Thu, 23 Oct 2014

12:00 - 13:00
L4

J.C. Maxwell's 1879 Paper on Thermal Transpiration and Its Relevance to Contemporary PDE

Marshall Slemrod
(University of Wisconsin - Madison)
Abstract
In his 1879 PRSL paper on thermal transpiration J.C.MAXWELL addressed the problem of steady flow of a dilute gas over a flat boundary. The experiments of KUNDT and WARBURG had demonstrated that if the boundary is heated with a temperature gradient , say increasing from left to right, the gas will flow from left to right. On the other hand MAXWELL using the continuum mechanics of his (and indeed our) day solved the ( compressible) NAVIER- STOKES- FOURIER equations for balance of mass, momentum, and energy and found a solution: the gas has velocity equal zero. The Japanese fluid mechanist Y. SONE has termed this the incompleteness of fluid mechanics. In this talk I will sketch MAXWELL's program and how it suggests KORTEWEG's 1904 theory of capillarity to be a reasonable “ completion” of fluid mechanics. Then to push matters in the analytical direction I will suggest that these results show that HILBERT's 1900 goal expressed in his 6th problem of passage from the BOLTZMANN equation to the EULER equations as the KNUDSEN number tends to zero in unattainable.
Tue, 10 Jun 2014

15:45 - 16:45
L4

What is the [Categorical] Weil Representation?

Shamgar Gurevich
(University of Wisconsin - Madison)
Abstract
The Weil representation is a central object in mathematics responsible for many important results. Given a symplectic vector space V over a finite field (of odd characteristic) one can construct a "quantum" Hilbert space H(L) attached to a Lagrangian subspace L in V. In addition, one can construct a Fourier Transform F(M,L): H(L)→H(M), for every pair of Lagrangians (L,M), such that F(N,M)F(M,L)=F(N,L), for every triples (L,M,N) of Lagrangians. This can be used to obtain a natural “quantum" space H(V) acted by the symplectic group Sp(V), obtaining the Weil representation. In the lecture I will give elementary introduction to the above constructions, and discuss the categorification of these Fourier transforms, what is the related sign problem, and what is its solution. The outcome is a natural category acted by the algebraic group G=Sp, obtaining the categorical Weil representation. The sign problem was worked together with Ofer Gabber (IHES).
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