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.

 

Pages

Subscribe to RSS - Colloquium