Informal Mathematical Physics Seminar

organized by Igor Krichever and Andrei Okounkov

Mondays, 5:30, Room 507 (except Oct. 21 and 28, when it will meet in room 622, 6pm-7:30pm)

Schedule of talks for Fall 2013:

Sept 23
Lie algebras and Kac polynomials
Oct 7, 6:00
Davesh Maulik
Motivic residues (note special time!)
Oct 14
Maria Angelica Cueto
Faithful tropicalization of the Grassmannian of planes
Oct 21, 6:00 in 622
Hiraku Nakajima
Kazhdan-Lusztig conjecture via quasi-maps and Uhlenbeck spaces
Oct 28, 6:00 in 622
Sachin Gautam
Analytic construction of the R-matrix of Yangians
Nov 11
Clément Hongler
Planar Ising model with magnetic field and E8 structure
Nov 18
Daniel Halpern-Leistner
Variation of geometric invariant theory quotients and autoequivalences of derived categories
Nov 25
Sachin Gautam
From dynamical difference equations to Casimir connections (after M. Balagović)
Dec 2
Daniel Shenfeld
Abelianization of Stable Envelopes in Symplectic Resolutions


October 14

Fix a complete non-Archimedean valued field K. Any subscheme X of (K^*)^n can be "tropicalized" by taking the (closure) of the coordinate-wise valuation. This process is highly sensitive to coordinate changes. When restricted to group homomorphisms between the ambient tori, the image changes by the corresponding linear map. This was the foundational setup of tropical geometry.

In recent years the picture has been completed to a commutative diagram including the analytification of X in the sense of Berkovich. The corresponding tropicalization map is continuous and surjective and is also coordinate-dependent. Work of Payne shows that the Berkovich space X^an is homeomorphic to the projective limit of all tropicalizations. A natural question arises: given a concrete X, can we find a split torus containing it and a continuous section to the tropicalization map? If the answer is yes, we say that the tropicalization is faithful. The curve case was worked  out by Baker, Payne and Rabinoff.

In this talk, we show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in Berkovich sense. Our proof is constructive and it relies on the combinatorial description of the tropical Grassmannian as a space of (generalized) phylogenetic trees by Speyer-Sturmfels.  We also show that both sets have piecewiselinear structures that are compatible with our homeomorphism.

This is joint work with M. Haebich and A. Werner (arXiv:1309.0450).

October 21

We construct integral forms of enveloping algebras, W-algebras
etc on equivariant intersection cohomology groups of quasi-maps, Uhlenbeck
spaces, etc. Then their character formulae are deduced from intersection
cohomology groups of fixed point sets.

October 28

The R-matrix R(u) was constructed by Drinfeld as a formal series in 1/u with coefficients from the tensor square of a Yangian. It satisfies the Yang--Baxter equation.

In this talk I will discuss the analytic properties of this formal series once evaluated on a pair of finite-dimensional representations. The results presented in this talk were obtained jointly with V. Toledano Laredo

November 11

The Ising model is a simple and natural model for ferromagnetism. In two dimensions, it has many symmetries, on lattice level,  and in the continuum limit (by Conformal Field Theory, Schramm-Loewner Evolution). It was suggested by Al. Zamolodchikov that the 2D Ising model with a magnetic field can be formulated in terms of an scattering theory with 8 types of particles. Numerical simulations are in perfect agreement with Zamolodchikov's prediction and recently there was experimental confirmation of this claim.

I will give an overview about what is known about this story, which involves complex analysis and probability, Conformal Field Theory (minimal models, W-algebras, coset models, etc), S-Matrix theory. Then I will outline what seems to be a reasonable strategy to make rigorous sense of it.

Based (mostly) on joint work in progress with Kalle Kytölä.

November 18

Geometric invariant theory provides a method of taking a variety with a reductive group action and producing a new "quotient" variety. I will describe a situation in which the derived category of coherent sheaves of the resulting GIT quotient has autoequivalences which don't come from automorphisms of the variety itself. These autoequivalences are always of a special form generalizing the notion of a spherical twist.

December 2

Stable envelopes, introduced by Maulik and Okounkov, form a basis for the equivariant cohomology of symplectic resolutions. We study the case of Nakajima quiver varieties, and relate the stable basis to that of the associated quotient by a maximal torus, obtaining a formula for the transition between the stable basis and the fixed point basis. As an application, we compute the transition matrix explicitly for the Hilbert scheme of points on the plane and obtain a formula expressing Schur polynomials in terms of Jack polynomials.