Back to Departmental Colloquium: Fall 2008
Departmental Colloquium
Date: Thursday, Oct 9, 2008
Time: 4:15PM
Location: JWB 335
Jason Starr
Stony Brook
Title |
The Weak Approximation Problem |
Abstract |
Given a system of polynomial equations in several variables and in 1 parameter, does there exist a “rational solution”, i.e., a family of solutions which is a rational function (fraction of polynomials) in the parameter? Do there exist enough rational solutions to approximate every power series solution in the parameter to arbtirary order? The first problem, or rather the problem of answering the first problem, is Hilbert’s 10th problem for $\mathbb{C}(t)$. It is expected there is no algorithm to answer the first problem. The second problem, the “Weak Approximation Problem”, conjecturally has a very simple answer: there are enough rational solutions precisely if after substituting a general value for the parameter, the corresponding system is “rationally connected”, i.e., every pair of solutions are common members of a family of solutions which are the output of a rational function. I will discuss the topological and number theoretic motivation of this conjecture, the evidence for the conjecture due to Hassett – Tschinkel, Hassett, Knecht and Colliot-Th'el`ene – Gille, and a new approach of Mike Roth and myself putting this conjecture in the larger context of “algebro-geometric analogues of topological obstruction theory”. |
Commutative Algebra
Topology