Graduate Student Seminar

Speaker 1: Joshua Hudson

Title

Capture features of long time dynamics of a system using the framework of a global attractor

Abstract

When studying a differential equation, often you want to know where a solution will tend to eventually, and don't care much about what happens in the middle. Sometimes the system is so complicated we can't hope to track individual solutions; in which case, is there anything we can conclude about where solutions can't or will end up? To examine these issues further, we employ the framework of the global attractor, which captures the features of the long time dynamics of the system when considering all bounded sets of initial data. We'll make some necessary assumptions on the differential equation as our starting point, and from there build up to defining what the global attractor for a dynamical system is, and examine some of its mathematical properties. At the end, we'll use the Lorenz System as an example to illustrate what can result.

Speaker 2: Juyoung Jeong

Title

Commutation Principle: A connection between a spectral cone and a permutation invariant cone

Abstract

A spectral cone is the cone that preserves the spectrality, that is, for any matrix A in the cone and any orthogonal matrix U, U^TAU is also in the same cone. A cone is said to be permutation invariant provided it is invariant under permutation matrices. In this talk, we consider a commutation principle which makes a connection between a spectral convex cone in the space of symmetric matrices and its underlying permutation invariant convex cone in the space of vectors. Especially, extending concepts of triangle in Euclidean geometry, we show, by using the commutation principle, the incenter/circumcenter of the spectral cone can be expressed in terms of the incenter/circumcenter of the corresponding permutation invariant cone. Also, we prove inradii/circumradii of two cones are the same. The commutation principle can be generalized to Euclidean Jordan algebras.