Overview
Title: Analyis and Numerics for the Kimura Diffusion Equation
Abstract: A standard forward model in population genetics is the Wright-Fisher model, which assumes random mating in a constant population. It models genetic drift, selection and mutation. An important infinite population limit is the Kimura diffusion equation. For a single gene with one 2 alleles the diffusion takes place on the unit interval, and with N alleles in an N-1 simplex. The 1-dimensional case is also important in the study of the allele frequency spectrum. The Kimura operator is degenerate at the boundary of the simplex, so its analysis falls outside of the standard elliptic/parabolic theory. I will discuss some of the analytic properties of the diffusion defined by this operator, joint with Rafe Mazzeo, and recent numerical work in the 1-dimensional case, joint with Jon Wilkening, which produces highly accurate solutions, even in very challenging situations.
Charles L. Epstein is a Senior Research Scientist in the Center for Computational Mathematics at the Flatiron Institute in New York. He is also the Thomas A. Scott Professor Emeritus at the University of Pennsylvania. Epstein was the founder and chair of the Graduate Group in Applied Mathematics and Computational Science at the University of Pennsylvania. His current research interests include Partial Differential Equations, Mathematical Physics, Population Genetics, Imaging Science, and Numerical Analysis; he has also worked in Hyperbolic Geometry, Univalent Function Theory, Several Complex Variables, Microlocal Analysis and Index Theory. He was a Sloan Foundation Fellow, and is a fellow of the American Association for the Advancement of Science, and the American Mathematical Society. He was a co-recipient of the Stefan Bergman Prize in 2016. He holds a Ph.D. and MS in Mathematics from New York University and an SB in Mathematics from the Massachusetts Institute of Technology.
The NSF-Simons National Institute for Theory and Mathematics in Biology Seminar Series aims to bring together a mix of mathematicians and biologists to foster discussion and collaboration between the two fields. The seminar series will take place on Fridays from 10am – 11am at the NITMB in the John Hancock Center in downtown Chicago. There will be both an in-person and virtual component.