Analysis and PDE Seminar: Kolmogorov Problem and Wiener-type Criteria in Potential Theory

Event details Date: Monday, August 24, 2026 Time: 4:10 PM to 5:00 PM Location: 736 Evans Hall Type: Lecture / Workshop About this event Speaker: Ugur G. Abdulla, Okinawa Institute of Science and Technology The central problem in the Analysis of PDEs is understanding the nature of singularities that reflect natural phenomena. This talk will present a full characterization of the fundamental boundary singularity, and equivalently, the unique solvability of the singular Dirichlet problem for the elliptic and parabolic PDEs. The results are threefold. We prove a new Wiener-type criterion for the “geometric” characterization of the removability of the fundamental singularity for arbitrary open sets in terms of the fine-topological thinness of the complementary set near the singularity point. In the special case when the surface of revolution forms the boundary of the open set near the singularity point, we establish a Kolmogorov-Petrovsky-type test to characterize the removability of the singularity and uniqueness. Finally, in the special case when a continuous graph locally represents the boundary of the open set, the minimal thinness criterion for the removability of the singularity is expressed in terms of the minimal regularity of the boundary manifold at the singularity point. From the probabilistic point of view, the criteria present an asymptotic law for conditional Brownian motion. In the topological context, the criteria present a full characterization of the neighborhood base of the boundary singularity point in the minimal-fine topology. In the more general framework, the talk will outline a program for the full characterization of the singularities formed by the elliptic and parabolic PDEs. Official event details: https://events.berkeley.edu/math/event/327273-analysis-and-pde-seminar-kolmogorov-problem-and