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

To sign up for dinner click here

Schedule of talks for Spring 2015:

Feb 2 Nick Proudfoot Intersection cohomology and quantum cohomology of conical symplectic resolutions
Feb 9, 10:10 Math 417 Ivan Loseu Categories O over quantized symplectic resolutions
Feb 9 David Jordan  How to integrate a quantum group over a surface
Feb 11, 10:10 Math 417 Ivan Loseu continues
Feb 16, 4:15
special time !
Allen Knutson Conormal varieties and the Temperley-Lieb algebra
Feb 23, 4:15-5:15,
Math 507
Alessandro Chiodo
special AG seminar
Feb 23
Leonid Chekhov
Chewing gums: degenerations of Riemann surfaces with holes; new laminations and new algebras of geodesic functions
March 2 Boris Khesin Pentagram maps and nondegenerate curves
March 9, 10:10 Math 417 Ivan Cherednik DAHA and torus knots I,II
March 9 Philippe Di Francesco first Minerva lecture (slides:lecture 1, lecture 3)
March 11, 10:10 Math 417 Ivan Cherednik continues
March 16 Spring Break  no seminar
March 23
Ivan Corwin
Stochastic quantum integrable systems
March 30 Andrei Negut Hilbert schemes, R-matrices and combinatorics
April 6 Galyna Dobrovolska Indecomposable objects and potentials over a finite field (I)
April 13 Galyna Dobrovolska Indecomposable objects and potentials over a finite field (II)
April 20, 10:10 Math 417 Andrei Negut continues, note special time
April 20 4:10-6:00,
Fabio Toninelli First Minerva Lecture, Height fluctuations in interacting dimers
April 27 Shamil Shakirov Refined Chern-Simons theory in genus two
May 4 Alina Marian Two- and four-dimensional instantons
May 11 Dmitry Galakhov Spectral Networks with Spin

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.


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. 

David Jordan

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.

Feb 15

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.

Feb 23

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.

March 2

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.

Ivan Cherednik

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).

March 23

 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.

March 30

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

April 6

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.

April 13

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.

April 20

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. 

April 27

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.

May 4
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.

May 11

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