Marc Härkönen
(Advisor: Prof. Anton Leykin)
will defend a doctoral thesis entitled
Dual representation of polynomial modules with applications to partial
differential equations
On
Friday, April 15 2022 at 11:00 a.m.
Skiles 006, or online
https://gatech.zoom.us/j/95997197594?pwd=RDN2T01oR2JlaEcyQXJCN1c4dnZaUT…
Abstract
In 1939, Wolfgang Gröbner proposed using differential operators to represent ideals in a
polynomial ring. Using Macaulay inverse systems, he showed a one-to-one correspondence
between primary ideals whose variety is a rational point, and finite dimensional vector spaces
of differential operators with constant coefficients. The question for general ideals was left
open. Significant progress was made in the 1960's by analysts, culminating in a deep result
known as the Ehrenpreis-Palamodov fundamental principle, connecting polynomial ideals and
modules to solution sets of linear, homogeneous partial differential equations with constant
coefficients.
This talk aims to survey classical results, and provide new constructions, applications, and
insights, merging concepts from analysis and nonlinear algebra. We offer a new formulation
generalizing Gröbner's duality for arbitrary polynomial ideals and modules and connect it to the
analysis of PDEs. This framework is amenable to the development of symbolic and numerical
algorithms. We also study some applications of algebraic methods in problems from analysis.
Committee
• Prof. Anton Leykin, School of Mathematics (advisor)
• Prof. Josephine Yu, School of Mathematics
• Prof. Grigoriy Blekherman, School of Mathematics
• Prof. Matthew Baker, School of Mathematics
• Prof. Dr. Jonas Hirsch, Mathematisches Institut, Universität Leipzig