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

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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

Feb 2

Given a symplectic resolution, Michael McBreen and I have conjectured that (roughly speaking) the q=1 specialization of the quantum cohomology of the resolved space is isomorphic to the intersection cohomology of the singular space. I will explain this conjecture in detail, and give some indication of how it is proved in certain examples.

Ivan Loseu

I will introduce categories O over quantization of symplectic resolutions that have a Hamiltonian torus action with finitely many fixed points following Braden, Licata, Proudfoot and Webster. Such a category depends on the choice of a generic one-parameter subgroup in the torus. It comes with a highest weight structure. I will introduce necessary definitions and then discuss two recent results of myself. First, that choosing a suitable non-generic one-parameter subgroup, one gets a standartly stratified structure on the category in the sense of Webster and myself. This structure will be used to establish the second main result: an action of a fundamental groupoid of a suitable hyperplane arrangement by derived equivalences on categories O corresponding to different choices of generic one-parameter subgroup. This proves a conjecture by Braden, Licata, Proudfoot and Webster.

We can think of the representation theory of the quantum group as
being assigned to a little patch of the plane, and understand the
tensor product as modelling collisions, and the braiding as invariance
of collisions under isotopies. A natural question then is: what
sort of gadget should be assigned to other surfaces?

This question is answered by the factorization homology of the
surface, which "integrates" the quantum group assigned to each little
patch into a new algebraic structure. I'll explain that to the
annulus, punctured torus, and closed torus this assigns the reflection
equation algebra, quantum differential operator algebra, and double
affine Hecke algebra, respectively, and I'll explain how this gives
new insights into things like quantum Fourier transforms. This
is joint work with David Ben-Zvi and Adrien Brochier.

If there's time/interest, I can discuss that this is part of a new
four-dimensional TFT we call Betti geometric Langlands, and can
discuss its extension to 3-manifolds in relation to quantum
A-polynomials and DAHA-Jones polynomials. That is also joint
with Noah Snyder.

The equations defining a matrix Schubert variety X_pi (in M_n, that
is, not GL_n/B) are the fairly obvious determinants [Fulton '92], and
one can use them as a Gr\"obner basis to degenerate X_pi to a union of
coordinate spaces, one for each reduced subword with product pi of the
"square word" in S_{2n} [K-Miller '05]. In particular all components
appear with multiplicity 1.

If we soup up X_pi to its Lagrangian conormal variety, and extend the
degeneration to a symplectic one on the cotangent bundle to M_n, the
components are now indexed by subwords of the square word in
Temperley-Lieb generators, and the multiplicity of a component is
2^{#loops}. Caveat: this only works if pi has a well-defined
Temperley-Lieb element associated, which is the condition that pi be
321-avoiding.

This is joint work with Paul Zinn-Justin, inspired by Maulik and
Okounkov's work on the stable basis of T^*(Gr(k,n)), which corresponds
to the case pi Grassmannian, and giving implications for the unstable
basis of conormal varieties.

We consider degenerations of Riemann surfaces occurring when
colliding two holes: we obtain laminations of new type that comprise
both (geodesic) lines ending at the bordered cusps obtained upon this
reductions, which are just cluster variables, or lambda-length, and
closed geodesic lines in the non reduced part of the Riemann surface.
The explicit coordinatization is provided by special limits of shear
coordinates on the original Riemann surface. We demonstrate how
Ptolemy relations are obtained in the limit when perimeters of two
holes go to infinity and give an explicit combinatorial description of
the obtained geodesic functions comprising both lines ending at cusps
and closed geodesic lines. Having at least one border cusp, we
establish an explicit algebraic relations between cluster varieties
and shear coordinates, which

provides a proof of the Laurent and positivity properties for these
cluster varieties. We obtain Poisson and quantum algebras of geodesic
functions and formulate the proper quantum ordering for the
corresponding cluster varieties. As an example we derive the quantum
algebra of general monodromy matrices for Dubrovin--Schlesinger
systems.

I will present all necessary definitions and notions during the talk.

Based on the work in preparation with M.Mazzocco.

A plane curve is called nondegenerate if it has no inflection points.
How many classes of closed nondegenerate curves exist on a sphere? We
are going to see how this geometric problem, solved in 1970,
reappeared along with its generalizations in the context of the
Korteweg-de Vries and Boussinesq equations. Its discrete version is
related to the 2D pentagram map defined by R.Schwartz in
1992. We will also describe its generalizations, pentagram maps on
polygons in any dimension and discuss their integrability properties.

This is a joint work with Fedor Soloviev.

These lectures will be devoted to DAHA-Jones polynomials (refined,
with an extra parameter) of iterated torus knots, including all
algebraic knots, with the focus on the so-called DAHA
superpolynomials, presumably coinciding with the Poincare polynomials
of the HOMFLYPT homology for algebraic knots

(equivalently, stable Khovanov-Rozansky polynomials). In string
theory, they are associated with the BPS states (the M_5-theory).

The first lecture will be mainly on DAHA of type A_1, including the
calculation of the refined Jones polynomial for trefoil from scratch.
This calculation can be readily extended to uncolored
DAHA-superpolynomials for the torus knots T(2n+1,2), coinciding with
those due to Evgeny Gorsky and others in terms of the rational DAHA.
This connection will be described in detail, though it remains
mysterious in spite of using (different) DAHA in both theories.

At the end of the first lecture, we will discuss the refined Verlinde
algebras, one of the most impressive application of DAHA at roots of
unity. They provide a priori link of the DAHA-Jones polynomials to
topology and actually were the starting point for Mina Aganagic and
Shamil Shakirov (their paper triggered my

approach). There is an interesting NT direction here, where PSL(2,Z)
and torus knots are replaced by the absolute Galois group (which will
be touched a bit).

The second lecture will be about general theory for arbitrary root
systems, with the main application in type A (the construction of
superpolynomials) and BCD (the hyperpolynomials, generalizing the

Kauffman polynomials in topology). A surprising application is the
justification of the formula for colored superpolynomials for
T(2n+1,2) suggested by physicists, using the theory of DAHA oftype
C-check-C_1 (directly related to the covers of CP^1 with 4 punctures).

If time permits we will discuss the iterated knots and plane curve
singularities. Conjecturally, DAHA provides the formulas for Betti
numbers of Jacobian factors of the latter, which are very difficult to
calculate in algebraic geometry. This is a great test of the maturity
of the new theory, closely related to the Oblomkov - Rasmussen -
Shende conjecture, which extends the OS-conjecture, generalized and
proved by Davesh Maulik. I will state the ORS conjecture (modulo the
definition of the weight filtration).

We describe recent work involving interacting particle systems
related to U_q(sl2) quantum integrable systems. This theory serves as
an umbrella for exactly solvable models in the Kardar-Parisi-Zhang
universality class, as well as provides new examples of such systems,
and new tools in their analysis.

In this talk, I wilk review the Maulik-Okounkov stable basis, which gives rise to R-matrices for various geometric integrable systems. Together with some of the attendees of this seminar, we've been discussing how to compute the change of stable basis matrix. I will review the setup of the problem in this talk, and explain how it connects to the various slope subalgebras of quantum toroidal gl_1

I report on a joint work with V. Ginzburg and R. Travkin, in which we prove a formula for an exponential sum over the set of absolutely indecomposable objects of a category satisfying certain conditions over a finite field in terms of geometry of the cotangent stack to the moduli stack of all objects of this category. In this talk I will concentrate on the case of the category of representations of a quiver with potential, reviewing the results of Crawley-Boevey and van den Bergh, as well as Hausel, Letellier, and Rodriguez Villegas, and considering in detail the example of the Calogero-Moser quiver with two vertices, one edge between by them, and a loop at one of the vertices.

I report on a joint work with V. Ginzburg and R. Travkin, in which we prove a formula for an exponential sum over the set of absolutely indecomposable objects of a category satisfying certain conditions over a finite field in terms of geometry of the cotangent stack to the moduli stack of all objects of this category. In this talk I will consider the general setting for the theorem, as well as its applications to counting absolutely indecomposable vector bundles with a parabolic structure on a projective curve and to counting irreducible l-adic local systems.

Perfect matchings of Z^2 (also known as non-interacting dimers on the square lattice) are an exactly solvable 2D statistical mechanics model. It is known that the associated height function behaves like a massless gaussian field, with the variance of height gradients growing logarithmically with the distance (see e.g. Kenyon, Okounkov, Sheffield '06). As soon as dimers mutually interact, the model is not solvable any more. However, tools from "constructive field theory" allow to prove that, as long as the interaction is small, the height field still behaves like a gaussian log-correlated field. In these 4 hours, I will try to explain the main ideas and some mathematical tools of the proof (Grassmann representation for the partition function, plus some ideas of "constructive renormalization group"). Work in collaboration with A. Giuliani and V. Mastropietro.

We will describe a representation of the mapping class group of a genus 2 surface, built of Macdonald polynomials. This is a deformation (refinement) of the Chern-Simons TQFT representation, and gives a deformation of Reshetikhin-Turaev knot invariants in genus 2. The corresponding knot operators form a genus 2 analog of the elliptic Hall algebra.

It is known that two-dimensional Yang-Mills theory, giving the universal intersection products of the moduli space of vector bundles on a smooth projective curve C, is entirely captured by a class of enumerative counts of holomorphic maps from C to suitable Grassmannians. In the talk I will describe the main ideas involved, and will further speculate that an analogous picture should hold in four dimensions. This would allow instanton counts on a ruled surface over C to be recovered from a theory of maps of C into the affine Grassmannian.

The spectral network technique give a nice approach to a study of Donaldson-Thomas invariants of Calabi-Yau threefolds and homological theory of semi-stable quiver representations via behavior of certain differential equation solutions. These theories are known to perform a wall-crossing phenomena when the invariants start to change discontinuously in the space of stability parameters across certain co-dimension 1 surfaces. We refine combinatorics of spetral networks with an extra grading to get so called motivic wall-crossing formulae and derive new difference equations for quiver Poincare polynomial generating functions. This refinement can also be seen as a "quantization" of the WKB method to solve ordinary differential equations.

Seminar arxiv: Fall
2014 Spring
2014 Fall
2013 Spring
2013 Fall
2012 Spring
2012