Glazebrook to Talk on “La Belle E ́poque of ODEs” Today

Abstract follows:

Following extensive developments in the theory of differential equations due to Sturm, Liouville, Abel and others, there was a further surge in activity toward the end of the nineteenth century that created substantial breakthroughs into other areas of mathematics. Much of this was inspired by the French school(s) centered around H. Poincare and E. Picard who took the theory of DEs into the realm of algebraic geometry and topology, thus creating an area of mathematics studied to the present day. Prolific contributors at that time included Fuchs, Schlesinger, Garnier, and Painleve (the latter was the French Prime Minister 1917, 1925).

In much of this work one discovers the ubiquitous ‘tau-function’, most notably in Painleve’s theory of transcendents’ which I will briefly discuss in the context of second order nonlinear ODEs. This is a topic that continues to have far-reaching consequences in mathematical physics (wave equations and ‘integrable systems’), algebraic geometry and representation theory. There is also a remarkable geometric-functional analysis slant to this theory which can be studied within the framework of ‘Convenient Geometry’, about which I will provide some background and some explanation, and reveal that ‘tau’ actually possesses a startling geometric property.