Self-Avoiding random motion

Speaker: 

Greg Lawler

Institution: 

University of Chicago

Time: 

Thursday, February 23, 2017 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

The self-avoiding walk (SAW) is a model for polymers that assigns equal probability to all paths that do not return to places they have already been. The lattice version of this problem, while elementary to define, has proved to be notoriously difficult and is still open. It is initially more challenging to construct a continuous limit of the lattice model which is a random fractal. However, in two dimensions this has been done and the continuous model (Schram-Loewner evolution) can be analyzed rigorously and  used to understand the nonrigorous predictions about SAWs.  I will survey some results in this area and then discuss some recent work on this ``continuous SAW''.

Open problems in Mean Field Games theory

Speaker: 

Wilfrid Gangbo

Institution: 

UCLA

Time: 

Thursday, January 26, 2017 - 4:00pm to 5:00pm

Host: 

Location: 

RH306

We present some of the recent results in Mean Field Games theory, especially the so–called master equation, backbone of the MFG the- ory. Despite the fact that the master equation is a non–local first order equation, we show how it is linked to metric viscosity solutions of a local Hamilton–Jacobi equation on the set of probability measures. (This talk is based on a joint work with A. Swiech). 

Invariants in the Bergman and Szeg\H o kernels in strictly pseudoconvex domains in $\mathbb C^2$

Speaker: 

Peter Ebenfelt

Institution: 

UCSD

Time: 

Thursday, November 17, 2016 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

 The Bergman and Szeg\H o kernels in a bounded domain $\Omega\subset \mathbb C^n$ are the reproducing kernels for the holomorphic functions in $L^2(\Omega,dV)$ and $L^2(\partial \Omega,d\sigma)$, respectively, where $dV$ denotes the standard Lebesgue measure in $\bC^n$ and $d\sigma$ a surface measure on the boundary $\partial\Omega$. Their restrictions to the diagonal are known to have asymptotic expansions of the form:

$$K_B\sim \frac{\phi_B}{\rho^{n+1}}+\psi_B\log\rho,\quad K_S\sim \frac{\phi_S}{\rho^{n}}+\psi_S\log\rho,$$

where $\phi_B,\phi_S,\psi_B,\psi_S\in C^\infty(\overline{\Omega})$ and $\rho>0$ is a defining equation for $\Omega$. The functions $\phi_B,\phi_S,\psi_B,\psi_S$ encode a wealth of information about the biholomorphic geometry of $\Omega$ and its boundary $\partial \Omega$. In this talk, we will discuss this in the context of bounded strictly pseudoconvex domains in $\mathbb C^2$ and pay special attention to the lowest order invariants in the log term and a strong form of a conjecture of Ramadanov.

Product rigidity results for group von Neumann algebras

Speaker: 

Thomas Sinclair

Institution: 

Purdue University

Time: 

Friday, November 4, 2016 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

Given a locally compact second countable group G, the group von Neumann algebra L(G) is the algebra associated to the invariant subspace decomposition of the left regular representation. It is a natural, and quite difficult, question to address how much of the group structure is recoverable from L(G). That is if two groups have isomorphic group von Neumann algebras what algebraic structure do the groups have in common? In the case of infinite discrete groups, we will explain how if G is a direct product of "indecomposable" groups, such as nonabelian free groups or nonelementary hyperbolic groups, then the product structure can be fully recovered from L(G). This is joint work with Ionut Chifan and Rolando de Santiago.

Towards the global bifurcation theory on the plane

Speaker: 

Yulij Ilyashenko

Institution: 

Cornell University, Higher School of Economics (Russia)

Time: 

Tuesday, October 11, 2016 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

The talk provides a new perspective of the global bifurcation theory on the plane. Theory of planar bifurcations consists of three parts: local, nonlocal and global ones. It is now clear that the latter one is yet to be created.

Local bifurcation theory (in what follows we will talk about the plane only) is related to  transfigurations of phase portraits of differential equations near their singular points. This theory is almost completed, though recently new open problems occurred. Nonlocal theory is related to bifurcations of separatrix polygons (polycycles). Though in the last 30 years there were obtained many new results, this theory is far from being completed.

Recently it was discovered that nonlocal theory contains another substantial part: a global theory. New phenomena are related with appearance of the so called sparkling saddle connections. The aim of the talk is to give an outline of the new theory and discuss numerous open problems. The main new results are: existence of an open set of structurally unstable families of planar vector fields, and of families having functional invariants (joint results with Kudryashov and Schurov). Thirty years ago Arnold stated six conjectures that outlined the future development of the global bifurcation theory in the plane. All these conjectures are now disproved. Though the theory develops in quite a different direction, this development is motivated by the Arnold's conjectures.

How Mathematical Models Can Provide Insight into Stopping Epidemics

Speaker: 

James Hyman

Institution: 

Tulane University

Time: 

Thursday, December 1, 2016 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

Public health workers are reaching out to mathematical scientists to use disease models to understand, and mitigate, the spread of emerging diseases. Mathematical and computational scientists are needed to create new tools that can anticipate the spread of new diseases and evaluate the effectiveness of different approaches for bringing epidemics under control. That is, these models can provide an opportunity for the mathematical scientists to collaborate with the public health community to improve the health of our world and save lives. The talk will provide an overview, for general audiences, of how these collaborations have evolved over the past decade. I will describe some recent advances in mathematical models that are having an impact in guiding pubic health policy, and describe what new advances are needed to create the next generation of models. Throughout the talk, I will share some of my personal experiences in used these models for controlling the spread of Ebola, HIV/AIDS, Zika, chikungunya, and the novel H1N1 (swine) flu. The talk is for a general audience.

 

The PDEs of mathematical finance

Speaker: 

Jerry Goldstein

Institution: 

University of Memphis

Time: 

Thursday, April 28, 2016 - 4:00pm

Location: 

RH 306

We will discuss three one space dimensional time dependent linear parabolic equations: the heat equation, the Black-Scholes equation (describing stock options) and the Cox-Ingersoll-Ross equation (describing bond markets).  New results will involve representation of the solution semigroups, chaotic properties of the semigroups, and a new kind of Feynman-Kac type representation of the solution for the CIR equation.

Random matrix type fluctuations: how to see them in the Ising model?

Speaker: 

S. Shlosman

Institution: 

CNRS, Marseille

Time: 

Thursday, March 10, 2016 - 4:00pm

Location: 

RH 306

I will talk about the Ising model -- the drosophila of the rigorous statistical physics. It turns out that some of the new phenomena which appear in modern mathematical physics can still be observed in the Ising model as well. 
One example which I will focus on is the size of typical fluctuations of the extended systems. If the size of the system is N, then the usual (Gaussian) fluctuations are of the order of N^{1/2}. Bit in the random matrix theory one sees the fluctuations of the order N^{1/3}. I will explain that one can see them already in the Ising model -- one just needs to know where to look.

Journey to the Center of the Earth

Speaker: 

Gunther Uhlmann

Institution: 

University of Washington

Time: 

Thursday, January 28, 2016 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

     We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also several applications in optics and medical imaging among others.
     The problem can be recast as a geometric problem: Can one determine a Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.
     We will also describe some recent results, join with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary. No previous knowledge of Riemannian geometry will be assumed. 

Birkhoff Conjecture and ''spectral rigidity'' of planar convex domains

Speaker: 

Vadim Kaloshin

Institution: 

Maryland University

Time: 

Thursday, January 21, 2016 - 4:00pm to 5:00pm

Host: 

Location: 

Rowland Hall 306

The classical Birkhoff conjecture states that the only integrable convex planar domains are circles and ellipses. In a joint work with A. Avila and J. De Simoi we show that this conjecture is true for perturbations of ellipses of small eccentricity. It turns out that the method of proof gives an insight into deformational spectral rigidity of planar axis symmetric domains and a partial answer to a question of P. Sarnak. The latter is a joint work with J. De Simoi and Q. Wei.

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