Ritt Lectures: From Solitary Waves to Ubiquitous Symmetry

Date: April 18, 19

Columbia University, New York

Abstract:One of the surprises of modern mathematics is the appearance of the Korteweg-de Vries Equation in the organization of new invariants of symplectic manifolds X (usually called quantum cohomology). Certain special differential equations (a subset of those known as integrable) have appeared in the physics literature on topological conformal field theory for over a decade. Kontsevich's Fields Medal is at least partly based on his verification of the formulas for X a point which are based on algebraic structures used for solving Korteweg-de Vries equation. The appearance of these equations in quantum cohomology is further reflected in the well known "Virasoro Conjecture". This asserts that the quantum cohomological invariants are fixed points of symmetries consisting of half a Virasoro algebra. These algebras are known to act on many mathematical structures, in particular on the solution sets of most integrable equations. There is little speculation or conjecture as to the reason for this truly amazing and unlikely mating of two entirely different subjects of integrable systems and topological invariants.

On the other hand, the Korteweg-de Vries equations themselves, which appeared in the 19th century to describe solitary water waves, have already played several roles in mathematics. The first lecture will be devoted to an elementary description of some of these earlier appearances. The second lecture will be devoted to the Virasoro symmetries and one explanation of their appearance in integrable systems. One goal of the talk is to interest the audience in deeper questions about connections between these two subjects and topology.

The lectures are intended for a broad mathematical audience rather than for specialists. A good part of the first lecture should be comprehensible to anyone who understands some differential equations. There is a nice web site at Herriot Watt University in Edinburgh which has some fun information about solitons. In particular, one of their history pages gives some information about John Scott Russell ,who appears to have been a fascinating man. I also like the particular photo from their collection of a water wave soliton best, although the pictures of the pseudospherical surfaces corresponding to the solutions of the Sine Gordon equation from Richard Palais' gallery of surfaces are also fun.

An introductory article The Symmetries of Solutions by Richard Palais contains more interesting history., some of which I will repeat in the talk. An article written by Chuu-Lian Terng and myself which appeared two years ago in the Notices of the American Mathematical Society gives an introduction to one method of viewing inverse scattering. I will define and explain the Virasoro actions, which can be understood as acting through the inverse scattering transform. This article might be good preparation for my explanation of the Virasoro actions.