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An Isaac Newton Institute Programme

Model Theory and Applications to Algebra and Analysis

Galois Theory of Paramaterized Linear Differential Equations and Linear Differential Algebraic Groups

26th May 2005

Author: Singer, Michael (North Carolina State University)

Abstract

I will describe a Galois theory of differential equations of the form

dY/dx = A(x,t_1, ... , t_n) Y

where A(x,t_1, ... , t_n) is an m x m matrix with entries that are functions of the principal variable x and parameters t_1, ... , t_n. The Galois groups in this theory are linear differential algebraic groups, that is, groups of m x m matrices (f_{i,j}(t_1, ..., t_n)) whose entries satisfy a fixed set of differential equations. For example, in this theory the equation

dy/d/x = (t/x) y

has Galois group

G = { (f(t)) | (log f(t))'' = 0} .

I will give an introduction to the theory of linear differential algebraic groups and the above Galois theory and discuss the place of isomonodromic famillies in this theory and the relation to Pillay's differential Galois theory. This is joint work with Phyllis Cassidy.