Mathematics Colloquium
Stephen C. Preston
Brooklyn College and CUNY Graduate Center
Title: The Geometric Approach to Partial Differential Equations
Date: Friday, December 6, 2019
Place and Time: Room 101, Love Building, 3:35-4:25 pm
Refreshments: Room 204, Love Building, 3:00 pm
Abstract. I will give a survey of those PDEs that can be viewed as geodesics on infinite-dimensional spaces. Vladimir Arnold observed in 1966 that an ideal fluid can be viewed as a geodesic on the group of volume-preserving diffeomorphisms of a domain, and he computed some of the sectional curvatures, showing that many of them are negative. Since then many other equations have found interpretation as geodesics, including the equation for inextensible whips, the Korteweg-de Vries equation, and other conservative PDEs. I will describe some of the finite-dimensional models (including the equations for a rigid body) along with the general aspects of finite-dimensional Riemannian geometry and what still works in infinite dimensions. Finally I will show how a new one-dimensional model of the Euler equation shares many of the same properties and also ties into the Teichmuller theory in complex analysis.