Mathematics Dept.
  Events
Micah Warren
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, 2:00pm-3:00pm

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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Walid Abou-Salem
UBC
Wed 16 Sep 2009, 4:00pm
Diff. Geom, Math. Phys., PDE Seminar
MATH 103
Dimensional reduction of the mean-field dynamics of bosons in strongly anisotropic harmonic potentials
MATH 103
Wed 16 Sep 2009, 4:00pm-5:00pm

Abstract

I discuss recent results on the spatial dimensional reduction of the effective mean-field dynamics of many-body 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 mean-field dynamics of many-body constraint quantum systems.
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Alexandre Munnier
UBC and Nancy 1
Tue 22 Sep 2009, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110
Controllability results for a fish-like swimming body
WMAX 110
Tue 22 Sep 2009, 3:30pm-4:30pm

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 non-standard 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, 3:30pm-4:30am

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 one-dimensional Schroedinger operator H_\gamma = -d^2/ds^2 + \kappa^2(s) acting on the space of square integrable 2\pi-periodic 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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Elton Hsu
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, 3:30pm-4:30pm

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 (pde-ers), 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 Moser-Onofri-Aubin inequality
WMAX 110
Tue 13 Oct 2009, 3:30pm-4:30pm

Abstract


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Ramon Zarate
UBC
Tue 27 Oct 2009, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
Inverse problems via variational methods
Tue 27 Oct 2009, 3:30pm-4:30pm

Abstract

We present a general variational method, involving self-dual variational calculus, for recovering non-linearities from prescribed solutions for certain types of PDEs which are not necessarily of Euler-Lagrange type, including parabolic equations. The approach can also be used for optimal control problems. The topological aspects involved, for the space of self-dual 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, 3:00pm-4:00pm

Abstract

I will discuss the scattering problem of mass-critical generalized KdV equation. We will see if the scattering of gKdV fails, then a minimal mass blow-up solution exist on the condition that scattering of mass-critical 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, 2:30pm-3:30pm

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 Ricci-bounded-below 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, 3:30pm-4:30pm

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 Max-K. von Renesse ('09).
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Reinhard Illner
U. of Victoria
Thu 12 Nov 2009, 3:00pm SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS mini-symposium in PDE); time changed
Traffic Flow and Traffic Jams: From Kinetic Theory to Functional Differential Equations
WMAX 110 (PIMS mini-symposium in PDE); time changed
Thu 12 Nov 2009, 3:00pm-4:00pm

Abstract

I will speak on certain kinetic and macroscopic models of traffic flow. After a review of the concept of a fundamental diagram the high-density 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 mini-symposium in PDE
Fourth order diffusion with geometric link to second order diffusion
WMAX 110: PIMS mini-symposium in PDE
Fri 13 Nov 2009, 1:00pm-2:00pm

Abstract

We describe a fourth order family generalizing the linear-mobility thin film equation on R^n.  In joint work with R. McCann we derive formally sharp converence rates to self-similarity, using a link to Denzler-McCann'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 fourth-order 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 mini-symposium in PDE
Dynamics of Kinematic Aggregation Patterns
MATX 1100: PIMS mini-symposium in PDE
Fri 13 Nov 2009, 2:00pm-3:00pm

Abstract


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Craig Cowan
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, 3:30pm-4:30pm

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)
Non-negatively cross-curved transportation costs
WMAX 110 (at PIMS)
Tue 24 Nov 2009, 3:30pm-4:30pm

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 Ma-Trudinger- 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, 3:30pm-4:30pm

Abstract

Consider a nonlinear Schr\"{o}dinger equation in $\mathbb{R}^3$ with a short-range 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 Tai-Peng Tsai.
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Stephen Gustafson
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, 3:30pm-4:30pm

Abstract

I will describe some background and recent results on singularity formation (and non-formation) for some simple, physical, and popular geometric PDE describing dynamics of maps into spheres -- the heat-flow, wave map, and Schroedinger map -- in the energy-critical 2D case. I'll try to keep it simple and accessible by illustrating the methods on a symmetric reduction of the heat-flow, leading to a single scalar PDE. 
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Leo Tzou
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, 3:30pm-4:30am

Abstract

We show that on a smooth compact Riemann surface with boundary (M_0, g) the Dirichlet-to-Neumann 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. DMS-0807502 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, 3:30pm-4:30am

Abstract

Traveling fronts are special solutions of reaction-diffusion 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
Well-posedness of stochastic PDEs
WMAX 110
Thu 4 Feb 2010, 3:30pm-4:30pm

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 space-time white noise by estimating the stochastic convolution via Ito calculus and showing its continuity via the factorization method. Next, we discuss
well-posedness of SKdV with multiplicative noise in $L^2$. In order to treat the non-zero 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 KdV-Burgers equation (SKdVB). We discuss how Fourier analytic technique can be applied to show LWP. If time permits, we discuss how one can obtain global well-posedness 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, 3:30pm-4:30pm

Abstract

 
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Dong Li
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, 3:30pm-4:30pm

Abstract

I will explain recent joint work with Xiaoyi Zhang on threshold solutions to critical nonlinear Schrodinger equations. These results are analogues of Liouville-type theorems in the dispersive setting. I will cover mainly the mass-critical case. Time permitting the energy-critical 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 Dirichlet-to-Neumann map, conformal geometry, and minimal surfaces
WMAX110
Tue 9 Mar 2010, 3:30pm-4:30pm

Abstract


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Nam Le
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, 3:30pm-4:30pm

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 Choi-Schoen estimate for minimal submanifolds. This is joint work with Natasa Sesum.
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Craig Cowan
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, 3:30pm-4:30am

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, 3:30pm-4:30pm

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, 3:30pm-4:30pm

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 width-sweepout construction of closed geodesics on closed Riemannian manifold to the Alexandrov setting, which provides a generalized version of the Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics. We will explain how the width-sweepout 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
Symmetry-breaking bifurcation in the Gross-Pitaevskii equation with a double-well potential
WMAX110
Tue 27 Apr 2010, 3:30pm-4:30pm

Abstract

We classify bifurcations of the asymmetric states from a family of symmetric states in the focusing (attractive) Gross-Pitaevskii equation with a symmetric double-well 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 non-degenerate 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 Paris-Dauphine
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, 3:00pm-4:00pm

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 1-rectifiable sets and the energy is the integral of a power $\theta^\alpha$ of their multiplicity $\theta$. I'll present an approximation by Gamma-convergence, 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 Modica-Mortola for the perimeter functional, where a double-well 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 double-well is replaced with a concave power, so that there is a sort of double-well 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, 10:30am-11:30am

Abstract

Usually, a nonlinear functional involving the Sobolev exponent does not satisfy the Palais-Smale 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 Kahler-Ricci flow
MATH 203 (<--- Room changed)
Mon 13 Sep 2010, 4:00pm-5:00pm

Abstract

We give a criterion under which a solution g(t) of the Kahler-Ricci 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 Monge-ampere equation on compact Kahler manifolds
MATX 1100
Mon 20 Sep 2010, 4:00pm-5:00pm

Abstract

We consider the complex Monge-Amp\`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 Monge-Amp\`ere equation when $F$ is in $W^{1, p_0}$.
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Ian Zwiers
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, 3:30pm-4:30pm

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 'log-log' rate. We prove that in the case of spin 1, there is comparable data that blows up with the vortex profile and the log-log rate. The case of spin 2 and 3 will be discussed. This is joint work with Gideon Simpson (Toronto)
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Tsuyoshi Yoneda
University of Victoria
Tue 5 Oct 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110
Ill-posedness of the 3D-Navier-Stokes equation and related topics
WMAX 110
Tue 5 Oct 2010, 3:30pm-4:30pm

Abstract


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Tai-Peng Tsai
UBC
Tue 19 Oct 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110
Asymptotics of small exterior Navier-Stokes flows with non-decaying boundary data
WMAX 110
Tue 19 Oct 2010, 3:30pm-4:30am

Abstract

 We prove the unique existence of solutions of the 3D incompressible Navier-Stokes equations in an exterior domain with small non-decaying 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 time-periodic, the spatial asymptotics of the time-entire solution is given by a Landau solution which is the same for all time. If the boundary data is time-periodic and the initial data is asymptotically discretely self-similar, the solution is asymptotically the sum of a time-periodic vector field and a forward discretely self-similar 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, 4:00pm-5:00pm

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 Karcher-Pinkall-Sterling, and by Kapouleas-Yang. (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, 3:30pm-4:30am

Abstract

We consider the situation in which a weak solution of the Navier-Stokes 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 Navier-Stokes 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, 3:30pm-4:30pm

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}chet-differentiable.
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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, 3:30pm-4:30pm

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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Vianney Combet
UBC
Tue 23 Nov 2010, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110
Multi-soliton solutions for the supercritical gKdV equations
WMAX 110
Tue 23 Nov 2010, 3:30pm-4:30pm

Abstract

We consider the problem of existence and uniqueness of multi-soliton solutions for the L²-supercritical generalized Korteweg-de Vries equation. We recall that a multi-soliton 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 1-soliton case, we will state the theorem of existence and uniqueness of an N-parameter family of N-solitons 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, 3:30pm-4:30pm

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, 3:30pm-4:30pm

Abstract

We will prove that solutions to the defocusing energy-critical Schrodinger equations are global in the hyperbolic space H^3. The relevance of the energy-critical 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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Dong Li
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, 3:30pm-4:30pm

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) space-time 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, 3:30pm-4:30pm

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, 3:30pm-4:30pm

Abstract


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University of Alabama at Birmingham
Thu 10 Feb 2011, 3:30pm SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
PIMS
Zero-velocity Lieb-Robinson bounds in the disordered xy-spin chain
PIMS
Thu 10 Feb 2011, 3:30pm-4:30pm

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 time-dependent 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 xy-spin chain to show on the physical level of rigor that the introduction of disorder will lead to zero-velocity Lieb-Robinson 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 two-valued minimal surface equation.
WMAX 110 (Note the time change to 3:00)
Tue 1 Mar 2011, 3:00pm-4:00pm

Abstract

We explore the question of whether there are nontrivial solutions to the two-valued minimal surface (2MSE) equation defined over the punctured plane. The 2MSE is a non-uniformly 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, 4:00pm-5:00pm

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 spectral-theoretic 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, 1:00pm-2:00pm

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: jump-diffusion 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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Jeff Viaclovsky
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, 11:00am-12:00pm

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 blow-up problem for the Euler equations and the Liouville type results for the fluid equations
WMAX110 (PIMS) --note the room change
Mon 25 Jul 2011, 2:00pm-3:00pm

Abstract

In the first part of the talk we discuss some new observations on the blow-up problem in the 3D Euler equations. We consider the scenarios of  the self-similar blow-ups and the axisymmetric blow-up. For the self-similar blow-up we prove a Liouville type theorem for the self-similar 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 Navier-Stokes 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 Aharonov-Bohm effect and the Calderon problem for connection Laplacians
WMAX 110 (PIMS) (Schedule and location subject to change)
Thu 8 Sep 2011, 3:30pm-4:30pm

Abstract

The Aharonov-Bohm 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.DMS-0807502.
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UBC
Tue 13 Sep 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)
A Self-dual Polar Factorization for Vector Fields
WMAX 110 (PIMS)
Tue 13 Sep 2011, 3:30pm-4:30pm

Abstract

We show that any non-degenerate 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 anti-symmetric convex-concave 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 self-dual 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 self-dual mass transport problem.
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Lu Li
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, 3:30pm-4:30pm

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 half-space 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 half-space 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 lower-order terms with bounded measurable coefficients.
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U. Paris-Sud, 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, 4:00pm-5:00pm

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. Paris-Sud, 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, 3:30pm-5:00pm

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, 3:30pm-4:30pm

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, 3:30pm-4:30pm

Abstract

In this talk, we will discuss about Li-Yau-Hamilton 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 Vlasov-Darwin system
WMAX 110 (PIMS)
Tue 18 Oct 2011, 3:30pm-4:30pm

Abstract

The relativistic Vlasov-Darwin (RVD) system is a kinetic model that describes the evolution of a collisionless plasma whose particles interact through the self-induced electromagnetic field. In contrast with the Vlasov-Maxwell system, the particle interaction is assumed to be a low-order 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. 68-79 ] to obtain uniqueness results for the Vlasov-Poisson 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, 3:30pm-4:30pm

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 Lane-Emden system with Hénon weights. 
 
References : B.-Ramos ANIHP 2009 - B.-dos Santos JDE 2010 - B.-Ramos-dos 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)
Liouville-type theorems for some elliptic equations and systems
PIMS (WMAX 110) (Notice the date change)
Tue 1 Nov 2011, 3:30pm-4:30pm

Abstract

In this talk, we consider the problem of non-existence of solutions for some basic elliptic equations and systems with weights. Starting with Henon-Lane-Emden system, we present a Liouville-type theorem for bounded solutions in dimension N=3 as well as the full Henon-Lane-Emden 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, 3:30pm-4:30pm

Abstract

For any smooth compact Riemannian 3-manifold 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 3-manifold. An interesting note is that no convexity assumption on the boundary is required. In this talk,  we will describe the min-max 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, 3:30pm-4:30pm

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, 2:00pm-3:00pm

Abstract

We prove that solutions to a fully nonlinear elliptic equation F(D^2u)=0 are classical outside a set of dimension at most n-epsilon, 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) (PDE-Math Biology Joint seminar)
Asymptotic Limit in a Cell Differentiation Model
WMAX 110 (PIMS) (PDE-Math Biology Joint seminar)
Tue 14 Feb 2012, 3:30pm-4:30pm

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 T-bet (x_1) and GATA-3 (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 T-cell 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, 2:00pm-3:00pm

Abstract

In this talk, I will give a survey on Harnack inequalities for solutions of second-order 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 Reaction-Diffusion Systems
WMAX 110 (PIMS)
Thu 1 Mar 2012, 3:45pm-4:45pm

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.)

Note for Attendees

Refreshments will be served in the PIMS Lounge from 3:30-3:45 p.m.
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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, 3:30pm-4:30pm

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, 2:00pm-3:00pm

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 time-irregular 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, 2:00pm-3:00pm

Abstract

We investigate a one-dimensional model describing the motion of liquid drops sliding down an inclined plane (the so-called quasi-static 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 non-constant 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 Multi-Solitons for a Nonlinear Schrödinger Equation
WMAX 110
Tue 3 Apr 2012, 3:30pm-4:30pm

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, 3:00pm-4:00pm

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, 3:00pm-4:00pm

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, many-to-one 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)
Blow-up of critical Besov norms at a Navier-Stokes singularity
ESB 2012 (in the new PIMS building)
Thu 6 Sep 2012, 3:30pm-4:30pm

Abstract

 In this talk we describe a generalization of the result of Escauriaza-Seregin-Sverak on blow-up of the L^3 norm at a Navier-Stokes singularity by establishing the blow-up 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 "dispersive-type" 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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Emil Wiedemann
UBC
Tue 18 Sep 2012, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)
Non-Uniqueness Phenomena for the 3D Euler Equations
ESB 2012 (in the new PIMS building)
Tue 18 Sep 2012, 3:30pm-4:30pm

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 non-unique weak solutions. Recently, De Lellis and Szekelyhidi demonstrated that this phenomenon can be viewed as an instance of the so-called h-principle, thereby providing a shorter and more general proof of the non-uniqueness. In this talk I will briefly review their method and then present some subsequent results, including global existence and non-uniqueness for 3D Euler, the approximation of measure-valued solutions by weak ones, and non-uniqueness for shear flow initial data.
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Gonzalo Dávila
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, 3:30pm-4:30pm

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, 3:30pm-4:30pm

Abstract

It is well known that an upper and lower bound on the Monge-Amp{\`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 Monge-Amp{\`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 Ma-Trudinger-Wang condition. This talk is based on a joint work with Young-Heon 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 Self-Similar Solutions of the Navier-Stokes Equations
ESB 2012 (in the new PIMS building)
Tue 9 Oct 2012, 3:30pm-4:30pm

Abstract

Extending the work of Jia and Sverak on self-similar solutions of the Navier-Stokes equations, we show the existence of large, forward, discrete self-similar 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 tree-strips of finite cone-type.
ESB 2012 (in the new PIMS building) TBA
Tue 16 Oct 2012, 3:30pm-4:30pm

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 infinite-dimensional 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 tree-strips. I will present a proof for the existence of absolutely continuous spectrum for models on tree-strips 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)
Non-differentiability locus of distance functions and Federer's curvature measures
ESB 2012 (in the new PIMS building)
Tue 23 Oct 2012, 3:30pm-4:30pm

Abstract

I will present an upper bound on the (d-1)-volume and covering numbers of a filtration of the non-differentiability 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, 3:30pm-4:30pm

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)
Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutions
ESB 4127 (PIMS video conference room)
Tue 11 Dec 2012, 3:30pm-4:30pm

Abstract

I will discuss uniqueness and Monge solution results for multi-marginal optimal transportation problems with a certain class of cost functions; this class arises naturally in multi-agent matching problems in economics.  This result generalizes a seminal result of Gangbo and \'Swi\c{e}ch on multi-marginal problems. I will also discuss some related observations about multi-marginal 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)
m-Liouville theorems for elliptic PDEs
ESB 2012 (in the new PIMS building)
Tue 8 Jan 2013, 3:30pm-4:30pm

Abstract

De Giorgi in 1978 conjectured that bounded and monotone solutions of the Allen-Cahn equation must be one-dimensional 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 m-dimensional for 0=<m=<N-1. This leads us to define the concept of “m-Liouville theorem” for PDEs. The motivation to this definition is the Liouville theorem (or 0-Liouville theorem) that we have seen in elementary analysis stating that bounded harmonic functions on the whole space must be constant (0-dimensional).


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, 3:30pm-4:30pm

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 multi-marginal mass transport
ESB 2012 (in the new PIMS building)
Tue 22 Jan 2013, 3:30pm-4:30pm

Abstract

 We expose and exploit a surprising relationship between elliptic gradient systems of PDEs and a multi-marginal Monge-Kantorovich optimal transport problem.  We show that the notion of an "orientable" elliptic system (Fazly-Ghoussoub) conjectured to imply that stable solutions are essentially 1-dimensional, is equivalent to the definition of a "compatible" cost function (Carlier-Pass), known to imply uniqueness and structural results for optimal measures to certain Monge-Kantorovich problems.  We use this equivalence to show that solutions to these elliptic PDEs, with appropriate monotonicity properties, are related to optimal measures in the Monge-Kantorovich 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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Annalisa Panati
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, 3:30pm-4:30pm

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, 3:30pm-4:30pm

Abstract

 In this talk, we discuss the Landau damping -- the asymptotic stability of the linearly stable homogeneous states of the Vlasov-Possion 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, 3:30pm-4:30pm

Abstract

 I will begin this talk by recalling the classic inequalities of Alexandrov-Fenchel and Polya-Szego for convex surfaces of 3-dimensional 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 mass-capacity inequality for black holes.
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Remi Schweyer
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 1-corotational energy critical harmonic heat flow
ESB 2012 (in the new PIMS building)
Tue 12 Mar 2013, 3:30pm-4:30pm

Abstract

After a short presentation of the equation of harmonic heat flow and corotational solutions, I am presenting a result of finite time blow-up 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, 3:30pm-4:30pm

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 low-degree 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 space-time 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, 3:30pm-4:30pm

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)
Quasi-static evolution in randomly perforated media
ESB 2012 (in the new PIMS building)
Tue 26 Mar 2013, 3:30pm-4:30pm

Abstract

We consider a quasi-static free boundary problem (the Hele-Shaw 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 Hele-Shaw 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 second-order elliptic and parabolic systems with boundary conditions.
ESB 2012 (in PIMS building)
Tue 23 Apr 2013, 3:30pm-4:30pm

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, 3:30pm-4:30pm

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 divergence-free 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 Ambrosio-Gigli-Savar\'e for the scalar heat equation, we provide a concept of "dissipative solutions" that enforces first the "weak-strong" 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, 3:30pm-4:30pm

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, 8:00am-6:00pm

Abstract

Schedule to be posted in the conference webpage:

http://www.pims.math.ca/scientific-event/ghoussoub


This conference brings together world-renowned 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 state-of-the-art 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, 3:30pm-4:30pm

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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Vitali Vougalter
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 semi-relativistic Schroedinger-Poisson system
MATX 1102
Thu 5 Sep 2013, 1:00pm-2:00pm

Abstract

We study the stationary states of the semi-relativistic Schroedinger-Poisson 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 energy-Casimir method. Moreover, we establish global well-posedness results for the semi-relativistic Schroedinger-Poisson system in appropriate functional spaces.
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Emil Wiedemann
UBC
Tue 17 Sep 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Relaxation of Variational Problems for Orientation-Preserving Maps
ESB 2012
Tue 17 Sep 2013, 3:30pm-4:30pm

Abstract

It is well-known 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 measure-valued. 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 measure-valued 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 orientation-preserving. 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 Monge-Kantorovich problems and polar decompositions of vector fields
ESB 2012
Tue 24 Sep 2013, 3:30pm-4:30am

Abstract

For any given integer N larger than 2, we show that every bounded measurable vector field is N-cyclically monotone up to a measure preserving N-involution. The proof involves the solution of a multidimensional symmetric Monge-Kantorovich 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 N-involutions and those Hamiltonians that are N-cyclically antisymmetric.
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Philippe Castillon
Montpellier / PIMS-UBC
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, 3:30pm-4:30pm

Abstract

Harmonic manifolds are those Riemannian manifolds whose harmonic functions satisfy the mean-value 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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Frédéric Robert
U. Lorraine /PIMS-UBC
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, 3:30pm-4:30pm

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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Xiaofeng Ren
George Washington U.
Tue 15 Oct 2013, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Double bubble and core-shell solutions in an inhibitory ternary system
ESB 2012
Tue 15 Oct 2013, 3:30pm-4:30pm

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 core-shell 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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Jun-Cheng Wei
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, 3:30pm-4:30pm

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 Costa-Hoffman-Meeks 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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Antoine Julien
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, 3:30pm-4:30pm

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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Yannick Sire
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, 3:30pm-4:30pm

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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Dong Li
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, 3:30pm-4:30pm

Abstract

 
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Hassan Jaber
Universite de Lorraine
Tue 21 Jan 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Hardy-Sobolev equations and related inequalities on compact Riemannian manifolds
ESB 2012
Tue 21 Jan 2014, 3:30pm-4:30pm

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(n-s)}{n-2} be the critical Hardy-Sobolev 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^{\crit-1}}{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 Blow-up around x_0, i prove that the sharp constant of the related Hardy-Sobolev inequalities on (M,g), which is equal to the one of the Euclidean Hardy-Sobolev 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, 3:30pm-4:30pm

Abstract

We consider the nonlinear fourth-order problem  \Delta^2 u=|u|^{p-1}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, 3:30pm-4:30pm

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 one-dimensional 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 multi-dimensional 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, 3:30pm-4:30pm

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, 3:30pm-4:30pm

Abstract

 

We show how to lift positive Ricci and almost non-negative curvatures from an orbit space M/G to the corresponding G-manifold, 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 Transport-based Model and Algorithms for Particle Image Velocimetry
ESB2012 (PIMS)
Tue 4 Mar 2014, 3:30pm-4:30pm

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, cross-correlation 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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Philippe Castillon
Montpellier / PIMS-UBC
Tue 11 Mar 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Spectral positivity on surfaces
ESB 2012
Tue 11 Mar 2014, 3:30pm-4:30pm

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 Monge-Ampere equation on Kahler manifolds
ESB 2012 (PIMS building) note the time and location change
Fri 21 Mar 2014, 3:30pm-4:30pm

Abstract

Complex Monge-Ampere (CMA) equation is of fundamental importance in Kahler geometry. We will discuss regularity results for two versions of complex Monge-Ampere 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 Monge-Ampere, which is known as a geodesic equation of the space of Kahler metrics.
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Christian Sadel
UBC
Tue 25 Mar 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Complex analytic, one-frequency cocycles
ESB 2012
Tue 25 Mar 2014, 3:30pm-4:30pm

Abstract

 
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Filip Rindler
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, 3:30pm-4:30pm

Abstract

Microlocal compactness forms (MCFs) are a new tool to study oscillations and concentrations in L^p-bounded 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 H-measures. Since in L^p-spaces 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 weak-to-strong 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 space-time [Joint with P. Isett (MIT) ]
ESB 2012 (PIMS building)
Mon 12 May 2014, 3:30pm-4:30pm

Abstract

In this talk, we will describe a construction of compactly supported solutions to the three-dimensional 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 Lellis-Székelyhidi, Buckmaster-De Lellis-Székelyhidi and Isett on construction of Hölder continuous dissipative Euler flows to the non-periodic setting. Our key technical innovation is a simple method for finding a compactly supported symmetric 2-tensor 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 c-Plateau problem for surfaces in space.
ESB 4127
Tue 13 May 2014, 11:00am-12:00pm

Abstract

The c-Plateau 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 surface-with-boundary, with \partial M_{c} = \sigma_{c}+\gamma where \sigma_{c} is a finite union of closed curves disjoint from \gamma, minimizing c-isoperimetric mass \mathbf{M}^{c}(M) := \text{area}(M)+c \cdot \text{length}(\partial M)^{2} amongst all M smooth orientable surfaces-with-boundary, 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 c-Plateau problem, formulated in the more general setting of integer two-rectifiable 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 Fermi-gas ground state
ESB 4127
Tue 13 May 2014, 3:30pm-4:30pm

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 bi-partite von-Neumann entanglement entropy of non-interacting fermions in multi-dimensional Euclidean space at zero temperature. Based on recent progress by one of us (A.V.S.) in semi-classical functional calculus for pseudo-differential 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'Aix-Marseille
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, 3:30pm-4:30pm

Abstract

The talk will be concerned with various optimization results for the principal eigenvalues of general second-order 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 Rayleigh-Faber-Krahn 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, 3:30pm-4:30pm

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 Monge-Amp\`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, 3:00pm-4:00pm

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.




Note for Attendees

[Notice the time and location change!]
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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, 3:30pm-4:30pm

Abstract

 
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John Ma
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, 3:30pm-4:30pm

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 self-shrinker (a local model for singularity), how it is formed in LMCF, and give some examples of Lagrangian self-shrinkers. Then I will discuss a recent work with Jingyi Chen concerning the space of all compact Lagrangian self-shrinkers in \mathbb C^2.
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Baptiste Devyver
UBC
Tue 23 Sep 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
General Hardy-type inequalities on manifolds.
ESB 2012
Tue 23 Sep 2014, 3:30pm-4:30pm

Abstract

Given a general second-order, 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, 2:30pm-3:30pm

Abstract

This talk concerns the geometry of shrinking Ricci solitons, a class of self-similar 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.

Note for Attendees

 Please note unusual time and room.
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Universite de Cergy-Pontoise
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, 3:30pm-4:30pm

Abstract

 We investigate various issues for stationary Kirchhoff systems in closed manifolds, such as the questions of existence, non-existence and compactness of solutions to the equations.
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Université Paris-Sud
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, 4:00pm-5:00pm

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.

Note for Attendees

 Please note unusual time and room.
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UBC
Tue 7 Oct 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Analytical properties for the Navier-Stokes equations and applications
ESB 2012
Tue 7 Oct 2014, 3:30pm-4:30pm

Abstract

Strong solutions to the 3D Navier-Stokes equations are known to exist locally-in-time 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 blow-up 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 blow-up speed for modified critical nonlinear Schrödinger equations
ESB 2012
Tue 14 Oct 2014, 3:30pm-4:30pm

Abstract

So far, only two blow-up regimes have been studied for NLS equations: the pseudo-conformal regime, where the blow-up speed is like $|t| ^{-1}$ and the log-log regime where the blow-up speed is like $|t|^{-1/2}$ with a log-log 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 L2-subcritical. We construct a minimal mass blowing up solution whose blow-up speed is  neither the log-log speed nor the pseudo-conformal 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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Tingting Huan
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, 3:30pm-4:30pm

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 Fisher-KPP nonlinearity. Also we prove the existence and nonexistence of traveling waves solutions for different ranges of the fractional power s for the generalized Fisher-KPP type model. 
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Mingfeng Zhao
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, 3:30pm-4:30pm

Abstract

In this talk, we will discuss the existence of the traveling wave solution for the Allen-Cahn equation involving the fractional Laplacians. Based on the existence of the standing waves for the balanced Allen-Cahn equation, we will use the continuity method to obtain the existence of the traveling waves for unbalanced Allen-Cahn 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, 3:30pm-4:30pm

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 Klein-Gordon, Schrodinger, and wave equations. The
dynamics include scatteing (to 0), blow-up, 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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Weiwei Ao
UBC
Tue 18 Nov 2014, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
On Non-topological Solutions of the rank 2 Chern-Simons System
ESB 2012
Tue 18 Nov 2014, 3:30pm-4:30pm

Abstract

In this talk, I will talk about the Chern-Simons equation arising from the study of  physics of high critical temperature superconductivity. A long-standing open problem is the existence of non-topological solutions. We proved the existence of non-topological solutions for the rank 2 Chern-Simons system. This is joint work with Professor Changshou Lin and Juncheng Wei
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Monica Musso
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, 3:30pm-4:30pm

Abstract

 We prove the existence of a sequence of nondegenerate, in the sense of Duyckaerts-Kenig-Merle, nodal nonradial solutions to the critical Yamabe problem or stationary energy-critical 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, 2:00pm-3:00pm

Abstract

 We propose  geometric flows to study the existence of a symplectic structure on an almost Hermitian manifold. We prove the short-time existence and uniqueness, and show some examples. 

Note for Attendees

 Note unusual day and time.
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Nicolaos Kapouleas
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, 3:30pm-4:30pm

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 self-shrinkers 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 four-manifolds 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, 3:30pm-4:30pm

Abstract

We show that for an immersed two-sided 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 two-sided 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, 3:30pm-4:30pm

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, 3:10pm-4:00pm

Abstract

 Directed polymer models are finite-temperature versions of first- and last-passage percolation on the lattice. In 1+1 dimensions, the free-energy of the directed polymer is conjecturally in the Tracy-Widom universality class at all finite temperatures. However, this has only been proven for a small class of polymers - the so-called solvable models that include Seppalainen's gamma polymers and the O'Connell-Yor semi-discrete polymer - with special sets of shapes and edge-weight 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 Poisson-Nernst-Planck type systems with cross-diffusion terms
ESB 2012
Tue 15 Sep 2015, 3:30pm-4:30pm

Abstract

The Poisson-Nernst-Planck (PNP) system is a well-known model of ion transport with many applications in biology, engineering and physics. Cross-diffusion terms may describe the exclusion of steric effects. In this lecture, I shall introduce cross diffusion terms from the Lennard-Jones potential and show the analytical results as follows:

1. Stability of 1D boundary layer solutions to original Poisson-Nernst-Planck (PNP) systems

2. Multiple solutions of PNP systems with steric effects

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Liangming Shen
UBC Math
Tue 22 Sep 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Unnormalized conical Kahler-Ricci flow
ESB 2012
Tue 22 Sep 2015, 3:30pm-4:30pm

Abstract

Conical Kahler metrics have become an interesting topic in Kahler geometry, and played an important role in the solution of Yau-Tian-Donaldson conjecture. In this talk, we make use of approximation method of Guenancia-Paun to extend Tian-Zhang's maximal existence result of Kahler-Ricci flow to conic case. Finally if possible, we can talk a little about C^{2,\alpha}-estimate for conical Kahler-Ricci 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, 3:30pm-4:30pm

Abstract

The leading asymptotic large-scale 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 large-scale term of this thermal EE turns out to be twice the first finite-size correction to the infinite-volume 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 zero-temperature limit, the leading large-scale 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 so-called area-law 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, 3:30pm-4:30pm

Abstract

I will consider the following classical  harmonic map flow from a general two-dimensional 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 blow-up solutions of Type II in general domains. We show  that type II blow-up solutions with blow-up  rate

(T-t)/\log^2 (T-t)

is stable and generic in arbitrary domains (without any symmetry).  I will also discuss the construction of  multiple blow-ups, reverse bubbling, bubbling trees, bubbling at infinity. As a by-product we can perform new geometric surgeries. (Joint work with Manuel del Pino and Juan Davila.) 

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Azahara de la Torre
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, 3:30pm-4:30pm

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 non-local 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
$$ (-\Delta)^\gamma u= c_{n, {\gamma}}u^{\frac{n+2\gamma}{n-2\gamma}}, u>0 \ \mbox{in} \ \r^n \backslash \{0\},$$
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 cross-diffusion system
ESB 2012
Tue 27 Oct 2015, 3:30pm-4:30pm

Abstract

We investigate the global time existence of smooth solutions for the Shigesada-Kawasaki-Teramoto system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion 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}n-Zygmund type  for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing Caffarelli-Peral perturbation technique together with a new two-parameter 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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Zachary Bradshaw
UBC Math
Tue 17 Nov 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Forward discretely self-similar solutions of the Navier-Stokes equations
ESB 2012
Tue 17 Nov 2015, 3:30pm-4:30pm

Abstract

For any discretely self-similar, incompressible initial data which is arbitrarily large in weak L^3, we construct a forward discretely self-similar solution to the 3D Navier-Stokes equations in the whole space. This also gives a third construction of self-similar solutions for any -1-homogeneous initial data in weak L^3,  improving those in by Jia-Sverak and Korobkov-Tsai 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 Tai-Peng Tsai.
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Matthew Coles
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, 3:30pm-4:30pm

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 non-standard since a relevant linear operator has a resonance.
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Changfeng Gui
University of Connecticut/University of Texas-San Antonio
Tue 1 Dec 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB2012
Saddle Solutions of Allen-Cahn Equation on the Plane.
ESB2012
Tue 1 Dec 2015, 3:30pm-4:30pm

Abstract

Allen-Cahn 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 Allen-Cahn equation on the plane. In particular, I will describe the variational characterization of these solutions as a mountain pass solutions.
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Robert Jerrard
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, 3:30pm-4:30pm

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 Hardy-Schrodinger operator
Tue 12 Jan 2016, 3:30pm-4:30pm

Abstract

I consider two different approaches for breaking scale invariance and restoring compactness for borderline variational problems involving the Hardy-Schrodinger 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 1-forms of a manifold using random samples
ESB 2012
Tue 2 Feb 2016, 3:30pm-4:30pm

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 1-forms. 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 1-forms.
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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 ill-posed ones
ESB 2012
Tue 15 Mar 2016, 3:30pm-4:30pm

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 ill-posed. However, our method allows to construct specific quasi-periodic 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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Yuzhao Wang
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, 3:30pm-4:30pm

Abstract

We will discuss the Cauchy problem for the cubic fourth order nonlinear Schrodinger equation (4NLS) on the circle. We first prove non-existence 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 well-posedness issue of the renoramilzed 4NLS (so called the Wick ordered W4NLS). In particular, by performing normal form reductions infinite many times, we prove well-posedness 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 two-dimensional Boussinesq equations with partial dissipation
ESB 4127
Thu 7 Apr 2016, 3:30pm-4:30pm

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 Rayleigh-Benard convection. Mathematically
the 2D Boussinesq equations serve as a lower-dimensional model of the 3D
hydrodynamics equations. In fact, the 2D Boussinesq equations retain some
key features of the 3D Euler and the Navier-Stokes 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 alpha-harmonic maps
ESB2012
Tue 12 Apr 2016, 3:30pm-4:30pm

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 alpha-harmonic maps.
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Vitali Vougalter
University of Toronto Mississauga
Tue 6 Sep 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Solvability of some integro-differential equations with anomalous diffusion
ESB 2012
Tue 6 Sep 2016, 3:30pm-4:30pm

Abstract

The work deals with the existence of solutions of an integro-differential 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, 3:30pm-4:30pm

Abstract

In this talk, we will discuss the Trudinger-Moser and Caffarelli-Kohn-Nirenberg 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, 3:30pm-4:30pm

Abstract

We consider the initial-value problem for the 3d Navier-Stokes 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 axi-symmetric solutions. The main tools are Nash-type estimates and certain monotone quantities. Uniqueness in the class of solutions which are not necessarily axi-symmetric remains a difficult open problem, which we plan to discuss briefly.  Joint work with Thierry Gallay.
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Seckin Demirbas
UBC
Tue 27 Sep 2016, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Gibbs' measure and almost sure global well-posedness for one dimensional periodic fractional Schrodinger equation
ESB 2012
Tue 27 Sep 2016, 3:30pm-4:20pm

Abstract

In this talk we will present recent local and global well-posedness 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 well-posedness 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 sub-Laplacian and Kohn Laplacian and Rigidity Theorems on pseudo-Hermitian CR manifolds
ESB 2012
Tue 4 Oct 2016, 3:30pm-4:30pm

Abstract

 In this talk, I will present  a CR-version of Lichnerowicz--Obata type theorem in a closed pseudo-Hermitian CR manifolds. It includes the lower bound estimates for the first positive eigenvalue for the both sub-Laplacian and Kohn Laplacian. I will also provide Obata type theorem associated to the sub-Laplacian and Kohn Laplacian on a closed pseudo-Hermitian 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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Laure Saint-Raymond
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, 2:00pm-3:00pm

Abstract


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Tobias Huxol
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, 3:30pm-4:30pm

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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Saikat Mazumdar
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, 3:30pm-4:30pm

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 Palais-Smale sequences as a sum of bubbles and adapt Lions's concentration-compactness 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, 3:30pm-4:30pm

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 Mu-Tao 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 Alexandrov-Fenchel 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, 3:30pm-4:30pm

Abstract

There is a well-known technique due to Schoen-Yau 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 2-convex embedded spheres
ESB 2012
Tue 8 Nov 2016, 3:30pm-4:30pm

Abstract

The space of smoothly embedded n-spheres in Rn+1 is the quotient space Mn:=Emb(Sn,Rn+1)/Diff(Sn). In 1959 Smale proved that M1 is contractible and conjectured that M2 is contractible as well, a fact that was proved by Hatcher in 1983.

While it is known that not all Mn are contractible, for n\get 3 no single homotopy group of Mn is known. Even knowing whether the Mn are path connected or not would be extremely interesting. For instance, if M3 is not path connected, the 4-d 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 Mn.
I will first illustrate how the theory of 1-d mean curvature flow (aka curve shortening flow) yields a very simple proof of Smale's theorem about the contractibility of M1.
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 2-convex 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, 2:00pm-3:00pm

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, 3:30pm-4:30pm

Abstract

We deal with a class of the semilinear elliptic equations of the Hénon-type 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 )
Mulit-to one-dimensional optimal transport
ESB Room 4127 (PIMS Videoconferencing Room )
Thu 8 Dec 2016, 4:00pm-5:00pm

Abstract

I will discuss joint work with Pierre-Andre Chiappori and Robert McCann on the Monge-Kantorovich 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, 3:30pm-4:30pm

Abstract

 
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Hao Shen
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, 3:30pm-4:30pm

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 Blume-Capel 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 beta-gamma system
MATH 126
Tue 24 Jan 2017, 3:00pm-4:00pm

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 beta-gamma 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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Harold Williams
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, 3:30pm-5:00pm

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 Gaiotto-Moore-Neitzke, Costello, and Kapustin-Saulina -- in short, coherent IC sheaves are the precise algebro-geometric counterparts of Wilson-'t Hooft line operators. The proof relies on techniques developed by Kang-Kashiwara-Kim-Oh 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 holomorphic-topological field theory). This is joint work with Sabin Cautis.

Note for Attendees

Refreshments will be served at 3:15pm in the MATH 125 Lounge Area.
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Martin Fraas
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, 3:30pm-4:30pm

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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Aaron Zeff Palmer
Cornell University
Wed 1 Feb 2017, 4:00pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (Note the unusual time: 4-5 pm on Wednesday, Feb 1. )
Incompressibility and Global Injectivity in Second-Gradient Non-Linear Elasticity
ESB 2012 (Note the unusual time: 4-5 pm on Wednesday, Feb 1. )
Wed 1 Feb 2017, 4:00pm-5:00pm

Abstract

This talk addresses how the calculus-of-variations is applied to non-linear elasticity.  In physically realistic classical models, energy-minimizing deformations may not be smooth enough to satisfy the variational Euler-Lagrange equations.  However, with a second-gradient model we guarantee sufficient regularity to rigorously prove energy-minimizers satisfy such an equation and maintain incompressibility and/or global injectivity. 

The constraints of incompressibility and self-contact introduce subtle challenges of infinite-dimensional non-linear 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 measure-valued surface traction for the self-contact 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, 3:30pm-4:30pm

Abstract

I will present an adiabatic theorem for the driven dynamics of ground state projections of a smooth family of many-body 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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I-Kun Chen
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, 3:30pm-4:30pm

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. I-Kun Chen is currently a Senior Lecturer at the Department of Applied Analysis and Complex Dynamical Systems, Kyoto University, http://www.acs.i.kyoto-u.ac.jp/en.html . He is visiting UBC between Feb 8-22, 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, 3:30pm-4:30pm

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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Lei Zhang
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, 3:30pm-4:30pm

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 Chang-shou 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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Nassif Ghoussoub
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, 3:30pm-4:30pm

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 Allen-Cahn and Free Boundary Problems
ESB 2012
Tue 21 Mar 2017, 3:30pm-4:30pm

Abstract

 I will report recent progress in De Giorgi's type conjectures (and beyond) for Allen-Cahn 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, 3:30pm-4:30pm

Abstract

We will discuss about the regularity of the Gauss curvature flow: the optimal C^{1,\frac{1}{n-1}} 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
Min-max minimal hypersurfaces with free boundary
ESB 2012
Tue 11 Apr 2017, 3:30pm-4:30pm

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 min-max method. I will explain the basic ideas behind the min-max 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, 1:00pm-2:00pm

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, 11:00am-12:00pm

Abstract

 We present a unified approach to quadratic asymptote of solutions to a class of fully nonlinear elliptic equations over exterior domains, including Monge-Ampere 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, 3:30pm-4:30pm

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 Tej-Eddine 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, 3:30pm-4:30pm

Abstract

Please note this talk is cancelled.
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Yong Liu
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 KP-I Lump Solution
ESB 2012
Tue 10 Oct 2017, 3:30pm-4:30pm

Abstract

 We prove that the lump solution of the classical KP-I 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 well-posedness and asymptotics of a type of Keller-Segel models coupled to fluid flow
ESB 2012
Tue 17 Oct 2017, 3:30pm-4:30pm

Abstract

We study chemotaxis equations coupled to the Navier-Stokes 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, 3:30pm-4:30pm

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
non-rigidity results of Li -Wang, and a new quasi local type inequality of
Lu-Miao. 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, 3:30pm-4:30pm

Abstract

Optimal stopping problems can be viewed as a problem to calculate the space and time dependent value function, which solves a nonlinear, possible non-smooth and degenerate, parabolic PDE known as an Hamilton-Jacobi-Bellman (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 non-degeneracy 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 Lagrangian-relaxation 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 ``piecewise-monotonic’' structure of the stopping set.
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Ali Hyder
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 Q-curvature
ESB 2012
Tue 14 Nov 2017, 3:30pm-4:30pm

Abstract


I will talk about the existence of solution to the Q-curvature 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 non-negative function and n>2. Geometrically, if u is a solution to \eqref{1} then Q is the Q-curvature 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 Q-curvature 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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Michal Kowalczyk
University of Chile
Tue 21 Nov 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Asymptotic stability for some nonlinear Klein-Gordon equations for odd perturbations in the energy space
ESB 2012
Tue 21 Nov 2017, 3:30pm-4:30pm

Abstract

 Showing asymptotic stability in one dimensional nonlinear Klein-Gordon 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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Sylvia Serfaty
Courant Institute, NYU
Thu 30 Nov 2017, 12:00pm SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB 4127 (PIMS Videoconference room)
Mean-Field Limits for Ginzburg-Landau vortices
ESB 4127 (PIMS Videoconference room)
Thu 30 Nov 2017, 12:00pm-1:00pm

Abstract

 Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued 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 Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.
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