【Plenary Lecture】Kolmogorov Problem and Wiener-type Criteria for the Removability of the Fundamental Singularity for the Elliptic and Parabolic PDEs

Speaker:  Prof. Ugur Abdulla （OIST)

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

Abstract: Wiener's celebrated result on the boundary regularity of harmonic functions is one of the most beautiful and delicate results of 20th-century mathematics. It has shaped the boundary regularity theory for elliptic and parabolic PDEs and has become a central result in the development of potential theory at the intersection of functional analysis, PDE, probability and measure theories. 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 Markov processes, 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.