Informal Mathematical Physics Seminar

organized by Igor Krichever and Andrei Okounkov

Mondays, 5:30, Room 507

Schedule of talks for Spring 2017:

 Feb 6 Changjian Su Chern-Schwartz-MacPherson classes, characteristic cycles and stable envelopes Feb 13 no seminar (?) Feb 20 Semen Artamonov A q,t-Integrable System on a Genus Two Surface Feb 27 Petr Pushkar Quantum K-theory of the Grassmannian and the Baxter Operator March 6 Shotaro Makisumi Modular Koszul duality for Kac–Moody groups March 13 spring break March 20 Barton Zwiebach 2:40 Pupin Hall Theory Center, 8th floor L∞ Algebras and Field Theory March 27 no seminar (?) April 3 Alexander Zamolodchikov CANCELLED April 5 Ivan Loseu Wednesday, Math 520 Modular category O for rational Cherednik algebras. April 10 Yuri Suris Pluri-Lagrangian structure as integrability of variational systems April 17 Ben Harrop-Griffiths The KdV equation and some generalizations April 24 Yakov Kononov Twist knots rectangular superpolynomials May 1 Noah Arbesfeld The index vertex, the refined vertex, and tautological classes on the Hilbert scheme of points
= note special day / time / place

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

February 6
The Chern-Schwartz-MacPherson (CSM) class theory is a Chern class theory for singular varieties, whose existence is established by MacPherson. Motivated by another construction of Ginzburg and the relation between the CSM classes of Schubert varieties and stable envelopes, we give another formula for the CSM classes for a general variety using the characteristic cycle associated to constructible functions. This is joint work with P. Aluffi, L. Mihalcea and J. Schuermann.

February 20
In my talk I will consider a quantum integrable system with two generic complex parameters q,t whose classical phase space is the moduli space of flat SU(2) connections on a genus two surface. This system and its eigenfunctions provide genus two generalizations of the trigonometric Ruijsenaars-Schneider model and Macdonald polynomials, respectively. I will show that the Mapping Class Group of a genus two surface acts by outer automorphisms of the algebra of operators of this system. Therefore this algebra can be viewed as a genus two generalization of A_1 spherical DAHA. (This is joint work with Shamil Shakirov)

February 27
In this talk I will define the quantum K-theory for Nakajima quiver varieties and show its connection to representation theory of quantum groups on the example of the Grassmannian. In particular, the Baxter operator will be identified by operators of quantum multiplication by quantum tautological classes. Quantum tautological classes will also be constructed and an explicit universal combinatorial formula for them will be shown. Based on a joint work with A.Smirnov and A.Zeitlin.

March 6
I will introduce a monoidal category of "free-monodromic tilting sheaves" on a Kac–Moody flag variety. The main result is that this category is equivalent to the monoidal category of parity sheaves on the Langlands dual Kac–Moody flag variety. This result implies the Riche–Williamson conjecture on characters of tilting modules of reductive groups. This is joint work with P. N. Achar, S. Riche, and G. Williamson.

March 20
I will review and develop the general properties of L∞ algebras focusing on the gauge structure of the associated field theories. Since these algebras also capture the structure of interactions, it is plausible that they can provide a classification of gauge invariant perturbative field theories.

April 5

We will explain some results regarding the modular representation theory (over a field of very large positive characteristic) of the type A rational Cherednik algebras. More specifically, we will introduce a modular analog of the category O and relate it to the usual characteristic 0 category O: roughly speaking, the latter is the associated graded of the former with respect to a suitable filtration. Time permitting (unlikely) we will also discuss wall-crossing functors. The talk is based on a work in progress. No prior knowledge of rational Cherednik algebras and their categories O will be assumed.

April 10
We propose a notion of a pluri-Lagrangian problem, which should be understood as an analog of multi-dimensional consistency for variational systems. This is a development along the line of research of discrete integrable Lagrangian systems initiated in 2009 by Lobb and Nijhoff, however having its more remote roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics and their quasiclassical limit, as well as in the theory of variational symmetries going back to Noether. We discuss main features of pluri-Lagrangian systems in dimensions 1 and 2, both continuous and discrete, along with the relations of this novel structure to more standard notions of integrability

April 17
The Korteweg-de Vries equation (KdV) is a nonlinear dispersive equation that arises as an asymptotic limit in numerous physical situations such as water waves and the propagation of waves in plasmas. In this talk we will introduce the KdV and several of its generalizations with an emphasis on the dynamical properties of solutions. If time, we will also introduce some quasilinear KdV equations with several remarkable properties.

April 24
We propose differential expansion formula for HOMFLY polynomials of the knot $4_1$ in arbitrary rectangular representation $R = [r^s]$ as a sum over all Young sub-diagrams $\lambda$ of $R$ with extraordinary simple coefficients $D_{\lambda^{t}}(r) D_\lambda(s)$ in front of $Z$-factors. Somewhat miraculously, these coefficients are made from quantum dimensions of symmetric representations of the groups $SL(r)$ and $SL(s)$ and restrict summation to diagrams with no more than $s$ rows and $r$ columns. They possess a natural $\beta$-deformation to Macdonald dimensions and produces positive polynomials, which can be considered as plausible candidates for the role of the rectangular superpolynomials. Both polynomiality and positivity are non-evident properties of arising expressions, still they are true. This extends the previous suggestions for symmetric and antisymmetric representations to arbitrary rectangular representations. As usual for differential expansion, there are additional gradings. In the only example, available for comparison -- that of the trefoil knot $3_1$, to which our results for $4_1$ are straightforwardly extended, -- one of them reproduces the "fourth grading" for hyperpolynomials. Factorization properties are nicely preserved even in the 5-graded case. We also discuss generalization to arbitrary twist knots.

May 1
The integrals of tautological classes of sheaves on the Hilbert scheme of points on a surface often arise in enumerative problems. We explain an approach to studying K-theoretic analogues of such expressions.
Namely, we review properties of the index vertex of Nekrasov and Okounkov used in their study of K-theoretic Donaldson-Thomas partition functions. We then explain the relation, due to Nekrasov and Okounkov, between their index vertex and the refined topological vertex of Iqbal, Kozçaz, and Vafa, and show how these results imply certain symmetries of generating series of equivariant Euler characteristics of tautological classes on the Hilbert scheme.