Speaker: Dr. Joshua Padgett, Department of Mathematical Sciences
Date: Wednesday, March 03, 2021, 4:00 PM – 5:00 PM
Title: Lecture Series on Pure and Applied Mathematics, Lecture #2: Algebra 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) abstract algebra, and (iii) geometry (this has changed since the first advertisement).
In this second talk, we will consider various aspects of overlap between abstract algebra and applied mathematics. As before, the emphasis will be on techniques which are useful in numerical analysis and computational mathematics. John Butcher demonstrated in the 1970s that the order theory and analysis of Runge-Kutta methods involve the manipulation of functions defined on rooted trees. This observation yielded a beautiful combinatorial technique for defining and analyzing numerical integration methods. Since Butcher’s original work, much has been learned regarding these techniques and has even allowed for the defining of numerical methods on manifolds (e.g., via Lie-Butcher series). In this talk, our focus will be on a simple motivation of the aforementioned method of Butcher by carefully demonstrating its use in the case of Taylor expansions. This will involve the introduction of the set of rooted trees (both non-planar and planar) and careful study of their associated binary operations. We will extend this to numerical methods and then demonstrate the surprising fact that these objects form a group. Depending upon time and interest of the audience, we will also outline some more interesting algebraic details such as the Hopf algebraic structure or Rota-Baxter algebraic of numerical integration methods. At the end of the talk we will present more interesting avenues of research in this direction and discuss how these algebraic properties have been integral in developing a deeper understanding of numerical integration methods for stochastic vector fields and vector fields on Lie groups.
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