Thu 19 Jan 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
seminar has been cancelled.

CANCELLED: Scaling limits of open ASEP and ferromagnetic Glauber dynamics

seminar has been cancelled.
Thu 19 Jan 2017, 3:30pm4:30pm
Abstract
We discuss two recent scaling limit results for discrete dynamics converging to stochastic PDEs. The first is the asymmetric simple exclusion process in contact with sources and sinks at boundaries, called Open ASEP. We prove that under weakly asymmetric scaling the height function converges to the KPZ equation with Neumann boundary conditions. The second is the Glauber dynamics of the BlumeCapel model (a generalization of Ising model), in two dimensions with Kac potential. We prove that the averaged spin field converges to the stochastic quantization equations. The main purpose of this talk is to discuss the general issues one needs to address when passing from discrete to continuum, the common challenge in the proofs of such scaling limit theorems, and how we overcome these difficulties in the two specific models. (Based on joint works with Ivan Corwin and Hendrik Weber)
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Max Planck Institute Bonn

Tue 24 Jan 2017, 3:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
MATH 126

Chiral differential operators and the curved betagamma system

MATH 126
Tue 24 Jan 2017, 3:00pm4:00pm
Abstract
Chiral differential operators (CDOs) are a vertex algebra analog of the associative algebra of differential operators. Originally introduced by mathematicians, Witten explained how CDOs arise as the perturbative part of the curved betagamma system with target X. I will describe recent work with Gorbounov and Williams in which we construct the BV quantization of this theory and use a combination of factorization algebras and formal geometry to recover CDOs. At the end, I hope to discuss how the techniques we developed apply to a broad class of nonlinear sigma models, including source manifolds of higher dimension.
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University of Texas at Austin

Thu 26 Jan 2017, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
MATH 102

Cluster Theory of the Coherent Satake Category

MATH 102
Thu 26 Jan 2017, 3:30pm5:00pm
Abstract
We discuss recent work showing that in type A_n the category of equivariant perverse coherent sheaves on the affine Grassmannian categorifies the cluster algebra associated to the BPS quiver of pure N=2 gauge theory. Physically, this can be understood as a statement about line operators in this theory, following ideas of GaiottoMooreNeitzke, Costello, and KapustinSaulina  in short, coherent IC sheaves are the precise algebrogeometric counterparts of Wilson't Hooft line operators. The proof relies on techniques developed by KangKashiwaraKimOh in the setting of KLR algebras. A key moral is that the appearance of cluster structures is in large part forced by the compatibility between chiral and tensor structures on the category in question (i.e. by formal features of holomorphictopological field theory). This is joint work with Sabin Cautis.
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Institute for Theoretical Physics, KU Leuven

Tue 31 Jan 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

On Products of Correlated Matrices Originating in the Statistical Structure of Quantum Mechanics

ESB 2012
Tue 31 Jan 2017, 3:30pm4:30pm
Abstract
Statistics of measurement outcomes in quantum mechanics is described by a map that associates a matrix to each possible measurement outcome. The probability of this outcome is then given by a trace of this matrix. Probability of a sequence of measurement outcomes is computed in the same way from a product of associated matrices. In this talk I will describe two results related to this setting. A theorem giving optimal conditions for uniqueness of the associated invariant measure on the projective sphere, and a theorem describing large deviation theory in the case when the matrices commute. The latter problem received lots of recent attention following experiments of S.~Haroche.
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Cornell University

Wed 1 Feb 2017, 4:00pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (Note the unusual time: 45 pm on Wednesday, Feb 1. )

Incompressibility and Global Injectivity in SecondGradient NonLinear Elasticity

ESB 2012 (Note the unusual time: 45 pm on Wednesday, Feb 1. )
Wed 1 Feb 2017, 4:00pm5:00pm
Abstract
This talk addresses how the calculusofvariations is applied to nonlinear elasticity. In physically realistic classical models, energyminimizing deformations may not be smooth enough to satisfy the variational EulerLagrange equations. However, with a secondgradient model we guarantee sufficient regularity to rigorously prove energyminimizers satisfy such an equation and maintain incompressibility and/or global injectivity.
The constraints of incompressibility and selfcontact introduce subtle challenges of infinitedimensional nonlinear analysis. I will discuss the techniques and assumptions that we use to prove the existence of a distributional pressure for the incompressibility constraint and a measurevalued surface traction for the selfcontact constraint. This work was part of my dissertation research done under the supervision of Professor Timothy J. Healey at Cornell University.
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University of Munich

Thu 2 Feb 2017, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
MATH 102 (note special day/room)

The adiabatic theorem for quantum spin systems

MATH 102 (note special day/room)
Thu 2 Feb 2017, 3:30pm4:30pm
Abstract
I will present an adiabatic theorem for the driven dynamics of ground state projections of a smooth family of manybody gapped quantum systems. The diabatic error is uniformly bounded in the volume of the interacting system. As an corollary, Kubo’s formula of linear response theory can be obtained in the thermodynamic limit.
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Kyoto University

Tue 14 Feb 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Singularity and regularity for the stationary solutions to linearized Boltzmann equations

ESB 2012
Tue 14 Feb 2017, 3:30pm4:30pm
Abstract
In Boltzmann equation, the interplay among free transport, collision, and boundary yields rich phenomena in regularity of solutions. In this talk, we will first introduce the logarithmic singularity both on macroscopic and microscopic variables due to the boundary. Then, we will discuss the regularity of stationary solutions in a convex domain. Finally, we will provide the analysis that realizes our observation.
Coffee and cookie will be provided before the seminar at the PIMS lounge.
Prof. IKun Chen is currently a Senior Lecturer at the Department of Applied Analysis and Complex Dynamical Systems, Kyoto University, http://www.acs.i.kyotou.ac.jp/en.html . He is visiting UBC between Feb 822, 2017.
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UBC Math

Tue 28 Feb 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Optimal Mass transport as a natural extension of classical mechanics to the manifold of probability measures

ESB 2012
Tue 28 Feb 2017, 3:30pm4:30pm
Abstract
I will describe how deterministic and stochastic dynamic optimal mass transports are to Mean Field Games what the classical calculus of variations offers to classical mechanics.
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University of Florida

Tue 7 Mar 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Local mass concentration and a priori estimate for singular rank 2 Toda systems

ESB 2012
Tue 7 Mar 2017, 3:30pm4:30pm
Abstract
A Toda system is a nonlinear second order elliptic system with exponential nonlinearity. It is very commonly observed in physics and has many ties with algebraic geometry. From analytic viewpoints it is challenging since the solutions do not have symmetry, maximum principles cannot be applied and the structures of global solutions are incredibly complicated. In this joint work with Changshou Lin, Juncheng Wei and Wen Yang, we use a unified approach to discuss all rank two singular Toda systems. First for local systems we prove that all weak limits of mass concentration belong to a very small finite set. Then for systems defined on compact Riemann surface we establish some new estimates. Our approach is a combination of delicate blowup analysis and fundamental tools from algebraic geometry.
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UBC Math

Tue 14 Mar 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Optimal Mass transport as a natural extension of classical mechanics to the manifold of probability measures II

ESB 2012
Tue 14 Mar 2017, 3:30pm4:30pm
Abstract
This is part II of the February 28 talk. Original abstract: I will describe how deterministic and stochastic dynamic optimal mass transports are to Mean Field Games what the classical calculus of variations offers to classical mechanics.
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UBC Math

Tue 21 Mar 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

On De Giorgi's Conjecture for AllenCahn and Free Boundary Problems

ESB 2012
Tue 21 Mar 2017, 3:30pm4:30pm
Abstract
I will report recent progress in De Giorgi's type conjectures (and beyond) for AllenCahn equation and some free boundary problems in the whole space.
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Columbia University

Tue 4 Apr 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (PIMS)

Regularity of the Gauss curvature flow

ESB 2012 (PIMS)
Tue 4 Apr 2017, 3:30pm4:30pm
Abstract
We will discuss about the regularity of the Gauss curvature flow: the optimal C^{1,\frac{1}{n1}} regularity of degenerate solutions with flat sides and the interior C^{\infty} regularity of strictly convex solutions.
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U.C. Santa Barbara

Tue 11 Apr 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Minmax minimal hypersurfaces with free boundary

ESB 2012
Tue 11 Apr 2017, 3:30pm4:30pm
Abstract
I will present a joint work with Martin Li. Minimal surfaces with free boundary are natural critical points of the area functional in compact smooth manifolds with boundary. In this talk, I will describe a general existence theory for minimal surfaces with free boundary. In particular, I will show the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. The minimal surfaces with free boundary were constructed using the minmax method. I will explain the basic ideas behind the minmax theory as well as our new contributions.
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Université de Montpellier, France

Fri 12 May 2017, 1:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB 4127 (PIMS videoconferencing room)

Prescribing the curvature of hyperbolic convex bodies

ESB 4127 (PIMS videoconferencing room)
Fri 12 May 2017, 1:00pm2:00pm
Abstract
The Gauss curvature of a convex body can be seen as a measure on the unit sphere (with some properties). For such a measure \mu , Alexandrov problem consists in proving the existence of a convex body whose curvature measure is \mu . In the Euclidean space, this problem is equivalent to an optimal transport problem on the sphere.
In this talk I will consider Alexandrov problem for convex bodies of the hyperbolic space. After defining the curvature measure, I will explain how to relate this problem to a non linear Kantorovich problem on the sphere and how to solve it.
Joint work with J\’er\^ome Bertrand.
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University of Washington

Mon 5 Jun 2017, 11:00am
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
MATH 126

Asymptotic behavior of solutions to Hessian equations over exterior domains

MATH 126
Mon 5 Jun 2017, 11:00am12:00pm
Abstract
We present a unified approach to quadratic asymptote of solutions to a class of fully nonlinear elliptic equations over exterior domains, including MongeAmpere equations (previously known), special Lagrangian equations, quadratic Hessian equations, and inverse harmonic Hessian equations. This is joint work with Dongsheng Li and Zhisu Li.
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Tokyo Institute of Technology

Tue 19 Sep 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

On uniqueness for the harmonic map heat flow in supercritical dimensions

ESB 2012
Tue 19 Sep 2017, 3:30pm4:30pm
Abstract
We examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension. It is shown that, generically, singular data can give rise to two distinct solutions which are both stable, and satisfy the local energy inequality. We also discuss how uniqueness can be retrieved. This is a joint work with Pierre Germain and TejEddine Ghoul.
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University of Tennessee, Knoxville

Tue 3 Oct 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

(this talk is cancelled)

ESB 2012
Tue 3 Oct 2017, 3:30pm4:30pm
Abstract
Please note this talk is cancelled.
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North China Electric University

Tue 10 Oct 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Nondegeneracy, Morse Index and Orbital Stability of KPI Lump Solution

ESB 2012
Tue 10 Oct 2017, 3:30pm4:30pm
Abstract
We prove that the lump solution of the classical KPI equation is nondegenerate and its Morex index is one. As a consequence, it is orbital stable.
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Yonsei University

Tue 17 Oct 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Global wellposedness and asymptotics of a type of KellerSegel models coupled to fluid flow

ESB 2012
Tue 17 Oct 2017, 3:30pm4:30pm
Abstract
We study chemotaxis equations coupled to the NavierStokes equations, which is a mathematical model describing the dynamics of oxygen, swimming bacteria (Bacillus subtilis) living in viscous incompressible fluids. It is, in general, not known if regular solutions with sufficiently smooth initial data exist globally in time or develop a singularity in a finite time. We discuss existence of regular solutions and asymptotics as well as temporal decays of solutions, under a certain type of conditions of parameters (chemotatic sensitivity and consumption rate) or initial data, as time tends to infinity.
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McGill

Tue 24 Oct 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

An isometric embedding problem and related geometric inequalities

ESB 2012
Tue 24 Oct 2017, 3:30pm4:30pm
Abstract
Solutions to the classical Weyl problem by Nirenberg and Pogorelov
play fundamental role in the notion of quasi local masses and positive quasi
local mass theorems in general relativity. An interesting question in
differential geometry is whether one can isometrically embed compact surfaces
with positive Gauss curvature to a general 3 dimensional ambient space. Of
particular importance is the anti de Sitter Schwarzchild space in general
relativity. We discuss some recent progress in this direction, the a priori
estimates for embedded surfaces in a joint work with Lu, the openness and
nonrigidity results of Li Wang, and a new quasi local type inequality of
LuMiao. We will also discuss open problem related to isometric embeddings to
ambient spaces with horizons.
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UBC

Tue 7 Nov 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Optimal Stopping with a Probabilistic Constraint

ESB 2012
Tue 7 Nov 2017, 3:30pm4:30pm
Abstract
Optimal stopping problems can be viewed as a problem to calculate the space and time dependent value function, which solves a nonlinear, possible nonsmooth and degenerate, parabolic PDE known as an HamiltonJacobiBellman (HJB) equation. These equations are well understood using the theory of viscosity solutions, and the optimal stopping policy can be retrieved when there is some regularity and nondegeneracy of solution.
The HJB equation is commonly derived from a dynamic programming principle (DPP). After adding a probabilistic constraint, the optimal policies no longer satisfy this DPP. Instead, we can reach the HJB equation by a method related to optimal transportation, and recover a DPP for a Lagrangianrelaxation of the problem. The resulting HJB equation remains coupled through the constraint with the optimal policy (and another parabolic PDE). Solving the HJB and recovery of the optimal stopping policy is aided by considering the ``piecewisemonotonic’' structure of the stopping set.
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UBC & Univ. Basel

Tue 14 Nov 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Conformal metrics on \mathbb{R}^n with arbitrary total Qcurvature

ESB 2012
Tue 14 Nov 2017, 3:30pm4:30pm
Abstract
I will talk about the existence of solution to the Qcurvature problem
\begin{align}\label{1}
(\Delta)^\frac n2 u=Qe^{nu}\quad\text{in }\mathbb{R}^n,\quad \kappa:=\int_{\mathbb{R}^n}Qe^{nu}dx<\infty,
\end{align}
where Q is a nonnegative function and n>2. Geometrically, if u is a solution to \eqref{1} then Q is the Qcurvature of the conformal metric g_u = e^{2u}dx^2 (dx^2 is the Euclidean metric on \mathbb{R}^n), and \kappa is the total Qcurvature of g_u.
Under certain assumptions on Q around origin and at infinity, we prove the existence of solution to \eqref{1} for every \kappa > 0.
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University of Chile

Tue 21 Nov 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Asymptotic stability for some nonlinear KleinGordon equations for odd perturbations in the energy space

ESB 2012
Tue 21 Nov 2017, 3:30pm4:30pm
Abstract
Showing asymptotic stability in one dimensional nonlinear KleinGordon equations is a notoriously difficult problem. In this talk I will describe an approach based on virial estimates which allows to prove it in case when only odd perturbations are allowed. In particular I will discuss asymptotic stability of the kink in the \phi^{^4} model.
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Courant Institute, NYU

Thu 30 Nov 2017, 12:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB 4127 (PIMS Videoconference room)

MeanField Limits for GinzburgLandau vortices

ESB 4127 (PIMS Videoconference room)
Thu 30 Nov 2017, 12:00pm1:00pm
Abstract
GinzburgLandau type equations are models for superconductivity, superfluidity, BoseEinstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complexvalued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic GinzburgLandau equation or the GrossPitaevskii (=Schrodinger GinzburgLandau) equation.
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University of Toronto

Thu 4 Jan 2018, 3:30pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (PIMS)

Free Discontinuities in Optimal Transport

ESB 2012 (PIMS)
Thu 4 Jan 2018, 3:30pm5:00pm
Abstract
Optimal maps in R^n to disconnected targets necessarily contain discontinuities (i.e.~tears). But how smooth are these tears? When the target components are suitably separated by hyperplanes, nonsmooth versions of the implicit function theorem can be developed which show the tears are hypersurfaces given as differences of convex functions  DC for short. If in addition the targets are convex the tears are actually C^{1,\alpha}. Similarly, under suitable affine independence assumptions, singularities of multiplicity k lie on DC rectifiable submanifolds of dimension n+1k. These are stable with respect to W_\infty perturbations of the target measure. Moreover, there is at most one singularity of multiplicity n. This represents joint work with Jun Kitagawa.
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UCDavis

Tue 9 Jan 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Variational Problems on Arbitrary Sets

ESB 2012
Tue 9 Jan 2018, 3:30pm4:30pm
Abstract
Let E be an arbitrary subset of R^n. Given real valued functions f defined on E and g defined on R^n, the classical obstacle problem asks for a minimizer of the Dirichlet energy subject to the following two constraints: (1) F = f on E and (2) F lies above g on R^n. In this talk, we will discuss how to use extension theory to construct (almost) solutions directly. We will also explain several recent results that will help lay the foundation for building a complete theory revolving around the belief that any variational problems that can be solved using PDE theory can also be dealt with using extension theory.
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UBC & PIMS

Tue 16 Jan 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

The space of minmax hypersurfaces

ESB 2012
Tue 16 Jan 2018, 3:30pm4:30pm
Abstract
We use LusternikSchnirelmann Theory to study the topology of the space of closed embedded minimal hypersurfaces on a manifold of dimension between 3 and 7 and positive Ricci curvature. Combined with the works of MarquesNeves we can also obtain some information on the geometry of the minimal hypersurfaces they found.
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Indiana University

Tue 6 Feb 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

The MongeAmpere eigenvalue problem, BrunnMinkowski inequality and global smoothness of the eigenfunctions

ESB 2012
Tue 6 Feb 2018, 3:30pm4:30pm
Abstract
In this talk, I will first introduce the MongeAmpere eigenvalue problem on general bounded convex domains and related analysis including the BrunnMinkowski inequality for the eigenvalue. Then I will discuss the recent resolution, in joint work with Ovidiu Savin, of global smoothness of the eigenfunctions of the MongeAmpere operator on smooth, bounded and uniformly convex domains in all dimensions. A key ingredient in our analysis is boundary Schauder estimates for certain degenerate MongeAmpere equations.
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McGill

Tue 6 Mar 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Blowingup solutions for critical elliptic equations on a closed manifold

ESB 2012
Tue 6 Mar 2018, 3:30pm4:30pm
Abstract
In this talk, we will look at the question of existence of blowingup solutions for smooth perturbations of energycritical elliptic nonlinear Schrödinger equations on a closed manifold. From a result of Olivier Druet, we know that in dimensions different from 3 and 6, a necessary condition for the existence of blowingup solutions with bounded energy is that the linear part of the limit equation agrees with the conformal Laplacian at least at one blowup point. I will present new existence results in situations where the limit equation is different from the Yamabe equation away from the blowup point. I will also discuss the special role played by the dimension 6. This is a joint work with Frederic Robert.
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University of California, Davis

Tue 20 Mar 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Stability of the superselection sectors of Kitaev’s abelian quantum double models

ESB 2012
Tue 20 Mar 2018, 3:30pm4:30pm
Abstract
Kitaev’s quantum double models provide a rich class of examples of twodimensional lattice models with topological order in the ground states and a spectrum described by anyonic elementary excitations. The infinite volume ground states of the abelian quantum double models come in a number of equivalence classes called superselection sectors. We prove that the superselection structure remains unchanged under uniformly small perturbations of the quantum double Hamiltonians. (joint work with Matthew Cha and Pieter Naaijkens)
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Arizona State University

Tue 27 Mar 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons

ESB 2012
Tue 27 Mar 2018, 3:30pm4:30pm
Abstract
I will discuss some recent joint work with Lu Wang in which we prove that a shrinking gradient Ricci soliton which agrees to infinite order at spatial infinity with a generalized cylinder along some end must be isometric to the cylinder on that end. When the shrinker is complete, it must be globally isometric to the cylinder or else to a Z_2quotient. This work belongs to a larger program aimed at obtaining a structural classification of complete noncompact shrinking solitons.
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Eberhard Karls University, Tuebingen

Tue 3 Apr 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Edge universality in interacting topological insulators

ESB 2012
Tue 3 Apr 2018, 3:30pm4:30pm
Abstract
In this talk, I will present universality results for the edge transport properties of interacting, 2d topological insulators. I will mostly focus on the case of quantum Hall systems, displaying single mode edge currents. After reviewing recent results for the bulk transport properties, I will present a theorem establishing the universality of the edge conductance and the emergence of spincharge separation for the edge modes. Combined with wellknown results for noninteracting systems, our theorem implies the validity of the bulkedge correspondence for a class of weakly interacting 2d lattice models, including for instance the interacting Haldane model. The proof is based on rigorous renormalization group methods, and on the combination of chiral Ward identities for the effective 1d QFT describing the infrared scaling limit of the edge currents, together with lattice Ward identities for the original lattice model. Joint work with G. Antinucci (UZH/Tuebingen) and with V. Mastropietro (Milan).
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Univ. Padova

Tue 10 Apr 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
MATH 225 **Special**

News on the MoserTrudinger inequality: from sharp estimates to the LeraySchauder degree

MATH 225 **Special**
Tue 10 Apr 2018, 3:30pm4:30pm
Abstract
The existence of critical points for the MoserTrudinger inequality for large energies has been open for a long time. We will first show how a collaboration with G. Mancini allows to recast the MoserTrudinger inequality and the existence of its extremals (originally due to L. Carleson and A. Chang) under a new light, based on sharp energy estimate. Building upon a recent subtle work of O. Druet and PD. Thizy, in a work in progress with O. Druet, A. Malchiodi and PD. Thizy, we use these estimates to compute the LeraySchauder degree of the MoserTrudinger equation (via a suitable use of the PoincaréHopf theorem), hence proving that for any bounded nonsimply connected domain the MoserTrudinger inequality admits critical points of arbitrarily high energy.
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University of Toronto

Tue 17 Apr 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012

Geometric Inequalities on Riemannian manifolds

ESB 2012
Tue 17 Apr 2018, 3:30pm4:30pm
Abstract
I will discuss some upper bounds for the length of a shortest periodic geodesic, and the smallest area of a closed minimal surface on closed Riemannian manifolds of dimension 4 with Ricci curvature between 1 and 1. These are the first bounds that use information about the Ricci curvature rather than sectional curvature of the manifold. (Joint with Nan Wu).
I will also give examples of Riemannian metrics on the 3disk demonstrating that the maximal area of 2spheres arising during an ``optimal" homotopy contracting its boundary cannot be majorized in terms of the volume and diameter of the 3disc and the area of its boundary. This contrasts with earlier 2dimensional results of Y. Liokomovich, A. Nabutovsky and R. Rotman and answers a question of P. Papasoglu. On the other hand I will show that such an upper bound exists if, instead of the volume, one is allowed to use the first homological filing function of the 3disc. (Joint with Parker GlynnAdey).
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UBC

Tue 4 Sep 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
MATH 105 [The NEW LOCATION!]

The Dyson Game (joint work with R. Carmona and M. Cerenzia)

MATH 105 [The NEW LOCATION!]
Tue 4 Sep 2018, 3:30pm4:30pm
Abstract
The Dyson game is an explicitly solvable N player dynamic game that admits Dyson Brownian motion as a Nash equilibrium. The game is motivated by the real world phenomenon found in the spacing of buses, parked cars and perched birds, which exhibit random matrix statistics (i.e. Dyson Brownian motion). We find the optimal repulsion parameter (universality class) of the equilibrium depends on the information available to the players, furthering the understanding of an open problem in random matrix theory proposed by Deift. The limiting mean field game has a local cost term, which depends on the optimal universality class due to the nontrivial asymptotic behavior of the players. We solve the mean field game master equation and the associated HamiltonJacobi equation on Wasserstein space exactly, and we discuss how generalizing our results will require answering novel questions on the analysis of these equations on infinite dimensional spaces.
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Utah State University

Tue 11 Sep 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
MATH 105

Superposition Formulas for Differential Systems and Darboux Integrability

MATH 105
Tue 11 Sep 2018, 3:30pm4:30pm
Abstract
The quintessential example of a Darboux integrable differential equation is the Liouville equation
u_{xy} = 2 {\rm e}^u,
whose general solution is given by
u= \log \frac{ f'(x) g'(y)}{(f(x)g(y))^2}.
Daboux integrability is classically related to the existence of intermediate integrals (or Riemann invariants) which in turn allow an explicit closed form formula to be derived for these equations.
Motivated by work of E. Vessiot, I will describe a differential geometric construction which provides a fundamental description of Darboux integrable systems in terms of superposition of differential systems and the quotient theory of differential systems by Lie groups.
The general theory will be discussed (I won't assume familiarity with differential systems), and demonstrated with examples. If time permits some interesting properties of these systems will be shown based on the existence of the quotient representation of Darboux integrable systems.
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Pusan National University and UBC

Tue 2 Oct 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
MATH 105

Solitons for the mean curvature flow and inverse mean curvature flow

MATH 105
Tue 2 Oct 2018, 3:30pm4:30pm
Abstract
Selfsimilar solutions and translating solitons are not only special solutions of mean curvature flow (MCF) but a key role in the study of singularities of MCF. They have received a lot of attention. We introduce some examples of selfsimilar solutions and translating solitons for the mean curvature flow (MCF) and give rigidity results of some of them. We also investigate selfsimilar solutions and translating solitons to the inverse mean curvature flow (IMCF) in Euclidean space.
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MIT

Tue 9 Oct 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
MATH 105

Minimal surfaces and the AllenCahn equation on 3 manifolds

MATH 105
Tue 9 Oct 2018, 3:30pm4:30pm
Abstract
The AllenCahn equation is a semilinear PDE that produces minimal surfaces via a certain singular limit. We will describe recent work proving index, multiplicity, and curvature estimates in the context of an AllenCahn minmax construction in a 3manifold. Our results imply, for example, that in a 3manifold with a generic metric, for every positive integer p, there is an embedded twosided minimal surface of Morse index p. This is joint with Otis Chodosh.
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UBC

Tue 23 Oct 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
MATH 105

Global NavierStokes flows for nondecaying initial data with slowly decaying oscillation

MATH 105
Tue 23 Oct 2018, 3:30pm4:30pm
Abstract
We consider the Cauchy problem of 3D incompressible NavierStokes equations for uniformly locally square integrable initial data. The existence of a timeglobal weak solution has been known, when the square integral of the initial datum on a ball vanishes as the ball goes to infinity. For nondecaying data, however, the only known global solutions are either for perturbations of constants or when the velocity gradients are in Lp with finite p. In this talk, I will outline how to construct global weak solutions for general nondecaying initial data whose local oscillations slowly decay.
This is a joint work with TaiPeng Tsai.
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Note for Attendees
Refreshments will be served in the PIMS Lounge from 3:303:45 p.m.