Date/Time : 27 Dec 2019 14:00
Venue : S17 #04-06 SR1
Speaker : Xiaoming Wang, Southern University of Science and Technology

We show that the Principle of Exchange of Stability holds for convection in a layer of fluids overlaying a porous media with proper interface boundary conditions and suitable assumption on the parameters. The physically relevant small Darcy number regime as well as the dependence of the convection on various parameters will be discussed. A theory on the dependence of the depth ratio of the onset of deep convection will be put forth together with supporting numerical evidence. A decoupled uniquely solvable, unconditionally stable numerical scheme for solving the system will be presented as well.

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Date/Time : 23 Oct 2019 14:00
Venue : S17 #04-06 SR1
Speaker : Bernard Shiffman, Johns Hopkins University

Independent random points on a compact Riemann surface tend to cluster, while the zeros of a random holomorphic section of a line bundle of high degree tend to repel on account of their correlations. I shall dis-cuss asymptotic formulas for the distribution and pair correlation of zeros of Gaussian random polynomials on C^n and, more generally, of random sections of powers of a positive line bundle on a compact Kahler manifold. These formulas depend on the off-diagonal asymptotic expansion of the Bergman kernel, which also yields central limit theorems for the zeros.

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Date/Time : 09 Oct 2019 14:00
Venue : S17 #04-06 SR1
Speaker : Hirofumi Osada, Kyushu University

I talk homogenization of diffusion in the two-dimensional Euclid space in a periodic Coulomb environment. That is, I consider a periodic point process in the plane. The diffusion has the repulsive interaction with the two-dimensional Coulomb potential with inverse temperature $\beta$ to each particle in the periodic point process. We first prove that the diffusion is diffusive with non-degenerated effective diffusion constant $\gamma$. We then remove one particle from the environment and consider the diffusive scaling limit of the diffusion. Then its new effective constant depends on inverse temperature $\beta$. It has a phase transition whose critical point is given explicitly in terms of the first effective diffusion constant $\gamma$ of the periodic homogenization problem. Using this result, we present explicit bounds for the critical point of the self-diffusion matrices of the strict two-dimensional Coulomb interacting Brownian motions for inverse temperature $\beta$. Moreover, we discuss the case d-dimensional case with d \ge 3.

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Date/Time : 21 Aug 2019 14:00
Venue : S17 #04-06 SR1
Speaker : Ma Xiaonan, Université Paris Diderot– Paris 7

We will review the different versions of ‘quantization commutes with reduction’ from Kaehler to symplectic manifolds. We will also consider the non-compact symplectic situation and our most recent result on Cauchy-Riemann version.

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Date/Time : 12 Jun 2019 14:00
Venue : S17 #04-06 SR1
Speaker : Weinan E, Princeton University

Modern machine learning has had remarkable success in all kinds of AI applications, and is also poised to change fundamentally the way we do physical modeling. In this talk, I will give an overview on some of the theoretical and practical issues that I consider most important in this exciting area. The first part of this talk will be focused on the fol-lowing question: How can we make use of modern machine learning tools to help build reliable and practical physi-cal models? Here we will address two issues (mostly using the example of molecular dynamics): (1) building machine learning models that satisfy physical constraints; (2) using microscopic models to generate the optimal data set.

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Date/Time : 15 Mar 2019 14:00
Venue : S17 #04-06 SR1
Speaker : Nicholas M. Katz, Princeton University

We will discuss recent joint work with Antonio Rojas Leon and Pham Huu Tiep about writing down explicit rigid local systems on the affine line in characteristic p whose monodromy groups are various “interesting” finite groups.

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Date/Time : 13 Mar 2019 14:00
Venue : S17 #04-06 SR1
Speaker : John Erik Fornaess, Norwegian University of Science and Technology

In complex analysis the most important domain is the unit disc. In fact all domains (at least simply connected and bounded) are biholomorphic, i.e. analytically equivalent, to the disc. In higher dimension, the natural analogue is the unit ball. But in higher dimension, the general domain is not biholomorphic to the ball. A basic question is then how well a general domain can be approximated by the ball. If we have a ball $B_r$ of radius $r<1$ contained in the unit ball $B_1$, then a domain $U$ with $B_r\subset U\subset B_1$ is said to be squeezed between the two balls. The larger we can choose $r,$ the closer the domain $U$ is to the ball. I will talk about this topic.

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Date/Time : 06 Mar 2019 14:00
Venue : S17 #04-06 SR1
Speaker : Georgios Daskalopoulos, Brown University

Loosely speaking, Holomorphic rigidity of Teichmueller space states that the action of mapping class group uniquely determines the Teichmueller space as a complex manifold. I will sketch the main ideas involved in proving such result. The method of proof is through harmonic maps and fits into the framework of geometric superrigidity. In addition, I may discuss a harmonic maps proof of both the high rank and the rank one superrigidity of the mapping class group proved via other methods by Farb-Masur and Yeung.

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Date/Time : 13 Feb 2019 14:00
Venue : S17 #04-06 SR1
Speaker : Yu Shih Hsien, National University of Singapore

A novel approach by combinations of classical mathematical analysis, probabilistic argument, and modern PDE together to give the fundamental solution of a heat equation with a bounded variation heat conductivity with a sharp 2nd order derivatives estimates.

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