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Galois theory of parameterised linear differential equations and linear differential algebraic groups

Presented by: 
M Singer North Carolina State University
Date: 
Thursday 26th May 2005 - 10:30 to 11:30
Venue: 
INI Seminar Room 2
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.

University of Cambridge Research Councils UK
    Clay Mathematics Institute The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons