Finiteness Theorems in Polynomial Rings with an Infinite Number of VariablesAlgebra & Discrete Mathematics
|Speaker:||Christopher Hillar, MSRI|
|Start time:||Fri, Nov 7 2008, 2:10PM|
In the study of biological, chemical, and statistical models, one motivating problem is to determine the algebraic relations between experimental measurements. In this regard, Sturmfels has asked whether, up to symmetry, there are finitely many of them that generate the others. The main difficulty here is that the rings of interest usually have an infinite number of variables (so that we do not have traditional Noetherianity). We discuss some of the mathematics underlying this problem. In particular, we present a natural framework for studying Groebner bases in (subalgebras) of infinite dimensional polynomial rings. This framework allows us to prove finiteness theorems in many of the situations that arise in applications. We shall also discuss how these ideas suggest that computation is possible in this setting, and we present some specific examples highlighting the underlying combinatorial aspects to this theory. (Joint with Seth Sullivant).