Speaker: Dr. Joshua Padgett, Department of Mathematical Sciences
Date: Wednesday, November 10, 2021, 4:00 PM – 4:50 PM, SCEN 604
Title: An exploration of approximating semigroups
Abstract: The concept of a semigroup was originally introduced in the early 20th century in an attempt to generalize results from group theory and also to study the multiplicative properties of an algebraic ring. However, after a century or so of intense study, we now realize that these objects naturally arise in other areas of mathematics. Of particular interest is the observation that, when this purely algebraic object is endowed with additional analytic structure, semigroups can be used to describe solutions of various differential equations posed in a variety of abstract settings. Due to this fact, the task of approximating solutions to differential equations can be reformulated into the problem of approximating (certain) semigroups. The goal of this talk is to introduce some basic ideas behind how semigroups arise in the study of differential equations and then explore how one can construct appropriate approximations to these semigroups. We will motivate the idea initially by focusing on the finite-dimensional setting (i.e., the case of the matrix exponential), but then touch on how things become more complicated in the more abstract setting (which includes partial differential equations). In addition, if time permits, we will discuss how approximations for differential equations on Lie groups and for differential equations posed in high-dimensional spaces need to be more carefully treated.
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