# Informal Mathematical Physics Seminar

organized by Igor Krichever and Andrei Okounkov

Mondays, 5:30, Room 507

Note a special lecture course on D-modules by A. Braverman, on Fridays 12-1:30, Math 507

Schedule of talks for Fall 2014:

 Sept 2, 2:30-4:00 special day & time Jake Solomon A reverse isoperimetric inequality for J-holomorphic curves Sept 8 no seminar Sept 15 Pavel Etingof Tensors of rank $\pi$ Sept 22 Albrecht Klemm Stable pair invariants on K3 and Mathieu Moonshine Sept 29 no seminar Oct 6 Si Li Perturbative renormalization and Index theorem Oct 13 Roma Bezrukavnikov Coherent sheaves on the Hilbert scheme revisited Oct 20 Aaron Pixton Primitive tautological relations Oct 27 Galyna Dobrovolska Counting indecomposables over a finite field Nov 3 Andrey Smirnov Quantum difference equations for Nakajima varieties Nov 10 Roma Bezrukavnikov continues Nov 17 Changjian Su Restriction formulas for stable basis in T^*G/B Nov 24 Dmitry Orlov Smooth and proper noncommutative schemes and phantoms Dec 1 staff Monodromy of the quantum difference equations for Nakajima varieties Dec 8 Mina Aganagic AGT, Free Field Representations and Triality

A note to the speakers: this is an informal seminar, meaning that the talks are longer than usual (1:30) and are expected to include a good introduction to the subject as well as a maximally accessible (i.e. minimally general & minimally technical) discussion of the main result. The bulk of the audience is typically formed by beginning graduate students. Blackboard talks are are particularly encouraged.

Abstracts

Sept 2

The length of the boundary of a J-holomorphic curve with Lagrangian boundary conditions is dominated by a constant times its area. This inequality is useful for showing the adic convergence of power series arising from A-infinity algebras associated with Lagrangian submanifolds. It gives rise to an invariant of Lagrangian submanifolds called the optimal isoperimetric constant. And it proves the compactness of certain moduli of J-holomorphic curves with Lagrangian boundary conditions in toric Calabi-Yau threefolds. This is joint work with Y. Groman.

Sept 15

Let $t$ be a complex number, and $V$ a complex vector space. I will explain how to define the tensor power $V^{\otimes t}$. This can be done canonically if we fix a nonzero vector in $V$. However, the result is not a vector space but rather an (ind-)object in the tensor category ${\rm Rep}(S_t)$, defined by P. Deligne as an interpolation of the representation category of the symmetric group $S_n$ to complex values of $n$. This category is semisimple abelian for $t\notin \bold Z_+$, but only Karoubian (=idempotent complete) for $t\in \bold Z_+$, in which case it projects onto the usual representation category of $S_n$. I will define the category ${\rm Rep}(S_t)$, and explain how Schur-Weyl duality works in this category when $t \notin \bold Z_+$. If time permits, I will explain what happens at integer $t$, which is more subtle and is due to Inna Entova-Aizenbud.

Oct 6
In this talk, I will build up the equivalence between the deformation quantization on a symplectic manifold and the perburbative BV quantization of a Chern-Simons type theory on its loop space. This allows us to borrow symmetries of deformation quantization to analyze the correlation functions of quantum observables. As an application, I will show that the homotopy of a rescaling symmetry within our QFT model leads to a simple proof of algebraic index theorem. This is joint work with Qin Li.

Oct 20
The tautological ring of the moduli space of stable curves is a subring of the Chow ring consisting of the cycles that arise naturally in geometry through forgetful and gluing morphisms. Relations in this ring can be pulled back to give recursion relations in Gromov-Witten theory. I will introduce the notion of a "primitive" tautological relation, one that cannot be derived from simpler relations using certain basic operations. We know very little about primitive tautological relations in genus 2 or greater, but I will describe one interesting infinite family of them.

Oct 27
We describe a different proof of the theorem of Hausel, Letellier, and Rodriguez-Villages on counting indecomposable quiver representations over a finite field, using character sheaves. If time permits, we show how similar ideas can be used for counting indecomposable vector bundles on an algebraic curve.

Nov 17
In this talk, I will give the restriction formulas for stable basis in T^*G/B, and generalize them to T^*G/P case. This allows us to identify the quantum multiplication of a divisor in equivariant quantum cohomology of T^*G/P.

Nov 24
This is going to be an introductory talk where I would like to discuss different properties of
noncommutative schemes such as smoothness, regularity and properness. I am also going to talk about different examples of quasi-phantoms and phantoms.