Informal Mathematical Physics Seminar
organized by Igor
Krichever and Andrei
Okounkov
Mondays, 5:30, Room 507
To sign up for dinner
click
here
Schedule of talks for Spring 2017:
=
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.
Seminar arxiv: Fall
2016 Spring
2016 Fall
2015 Spring
2015 Fall
2014 Spring
2014 Fall
2013 Spring
2013 Fall
2012 Spring
2012