# Informal Mathematical Physics Seminar

organized by Igor Krichever and Andrei Okounkov

Mondays, 4:45, Room 520

Schedule of talks for Spring 2013:

 January 30 Ivan Ip Positive Representations of Split Real Quantum Groups February 4 Andrei Negut February 11 Paul Johnson Generating functions for Hilbert schemes of points on orbifold surfaces February 18 Sabin Cautis Clasp technology to knot homology via affine Grassmannian February 25 Paul Zinn-Justin Discrete holomorphicity and quantized affine algebras March 4 Anton Zabrodin March 11 Ivan Loseu March 25 Nicolai Reshetikhin March 26 Roman Bezrukavnikov 10:00 AM  Cherednik algebra in char p April 1 Richard Kenyon Dimers and integrability April 8 Grisha Mikhalkin April 15 Fredrik Johansson Viklund 5:00 special time ! On the geometry of random Loewner chains April 22 Jérémie Bouttier Distances in random planar maps and discrete integrability April 29 Anton Khoroshkin 4:00 special time !  Highest weight categories and Orthogonal polynomials May 6 Ivan Kostov Scalar product of Bethe vectors in XXX spin chain and domain wall partition functions May 13 Dario Beraldo On the chiral Whittaker category of the affine Grassmannian May 20 Sachin Gautam Tensor isomorphism between Yangians and quantum loop algebras

Abstracts

April 15. I will discuss Loewner's differential equation in connection with planar random growth models. The main focus will be on the SLE Loewner chains and some of their geometric properties. The talk is intended as an introduction and survey of the topic.

April 22. Metric properties of random maps (graphs embedded in surfaces) have
been subject to a lot of recent interest. In this talk, I will review
a combinatorial approach to these questions, which exploits bijections
between maps and some labeled trees. Thanks to an unexpected
phenomenon of discrete integrability'', it is possible to enumerate
exactly maps with two or three points at prescribed distances, and
more. I will then discuss probabilistic applications to the study of
the Brownian map (obtained as the scaling limit of random planar maps)
and of uniform infinite planar maps (obtained as local limits). If
time allows, I will also explain the combinatorial origin of discrete
integrability, related to the continued fraction expansion of the
so-called resolvent of the one-matrix model. Based on joint works with
E. Guitter and P. Di Francesco.

April 29.  I will explain the approach to the representation theory of
nonsemisimple Lie algebras with anti-involution
based on the notion of highest weight categories.
These notion formalises the existence of the BGG duality
first discovered by  Bershtein-Gelfand-Gelfand in the case of category O.
I will explain the necessary and sufficient condition for the
category of modules over nonsemisimple Lie algebra to be a highest weigh category.
As a corollary I will show that the characters of corresponding
Verma modules are Macdonald polynomials.

May 13. The goal of geometric Langlands is to express the category of D-modules on $Bun_G$ (the stack of $G$-bundles on a compact Riemann surface $X$) in terms of the stack of local systems for the Langlands dual group $\check{G}$.
It has been understood by physicists and mathematicians that this program could be achieved by local-to-global techniques, i.e. by quantum field theories. In this talk I wish to discuss a result in that direction: a \emph{chiral} equivalence between $Rep(\check{G})$ and the Whittaker category of the affine Grassmannian $Gr_G$. By definition, the latter category consists of those D-modules on $Gr_G$ that are $N(\!( t)\!)$-equivariant against a non-degenerate character and it coincides with the global Whittaker category defined by Frenkel-Gaitsgory-Vilonen.

May 20. The aim of this talk is to establish an explicit relation between
Yangians and quantum loop algebras, as {\em Hopf} algebras. More precisely, we will
show that a certain subcategory of finite--dimensional representations of the Yangian
is isomorphic, as a tensor category, to the category of finite--dimensional
representations of the quantum loop algebra.

The isomorphism between these two categories is governed by the monodromy of
an abelian difference equation. Moreover, the twist relating the tensor products
turns out to be a solution of an abelian version of the qKZ equations of Frenkel
and Reshetikhin.

These results are part of an ongoing project, joint with V. Toledano Laredo.