Postdoc Talks/Lunch

Date: October 19, 2022

Time: 11:30AM - 12:45PM

Location: BLOC 302

Description: 11:30a.m.
Dr. Matthias Hofmann
Title: Clustering problems on quantum graphs

Abstract: We introduce quantum graphs and motivate their relevance in the context of networks. We compare known results for eigenvalue problems to counterparts for metric graphs. These eigenvalue problems can be related to competing species problems and can be described via spectral minimal partition problems, recently introduced in Kennedy et al [CVPDE 60 (2021), 61]. We present existence results for spectral minimal partitions and show sharp lower and upper estimates for various spectral minimal energies, estimates between these energies and eigenvalues of the Laplacian and discuss their asymptotic behaviour. Other ways to identify clusters in networks can be obtained by identifying nodal domains of eigenfunctions of appropriate operators. In this context we present Pleijel and Weyl-type theorems on the asymptotics of the number of nodal domains ν_n of the n-th eigenfunction(s) of a broad class of operators of Schrödinger type. Among other things, these results characterize the accumulation points of the sequence {ν_n/n}_(n∈N), which are shown always to form a finite subset of (0,1].

Abstract: This talk discusses generalized Nash equilibrium problems that are given by rational functions. Rational expressions for Lagrange multipliers and feasible extensions of KKT points are introduced to compute generalized Nash equilibria (GNEs). We give a hierarchy of rational optimization problems to solve rational generalized Nash equilibrium problems. The Moment-SOS relaxations are applied to solve the rational optimization problems.

Abstract: Entropy is an important numerical isomorphism invariant of pmp dynamical systems. Orbit equivalence is a relation between pmp dyna