Many effective physical theories are based on geometric structures that are, at least locally, quite simple. Was it inevitable that there would be simple geometric structures that expressed the relevant physical laws or is is unreasonable that we should be so lucky? Illustrations from Newton's Laws, Riemann's fundamental ideas, Einstein's general theory of relativity, Jesse Douglas' questions about the inverse problem, Lie's question about metrics representing a given set of `straight lines'. ## UNIVERSITY OF PENNSYLVANIA

## Department of Mathematics

Spring 2008 -Hans Rademacher Lectures in Mathematics

Robert Bryant

Duke University,

Mathematical Sciences Research Institute

&

University of California, Berkeleywill speak on

## Differential Equations and Geometric Structures

The unreasonable effectiveness of geometric methods

Tuesday....April 22, 2008.....4:00pm...A8 DRL

Geometric flows have been considered as an approach to constructing solutions of the special holonomy equations. In this talk, I will examine the nature of these `flows', in particular, their well-posedness and stability. Geometric flows and special holonomy

Wednesday....April 23, 2008....4:00pm...A6 DRL

[NOTE: Lecture will be broken into two 45-minute talks. The first 45 minutes will be of more general interest; the second 45 minutes will be more detailed.]

Several problems in differential geometry can be expressed as the problem of finding a so-called `Hessian representation' of a symmetric differential k-form. I will introduce these problems and discuss the problems in differential equations that they raise. which appear to require a number of different techniques to address. The Hessian representation problem

Thursday.....April 24, 2008....4:00pm...A8 DRL

[NOTE: Lecture will be broken into two 45-minute talks. The first 45 minutes will be of more general interest; the second 45 minutes will be more detailed.]

List of Previous Rademacher Lecturers Lectures will be held in the David Rittenhouse Laboratory,

S.E. corner of 33rd and Walnut Streets, Philadelphia, PA.

Tea: 4E17 DRL at 3:30pm.For further information, please call the Department of Mathematics at the University of Pennsylvania - 215-898-8627.