Abstract

The existence of a height pairing on the equivalence relation defining Bloch's higher cycle groups is a surprising consequence of some recent joint work by myself and Xi Chen on a nontrivial K_1-class on a self-product of a general K3 surface. I will explain how this pairing comes about.

Abstract

Abstract

In 1757, Euler presented to the Berlin Academy of Sciences the basic equations of fluid mechanics.  As pointed out by V.I. Arnold in 1966, the Euler equations for incompressible fluids have a very simple geometric interpretation that combines the concept of geodesics and the concept of volume preserving maps.  The later concept is very simple and nothing but a continuous version of the discrete and more elementary concept of permutation.  Conversely, the Euler equations have a natural discrete counterpart in terms of permutation and combinatorial optimization, which establishes a direct link with the mathematical theory of "optimal transport". This theory, that goes back to Monge 1781 and has been renewed by Kantorovich since 1942, is nowadays a flourishing field with many applications, in natural sciences, economics, differential geometry and analysis.

About the Niven Lectures: Ivan Niven was a famous number theorist and expositor; his textbooks won numerous awards, have been translated into many languages and are widely used to this day.  Niven was born in Vancouver in 1915, earned his Bachelor's and Master's degrees at UBC in 1934 and 1936 and his Ph.D. at the University of Chicago in 1938.  He was a faculty member at the University of Oregon from 1947 until his retirement in 1982.  The annual Niven Lecture Series, held at UBC since 2005, is funded in part through a generous bequest from Ivan and Betty Niven to the UBC Mathematics Department.

Details

The Niven Lecture (11:30a.m.-12:30p.m.) precedes the Graduation Reception.

Abstract

 Motivated by seeking stationary solutions to the Euler equations with prescribed vortex topology, H.K. Moffatt has described in the 80s a diffusion process, called "magnetic relaxation", for 3D divergence-free vector fields that (formally) preserves the knot structure of their integral lines. (See also the book by V.I. Arnold and B. Khesin.)
The magnetic relaxation equation is a highly degenerate parabolic PDE which admits as equilibrium points all stationary solutions of the Euler equations. Combining ideas from P.-L. Lions for the Euler equations and Ambrosio-Gigli-Savar\'e for the scalar heat equation, we provide a concept of "dissipative solutions" that enforces first the "weak-strong" uniqueness principle in any space dimensions and, second, the existence of global solutions at least in two space dimensions.