# John Harnad

Concordia University and Centre de recherches mathématiques

February 29, 2008 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **John Harnad (Concordia University and Centre de recherches mathématiques)**

What do the following have in common? - Irreducible characters of Lie groups (e.g., Schur functions) - Riemann's theta function on the Jacobian of a genus g Riemann surface - Deformation classes of random matrix integrals - Weights on path spaces of partitions, generating "integrable" random processes random tilings, and growth processes - Generating functions for Gromov-Witten invariants - Generating functions for classical and quantum integrable systems, such as the KP hierarchy (What have we left out? L-functions? Take their Mellin transforms.) In this talk, I will show how all the above may be seen as special cases of one common object: the "Tau function". This is a family of functions introduced by Sato, Hirota and others, originally in the context of integrable systems. They are parametrized by the points of an infinite dimensional Grassmann manifold, and depend on an infinite sequence of variables (t_1, t_2, ...), real or complex, continuous or discrete. They satisfy an infinite set of bilinear differential (or difference) relations, which can be interpreted as the Plucker relations defining the embedding of this "universal" Grassmann manifold into an exterior product space (called the "Fermi Fock space" by physicists) as a projective variety. This involves the "Bose-Fermi equivalence", which follows from interpreting the t-variables as linear exponential parameters of an infinite abelian group that acts on the Grassmannian and Fock space. A basic tool, which is part and parcel of the Plucker embedding, is the use of fermionic "creation" and "annihilation" operators. The tau function is obtained as a "vacuum state matrix element" along orbits of the abelian group. This is language that is familiar to all physicists, but little used by mathematicians, except for those, like Kontsevich, Witten, Okounkov (or, in earlier times, Cartan, Chevalley, Weyl), who know how to get good use out of it.

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