Skip to content
Texas A&M University
Mathematics

Events for 10/08/2020 from all calendars

Mathematical Physics and Harmonic Analysis Seminar

iCal  iCal

Time: 10:00AM - 11:00AM

Location: Zoom

Speaker: Delio Mugnolo, University of Hagen

Title: Bi-Laplacians on graphs: self-adjoint extensions and parabolic theory

Abstract: Elastic beams have been studied by means of hyperbolic equations driven by bi-Laplacian operators since the early 18th century: several properties of the corresponding parabolic equation on the whole Euclidean space have been discovered since the 1960s by Krylov, Hochberg, and Davies, among others. On a bounded domain or a metric graph, the bi-Laplacian is generally not anymore acting as a squared operator, though: this strongly affects the features of associated PDEs.

I am going to present a full characterization of self-adjoint extensions of the bi-Laplacian, focusing on a class of realizations that encode dynamic boundary conditions. Maximum principles of parabolic equations will also be discussed: after a transient time, I am going to show that solutions often display Markovian features.

This is joint work with Federica Gregorio.


Groups and Dynamics Seminar

iCal  iCal

Time: 12:00PM - 1:00PM

Location: online

Speaker: Victor Guba, Vologda State University

Title: On the Ore condition for the group ring of R. Thompson's group F

Abstract: Let $R=K[G]$ be a group ring of a group $G$ over a field $K$. The Ore condition says that for any $a,b\in R$ there exist $s,t\in R$ such that $as=bt$ and $s\ne0$ or $t\ne0$. It always holds whenever $G$ is amenable. Recently it was shown that for R.\,Thompson's group $F$ the converse is also true. So the famous amenability problem for $F$ is equivalent to the question on the Ore condition for the same group. It is easy to see that the problem on the Ore condition is equivalent to the same question for the monoid ring $K[M]$, where $M$ is the monoid of positive elements of $F$. In this paper we reduce the problem for the case when $a$, $b$ are homogeneous elements of the same degree in the monoid ring. We study the case of degree $1$ and find minimal solutions of the Ore equation. For the case of degree $2$, we study the case of linear combinations of monomials from $S=\{x_0^2,x_0x_1,x_0x_2,x_1^2,x_1x_2\}$. We show that this set is not doubling, that is, there are nonempty finite subsets $X\subset M\subset F$ such that $|SX| < 2|X|$. In particular, Ore condition holds for linear combinations of these monomials. The case of monomials of higher degree is open as well as the case of degree $2$ for the monomials on $x_0,x_1,...,x_m$, where $m\ge3$. Recall that negative answer for one of these cases will immediately imply non-amenability of $F$.