Discrete models of geometry and the infinity algebras of topology - Dennis Sullivan

Dennis Sullivan (Stony Brook University/CUNY Graduate Center)
April 29, 2010

Special Colloquium in Honor of Dennis Sullivan Winning the 2010 Wolf Foundation Prize in Mathematics

Cauchy's angle defect at codimension two faces is an excellent discrete model of scalar curvature, but discrete models of Ricci and Riemann curvature have been elusive. Poincare's coboundary operator on cochains is a perfect discrete model of exterior d acting on forms but Steendrod and Whitney's cochain cup product as a model of wedge product requires symmetrization and higher corrections to be a good model. Lattice field translations along edges and the subsequent mondromies around two faces need the addition of higher terms in order to model bundles with connection. Poincare's dual cell constructions require higher terms to model the Hodge star operator. The yoga of infinity algebras from algebraic topology provides a language and a unifying concept of effective theories for these higher structures. In every instance the higher terms combine to form a solution of a Maurer Cartan equation. The hope is that such discretizations based on this concept of effective theories may be useful in geometry like finding good discrete models of the Ricci and Riemann curvature and in PDEs like generating coherent computational algorithms for fluid motion at every scale.