Series: Mathematics Colloquium

Date: Thursday, September 27, 2001

Time: 4:30 - 5:30 PM

Place: 102 McAllister Building

Host: Sergei Tabachnikov

Refreshments: 4:00 - 4:30 PM, in 212 McAllister

Speaker: Valentin Ovsienko, Center for Theoretical Physics, Luminy

Title: Differential Operators and Differential Geometry

Abstract: 

Differential geometry was interpreted by Klein as a study of a Lie
group action on a manifold.  The basic spaces, such as spaces of
functions, tensor fields, etc are then viewed as modules over the Lie
group.  The choice of the group of symmetry characterizes the
geometrical context; this group can be small as the group of
isometries in Riemannian geometry or huge as the full group of
diffeomorphisms.  The aim of this talk is to discuss the geometric
properties of linear differential operators.  The first classical
example, the Sturm-Liouville operator, $d^2/dx^2 + u(x)$, is itself a
rich universe and was a subject of profound research (Kuiper, ...,
Kirillov, ...).  I will dwell on this and other examples in order to
give a overview of the whole field.  I will, finally, mention some
recent developments and new ideas and identify this field among other
branches of geometry and mathematical physics.