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Lunch will be provided only if you are writing the Analysis exam.  Lunch is served 12:00-1:00 pm in Math 125.
If you have food allergies, please let Lee know.

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Details

Abstract

I will show a simple method to decompose the Gaussian free field associated to a (weighted) graph or manifold into a sum of finite range Gaussian fields, which are fields that are smoother than the original field and have spatially localized correlations.

Abstract

We consider three counter-example: A non-smooth group which has a fppf-torsor that is not étale, similarly for a formally smooth group. We also consider how to differentiate between free and non-free actions for µ_p.

Abstract

We construct noninvertible maps on every compact surface, and study their chaotic properties from both the measure theoretic and topological points of view (joint work with Jane Hawkins).

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

Abstract

In this talk, I would like to discuss some recent progress in analysis and geometry of optimal transport maps that arise when mass distributions are matched in most cost efficient way. Especially, continuity of optimal maps will be addressed, which is related to nonlinear partial differential equations as well as Riemannian geometry.

Abstract

Let X be a Calabi-Yau threefold. We study the symmetric trilinear form on the integral second cohomology group of X defined by the cup product. Our study is motivated by C.P.C. Wall's classification theorem, which roughly says that the diffeomorphism class of a spin sixfold is determined by the trilinear form. We investigate the interplay between the Chern classes and the trilinear form of X, and demonstrate some numerical relations between them. If time permits, we also discuss some properties of the associated cubic form. This talk is based on a joint work with P.H.M. Wilson.

Abstract

We analyze stochastic completeness, or non-explosiveness, of the variable-speed
random walk (VSRW) on weighted graphs.  We prove a criterion relating volume
growth in an adapted metric to stochastic completeness of the VSRW.  This
criterion is analogous to the optimal result for Riemannian manifolds and is
shown to be sharp.  The proof is accomplished through the construction of a
Brownian motion on a  metric graph which behaves similarly to the VSRW under
consideration.  Results of Sturm on stochastic completeness for local Dirichlet
spaces are then applicable to this Brownian motion, and non-explosiveness of
the Brownian motion is shown to imply non-explosiveness of the VSRW.

Abstract

It was a result of Greenlees and Sadofsky that classifying spaces of finite groups satisfy a Morava K-theory version of Poincare duality, which was proved by showing the contractibility of the corresponding Tate spectrum. In this series of two talks, I will explain the proof, discuss its generalization to quotient orbifolds and consequences with examples. Some background in equivariant stable homotopy theory will be given. If time permits, I will also explain why the duality map can be viewed as coming from a Spanier-Whitehead type construction for differentiable stacks.

Abstract

 We will discuss some results concerning the representability of torsors. In the case of an abelian scheme, we will explain how the properties "X is representable" and "the class of X in the first cohomology group is torsion" are related. This will lead us to the construction of an fppf sheaf torsor which is not representable.

Abstract

A formula for the dimension of the space of cuspidal modular forms on Γ₀(N) of even weight k≥2 has been known for several decades. More recent but still well-known is the Atkin-Lehner decomposition of this space of cusp forms into subspaces corresponding to newforms on Γ₀(d) of weight k, as d runs over the divisors of N. A recursive algorithm for computing the dimensions of these spaces of newforms follows from the combination of these two results, but it is desirable to have a formula in closed form for these dimensions. In this talk we describe such a closed-form formula, not only for these dimensions, but also for the corresponding dimensions of spaces of newforms on Γ₁(N) of weight k≥2. This formula is much more amenable to analysis and to computation. For example, we derive asymptotically sharp upper and lower bounds for these dimensions, and we compute their average orders. We also establish sharp inequalities for the special case of weight-2 newforms on Γ₀(N), and we report on computations of these dimensions. For example, we can find the complete list of all N such that the dimension of the space of weight-2 newforms on Γ₀(N) is less than or equal to 100; previous such results had only gone up to 3.

Abstract

Beginning with the work of J. Hersch for the two sphere and that of P. Li and S. T. Yau for more general surfaces, the question of determining surfaces of fixed area that maximize the first eigenvalue has been actively studied. In this talk I will describe recent work with R. Schoen concerning extremal eigenvalue questions for surfaces with boundary. In both cases the eigenvalue problems are related to minimal surface questions. For closed surfaces these are minimal surfaces in spheres while for surfaces with boundary they are related to minimal surfaces in the ball satisfying a natural boundary condition. I will describe some results on determining optimal surfaces. I will also describe some recent work with Martin Li on compactness of the space of free boundary minimal surfaces with a fixed topological type.

Abstract

The Gopakumar-Marino-Vafa formula, proven almost ten years ago, evaluates certain triple Hodge integrals on moduli spaces of curves in terms of Schur functions.  It has since been realized that the GMV formula is a special case of the Gromov-Witten/Donaldson-Thomas correspondence for Calabi-Yau threefolds.

In this talk, I will introduce an orbifold generalization of the GMV formula which evaluates certain abelian Hodge integrals in terms of loop Schur functions.  I will introduce local Z_n gerbes over the projective line and show how the gerby GMV formula can be used to prove the GW/DT correspondence for this class of orbifolds.  With the remaining time, I will sketch the main ideas in the proof of the formula and discuss generalizations to other geometries.

Abstract

A three-phase contact line forms when a gas-liquid interface intersects a solid substrate, and a moving contact line presents a well-known singularity that cannot be computed using the conventional Navier-Stokes formalism. I will discuss the use of a diffuse-interface model for computing moving contact lines. The Cahn-Hilliard diffusion is known to regularize the singularity and makes possible a continuum-level computation. But relating the results to physical reality is subtle. I will show numerical results that suggest a well-defined sharp-interface limit, with a finite contact line speed that can be related to measurements. Then I will discuss applications including enhanced slip on textured substrates and propulsion of water striders on the air-water interface.

Abstract

Abstract: It is by now a classical result that sets of positive density of R^2 contain all large distances. It was however observed by Bourgain that the analogue result does not hold for 3 term progressions (3 equally spaced points along a line) in any dimensions. Our aim is to discuss an approach to show that such results are still possible if one changes the metric, for example using the l^4 metric. If time permits we'll mention other possible point configurations. This is ongoing joint work with Brian Cook and Malabika Pramanik.

Abstract

 

During the last two decades, semidefinite optimization has grown into a significant field of research with applications in many diverse areas such as graph theory, distance geometry, combinatorial optimization, low-rank matrix completion, and polynomial optimization. In this talk, I will discuss my work in two of these areas, namely distance geometry and combinatorial optimization. In distance geometry, Euclidean distance matrices (EDMs) have recently received revived interest due to their use in modern applications such as sensor network localization, protein structure determination, and machine learning. The second reason for this revived interest is the fact that we now have semidefinite optimization solvers that we can use to solve problems involving EDMs (however, we are limited in the size of problems we can solve efficiently due to the high complexity of these semidefinite solvers). I will discuss my theoretical contribution relating cliques in the graph of a partial EDM to identifying a reduced problem formulation, and how I used this result to develop numerical methods to solve large-scale instances in each of the three modern applications mentioned above. In combinatorial optimization, semidefinite optimization is used to efficiently compute high-quality bounds to many difficult (in fact, NP-hard) problems, such as Max-Cut and binary quadratic optimization. This has led to the development of state-of-the-art branch-and-bound methods for solving such problems to optimality. I will discuss my work on a bounding procedure for Max-Cut which has been obtained by adding a regularization term to the standard semidefinite bound — this allows us to use basic numerical tools (a eigenvalue decomposition method, and a quasi-Newton optimization method) to compute high-quality semidefinite bounds efficiently. I will show how this new bounding procedure gives a significant improvement over the current state-of-the-art method.

Abstract

We will discuss a chapter of a book on Delay differential equations by Hal Smith. This chapter can be downloaded from the internet at the following link provided. The chapter is called "Delayed Negative Feedback: A warm-up", and it will introduce some of the basics of delay differential equations and how to obtain some analytical results about their stability.

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.

Abstract

The random interlacements (at level u) is a one parameter family of random subsets of Z^d (d>=3), introduced recently by A.-S. Sznitman, which arises as the 
local limit of the trace of a simple random walk on a d-dimensional torus, when the size of the torus goes to infinity. The parameter u controls the density of 
the interlacement. The vacant set at level u (i.e. the complement set of the random interlacement at level u) undergoes non-trivial percolation phase transition 
as u varies. We study the supercritical phase and show that finite connected components of the vacant set are "small" for all d>=3 if u is small enough. Our 
method is markedly different from that of A. Teixeira (2011), which gives the analogous result for d>=5.

Abstract

This is joint work with Tetsuya Ito. Fractional Dehn twist coefficient (FDTC), defined by Honda-Kazez-Matic, is an invariant of mapping classes. In this talk we study FDTC by using open book foliation method, then obtain results in topology, geometry, and contact geometry of the open-book-manifold of a mapping class.

Abstract

Abstract

Let p be a prime, r >= 3, and n_i = p^{a_i} for positive integers a_1,...,a_r. Set G = GL_{n_1} x ... x GL_{n_r}, and let \mu be a central subgroup of G. The Galois cohomology set H^1(K, G/\mu) classifies r-tuples of central simple algebras satisfying linear equations in the Brauer group Br(K). We study the essential dimension of G/\mu by constructing the 'code' associated to /mu.

Abstract

This is the continuation of my previous talk. 

It was a result of Greenlees and Sadofsky that classifying spaces of finite groups satisfy a Morava K-theory version of Poincare duality, which was proved by showing the contractibility of the corresponding Tate spectrum. In this series of two talks, I will explain the proof, discuss its generalization to quotient orbifolds and consequences with examples. Some background in equivariant stable homotopy theory will be given. If time permits, I will also explain why the duality map can be viewed as coming from a Spanier-Whitehead type construction for differentiable stacks.

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.

Abstract

We will explain the basic definitions of Chow groups, correspondences and Chow motives. This will be the first talk in a series to help us study the incompressibility of certain varieties.

Abstract

UBC/UMC is the Undergraduate Mathematics Colloquium at UBC.

These talks are for undergraduates interested in mathematics and mathematical research. They are put on by professors, senior graduate students and visitors to the department. Graduate students are also invited to attend.

Title: Topological Data Analysis

Abstract: This is not an applied math talk. On the contrary: I will show how the abstract tools of Algebraic Topology can be used to assemble low-dimensional representations of data into high-dimensional structures. An example of such an assembly would be the brain's inference of a 3D environment from a 2D image from each eye.

Abstract

The Sylvester-Gallai Theorem states that, given any set P of n points in the plane not all on one line, there is at leads one line through precisely two points of P. Such a line is called an ordinary line. How many ordinary lines must there be? The Sylvester-Gallai Theorem says that there must be at leads one but, in recent joint work with T. Tao, we have shown that there must be at least n/2 if n is even and at least 3n/4 - C if n is odd, provided that n is sufficiently large. These results are sharp. My plan in the talk is to give an overview of this problem and of our work towards its solution.

Abstract

The Euclidean division is a basic tool when dealing with the ordinary integers. It does not extend to rings of integers of algebraic number fields in general. It is natural to ask how to measure the "deviation" from the Euclidean property, and this leads to the notion of Euclidean minimum. The case of totally real number fields is of special interest, in particular because of a conjectured upper bound (conjecture attributed to Minkowski). The talk will present some recent results, obtained jointly with Piotr Maciak.

Abstract

The study of quadratic forms is a classical  and important topic of algebra and number theory. A natural example is the trace form of a finite  Galois extension. This form has the additional property of being invariant under the Galois group, leading to the notion of "self-dual nornal basis", introduced by Lenstra. The aim of this talk is to give a survey of this area, and to present some recent joint results with Parimala and Serre.