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Thu Sep 13

Colloquium

3:35pm - Vincent Hall 16
Singularities of the Ricci flow and Ricci solitons
Huai-Dong Cao, Lehigh University, Ordway Visitor

Understanding formation of singularities has been an important subject in the study of the Ricci flow and other geometric flows. It turns out generic singularities in the Ricci flow are modeled on shrinking Ricci solitons. In this talk, I will discuss some of the recent progress on classifications of shrinking Ricci solitons and their stability/instability with respect to Perelman's $\nu$-entropy.

Tue Sep 25

Colloquium

3:35pm - Vincent Hall 16
Microlocal codimension-three conjecture
Kari Vilonen, University of Melbourne, Ordway Visitor

Special functions (or distributions) can be understood and analyzed in terms of the systems of differential equations they satisfy. To this end, a general theory of systems of linear (micro) differential equations was developed by the Sato school in Kyoto. This point of view, in its various incarnations, is now ubiquitous in many parts of mathematics. For example, in the geometric Langlands program and representation theory it allows us to replace functions and group representations by geometric objects, perverse sheaves or D-modules. It has been well-known for a long time that the description of these objects gains more symmetry when one passes to the cotangent bundle. We explain the shape of the general microlocal structure of these objects and discuss, in particular, the role played by the codimension-three conjecture which was proved by Masaki Kashiwara and the speaker a few years ago.

Thu Oct 11

Colloquium

3:35pm - Vincent Hall 16
Propagation of bistable fronts through a perforated wall
Hiroshi Matano, Meiji University, Tokyo, Ordway Visitor

We consider a bistable reaction-diffusion equation on ${\bf R}^N$ in the presence of an obstacle $K$, which is a wall of infinite span with periodically arrayed holes. More precisely, $K$ is a closed subset of ${\bf R}^N$ with smooth boundary such that its projection onto the $x_1$-axis is bounded, while it is periodic in the rest of variables $(x_2,\ldots, x_N)$. We assume that ${\bf R}^N \setminus K$ is connected. Our goal is to study what happens when a planar traveling front coming from $x_1 = +\infty$ meets the wall $K$.

We first show that there is clear dichotomy between `propagation' and `blocking'. In other words, the traveling front either completely penetrates through the wall or is totally blocked, and that there is no intermediate behavior. This dichotomy result will be proved by what we call a De Giorgi type lemma for an elliptic equation on ${\bf R}^N$. Then we will discuss sufficient conditions for blocking, and those for propagation. This is joint work with Henri Berestycki and Francois Hamel.

Thu Nov 15

Colloquium

3:35pm - Vincent Hall 16
(Log)-Epiperimetric Inequality and the Regularity of Variational Problems
Luca Spolaor, MIT

In this talk I will present a new method for studying the regularity of minimizers to variational problems. I will start by introducing the notion of blow-up, using as a model case the so-called Obstacle problem. Then I will state the (Log)-epiperimetric inequality and explain how it is used to prove uniqueness of the blow-up and regularity results for the solution near its singular set. I will then show the flexibility of this method by describing how it can be applied to other free-boundary problems and to (almost)-area minimizing currents. Finally I will describe some future applications of this method both in regularity theory and in other settings.

Tue Nov 27

Colloquium

3:35pm - Vincent Hall 16
Surface bundles, monodromy, and arithmetic groups
Bena Tshishiku, Harvard University

Fiber bundles with fiber a surface arise in many areas including hyperbolic geometry, symplectic geometry, and algebraic geometry. Up to isomorphism, a surface bundle is completely determined by its monodromy representation, which is a homomorphism to a mapping class group. This allows one to use algebra to study the topology of surface bundles. Unfortunately, the monodromy representation is typically difficult to "compute" (e.g. determine its image). In this talk, I will discuss some recent work toward computing monodromy groups for holomorphic surface bundles, including certain examples of Atiyah and Kodaira. This can be applied to the problem of counting the number of ways that certain 4-manifolds fiber over a surface. This is joint work with Nick Salter.

Thu Nov 29

Colloquium

10:00am - Vincent Hall 570
The role of Energy in Regularity
Max Engelstein, MIT

The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.

However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.

We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur

Thu Nov 29

Colloquium

3:35pm - Vincent Hall 16
Interpolative decomposition and its applications
Lexing Ying, Stanford, Ordway Visitor

Interpolative decomposition is a simple and yet powerful tool for approximating low-rank matrices. After discussing the theory and algorithm, I will present a few new applications of interpolative decomposition in numerical partial differential equations, quantum chemistry, and machine learning.

Thu Dec 06

Colloquium

3:35pm - Vincent Hall 16
Number theory over function fields and geometry
Will Sawin, Columbia University

The function field model involves taking problems in
classical analytic number theory and replacing the integers with
polynomials over a finite field. This preserves most of the complexity
of these problems while giving them a stronger relationship to
geometry, allowing new techniques to be applied. We will explain how
this works using a recent example involving sums of the divisor
function where, thanks to good luck, particularly simple methods can
prove a particularly powerful result.

Tue Jan 22

Colloquium

2:30pm - Vincent Hall 16
Multiscale Problems in Cell Biology
Chuan Xue, Ohio State University

Complex biological systems involve multiple space and time scales. To get an integrated understanding of these systems involves multiscale modeling, computation and analysis. In this talk, I will discuss two such examples in cell biology and illustrate how to use multiscale methods to explain experimental data. The first example is on chemotaxis of bacterial populations. I will present recent progress on embedding information of single cell dynamics into models of cell population dynamics. I will clarify the scope of validity of the well-known Patlak-Keller-Segel chemotaxis equation and discuss alternative models when it breaks down. The second example is on the axonal cytoskeleton dynamics in health and disease. I will present a stochastic multiscale model that gave the first mechanistic explanation for the cytoskeleton segregation phenomena observed in many neurodegenerative diseases.

Thu Jan 31

Colloquium

3:35pm - Vincent Hall 16
From infinite random matrices over finite fields to square ice
Leonid Petrov, University of Virginia

Asymptotic representation theory of symmetric groups is a rich and beautiful subject with deep connections with probability, mathematical physics, and algebraic combinatorics. A one-parameter deformation of this theory related to infinite random matrices over a finite field leads to randomization of the classical Robinson-Schensted correspondence between words and Young tableaux. Exploring such randomizations we find unexpected applications to six vertex (square ice) type models and traffic systems on a 1-dimensional lattice.

Tue Feb 05

Colloquium

3:35pm - Vincent Hall 16
A Multiscale/Multiphysics Coupling Framework for Heart Valve Damage
Yue Yu, Lehigh University

Bioprosthetic heart valves (BHVs) are the most popular artificial replacements for diseased valves that mimic the structure of native valves. However, the life span of BHVs remains limited to 10-15 years, and the mechanisms that underlie BHVs failure remain poorly understood. Therefore, developing a unifying mathematical framework which captures material damage phenomena in the fluid-structure interaction environment would be extremely valuable for studying BHVs failure. Specifically, in this framework the computational domain is composed of three subregions: the fluid (blood) , the fracture structure (damaged BHVs) modeled by the recently developed nonlocal (peridynamics) theory, and the undamaged thin structure (undamaged BHVs). These three subregions are numerically coupled to each other with proper interface boundary conditions.

In this talk, I will introduce two sub-problems and the corresponding numerical methods we have developed for this multiscale/multiphysics framework. In the first problem the coupling strategy for fluid and thin structure is investigated. This problem presents unique challenge due to the large deformation of BHV leaflets, which causes dramatic changes in the fluid subdomain geometry and difficulties on the traditional conforming coupling methods. To overcome the challenge, the immersogeometric method was developed where the fluid and thin structure are discretized separately and coupled through penalty forces. To ensure the capability of the developed method in modeling BHVs, we have verified and validated this method. The second problem focuses on the nonlocal Neumann-type interface boundary condition which plays a critical role in the fluid—peridynamics coupling framework. In the nonlocal models the Neumann-type boundary conditions should be defined in a nonlocal way, namely, on a region with non-zero volume outside the surface, while in fluid—structure interfaces the hydrodynamic loadings from the fluid side are typically provided on a sharp co-dimensional one surface. To overcome this challenge, we have proposed a new nonlocal Neumann-type boundary condition which provides an approximation of physical boundary conditions on a sharp surface, with an optimal asymptotic convergence rate to the local counter part. Based on this new boundary condition, we have developed a fluid—peridynamics coupling framework without overlapping regions.

Tue Mar 05

Colloquium

3:35pm - Vincent Hall 16
Cohomology of Shimura varieties
Sug Woo Shin, U.C. Berkeley

Shimura varieties are a certain class of algebraic varieties over number fields with lots of symmetries, introduced by Shimura and Deligne nearly half a century ago. They have been playing a central role in number theory and other areas. Langlands proposed a program to compute the L-functions and cohomology of Shimura varieites in 1970s; this was refined by Langlands-Rapoport and Kottwitz in 1980s. I will review some old and recent results in this direction.

Thu Mar 14

Colloquium

3:35pm - Vincent Hall 16
Colloquium
Jose Figueroa-Lopez, Washington University
Tue Apr 16

Colloquium

3:35pm - Vincent Hall 16
Colloquium
Ordway Visitor Christopher Skinner, Princeton University
Thu Apr 18

Colloquium

3:35pm - Vincent Hall 16
Colloquium
Vlad Markovic, Caltech
Tue Apr 30

Colloquium

3:35pm - Vincent Hall 16
Colloquium
Jeffrey Hoffstein, Brown University
Thu May 02

Colloquium

3:35pm - Vincent Hall 16
Colloquium
Thomas Hou, Ordway Vsitor, Caltech