2020-2021

Fall 2020

The analysis seminar will be held online, typically on Thursdays at 3-4PM

Ihsan Topaloglu (Virginia Commonwealth University)

Thursday October 8 at 3-4PM via Zoom

Minimality of polygons in a nonlocal anisotropic isoperimetric problem

In this talk I will discuss the minimization of an energy functional given by the sum of a crystalline perimeter and a nonlocal interaction of Riesz type, under volume constraint. I will show that, in the small mass regime, if the polygonal Wulff shape of the anisotropic perimeter has certain symmetry properties, then it is the unique global minimizer of the total energy. I will also present a rigidity result for the structure of (local) minimizers in two dimensions. This is a joint work with M. Bonacini and R. Cristoferi.

Hyogo Shibahara (University of Cincinnati)

Thursday October 22 at 3-4PM via Webex

Gromov-Hausdorff distance with boundary and its application to Gromov hyperbolic spaces and uniform spaces

Bonk, Heinonen, and Koskela proved the connection between Gromov hyperbolic spaces and uniform spaces by considering suitable metric deformations, uniformization and quasihyperbolization. In this talk, we will discuss the stability of this connection under convergence of metric spaces. Since uniform spaces are noncomplete metric spaces, we will introduce a notion of Gromov-Hausdorff distance with boundary for a certain class of noncomplete metric spaces, which allows us to distinguish two metric spaces whose completions coincide with each other.

Leonid Slavin (University of Cincinnati)

Thursday October 29 at 3-4PM via TBA

A sharp BMO-BLO bound for the martingale maximal function

We construct the exact Bellman function for the BMO-BLO action of the natural martingale maximal function for continuous-time martingales. (BMO and BLO stand for "bounded mean oscillation" and "bounded lower oscillation," respectively; a natural maximal function is the one without the absolute value in the average). As consequences, we show that the BMO-BLO norm of the operator is 1 and also obtain a sharp weak-type inequality, which can be integrated to produce a broad range of sharp phi-estimates.

In an earlier work we found the corresponding Bellman function for alpha-regular discrete-time martingales, including the dyadic martingale. I will discuss the essential differences between the two cases. This is joint work with Adam Osekowski and Vasily Vasyunin.

University of Cincinnati: Analysis and PDE Seminar (2019-2020)

If you are interested in giving a talk contact Gareth Speight (Gareth.Speight<at>uc.edu).

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