Speaker: Dr. Joshua Padgett, Department of Mathematical Sciences
Date: Wednesday, February 03, 2021, 4:00 PM – 5:00 PM
Title: Lecture Series on Pure and Applied Mathematics, Lecture #1: Analysis and Applied Mathematics
Abstract: The purpose of this lecture series is to illustrate some of the common threads which exist between applied mathematics and various areas in pure mathematics. We will consider three broad areas of intersections: i) analysis, (ii) algebra and geometry, and (iii) topology.
In this first talk, we will consider various aspects of overlap between analysis and applied mathematics — with a particular emphasis on the connections to numerical analysis and computational mathematics. There has been a long history of using classical and functional analysis in applied mathematics (cf., e.g., the classical textbook by Kato), however, sometimes this history or the fact that there are still numerous open problems in this direction can be overlooked. This talk will primarily focus the use of semigroup and operator theory in computational mathematics (but we may mention other directions, if time permits). We will illustrate this use by considering the issues of stability of numerical time-stepping methods, rigorous analysis of operator splitting methods, and determining the existence of certain types of spectra for self-adjoint operators. This latter example is especially useful in the study of the so-called Anderson localization problem in plasma physics.
In order to make these talks accessible to a large audience, we will start by motivating each situation with a specific example and then begin the abstraction process after exploring the shortcoming of standard approaches. These talks may also be of interest to pure mathematicians, as they may demonstrate novel directions of applications for pure mathematics research.
Bio: Josh Padgett is an Assistant Professor in the Department of Mathematical Sciences at the University of Arkansas. Prior to joining the University of Arkansas, Josh was a postdoc in the Department of Mathematics and Statistics at Texas Tech University. Josh earned his Ph.D. in mathematics at Baylor University under the supervision of Qin Sheng. His undergraduate research focused on the metabolic features of tumor cells and the occurrence of the so-called Warburg effect. Josh is also an affiliated faculty member at the Center for Astrophysics, Space Physics, and Engineering Research and is a honorary adjunct faculty at Texas Tech University. His research lies at the intersection of pure, applied, and computational mathematics. Current research interests include applied mathematics, numerical analysis, geometric and Lie group integration methods, mathematics of deep learning, operator splitting methods, algebraic structures of numerical methods, fractional differential equations, stochastic differential equations, and the use of spectral and operator theory in theoretical and experimental physics.
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