Events for 04/22/2025 from all calendars

Nonlinear Partial Differential Equations

Time: 3:00PM - 4:00PM

Location: Blocker 302

Speaker: Ugur G. Abdulla, Okinawa Institute of Science and Technology, Japan

Title: Kolmogorov Problem and Wiener-type Criteria for the Removability of the Fundamental Singularity for the Elliptic and Parabolic PDEs

Abstract: This talk will address the major problem in the Analysis of PDEs on the nature of singularities reflecting the natural phenomena. I will present my solution of the "Kolmogorov's Problem" (1928) expressed in terms of the new Wiener-type criterion for the removability of the fundamental singularity for the heat equation. The new concept of regularity or irregularity of singularity point for the parabolic (or elliptic) PDEs is defined according to whether or not the caloric (or harmonic) measure of the singularity point is null or positive. The new Wiener-type criterion precisely characterizes the uniqueness of boundary value problems with singular data, reveal the nature of the harmonic or caloric measure of the singularity point, asymptotic laws for the conditional Brownian motion, and criteria for thinness in minimal-fine topology. The talk will end with the description of some outstanding open problems and perspectives of the development of the potential theory of nonlinear elliptic and parabolic PDEs.

Free Probability and Operators

Time: 4:00PM - 5:00PM

Location: BLOC 117

Speaker: Octavio Arizmendi, CIMAT

Title: The S-Transform in Free Probability

Abstract: An important analytical tool for computing the free multiplicative convolution of two probability measures is Voiculescu's S-transform. Several works have progressively extended its domain of applicability. It was first introduced by Voiculescu (1987) for distributions with nonzero mean and compact support, and later studied by Bercovici and Voiculescu (1993) in the case of probability measures on ℝ≥0 with unbounded support. Subsequently, Raj Rao and Speicher (2007) defined an S-transform for measures with zero mean and all moments finite. Their approach is combinatorial, using Puiseux series, and cannot be extended to the general case. Arizmendi and Pérez-Abreu (2009) considered the case of symmetric measures with unbounded support. In this talk, we will explain a recent result with Hasebe and Kitagawa, in which we extend Voiculescu's S-transform to the case of general probability measures on ℝ with possibly unbounded support. Our approach is analytical and is based on a reformulation using the T-transform, defined as the reciprocal of the S-transform. Unlike previous works, our construction does not require any assumptions on symmetry, mean, or boundedness of the support. This new definition generalizes previous approaches while preserving key properties such as the existence of the limit and non-vanishing in a natural domain.

Math Club Meeting : Movie Night: Hidden Figures

Time: 7:00PM - 9:00PM

Location: BLOC 306

Description: Three brilliant African-American women at NASA -- Katherine Johnson, Dorothy Vaughan and Mary Jackson -- serve as the brains behind one of the greatest operations in history: the launch of astronaut John Glenn into orbit, a stunning achievement that restored the nation's confidence, turned around the Space Race and galvanized the world. There will be refreshments provided, and the meeting will satisfy credit for MATH 170.