Informal Mathematical Physics Seminar

organized by Igor Krichever and Andrei Okounkov

Mondays, 5:30, Room 507

Schedule of talks for Spring 2016:

 Jan 25 Michael McBreen P=W for nodal curves, in rank one Feb 1 Michael Finkelberg Towards a cluster structure on trigonometric zastava Feb 8 Hiraku Nakajima Cherkis' bow varieties and Coulomb branches of quiver gauge theories of affine type A Feb 15 11:40 Felix Janda note special time Double ramification cycles Feb 15 Yongbin Ruan Nonabelian gauged linear sigma model Feb 17 11:40 Emily Clader note special day & time Pixton's double ramification cycle relations Feb 22 Vasily Pestun Quiver W-algebras Feb 29 Marta Mazzocco Quantum cluster algebras from geometry Mar 7 Alexander Bobenko Quadrilateral surfaces, see also an image Mar 14 no seminar spring break Mar 21, 11:40 Ivan Loseu note special time Deformations of symplectic singularities and the orbit method Mar 21 Scott Sheffield Minerva lecture Universal Randomness in 2D Mar 23, 11:40 Ivan Loseu note special day & time Combinatorial wall-crossing Mar 28 Emil Prodan Fitting the topological insulators in Alain Connes' noncommutative geometry program April 4 Qi You Categorification of small quantum groups April 11 Petr Pushkar Bethe Ansatz, Baxter Operator and the Vertex on the Grassmannian April 18 Ivan Danilenko DAHA and iterated torus knots April 20, 11:40 Julia Pevtsova note special day & time Support, cosupport and tensor ideals in modular representation theory April 25 Dmitry Korb Dynamical Weyl Group and resolutions of Affine Grassmanians April 27, 10:00 Pavlo Gavrilenko note special day & time The isomonodromy tau-function and conformals blocks May 2 Andrey Smirnov Rationality of capped descendants in quantum K-theory of Nakajima varieties

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

January 25
Joint work with Zsuzsanna Dancso and Vivek Shende. The non-abelian Hodge correspondence identifies local systems on a smooth curve with Higgs bundles on the same curve. According to the P=W conjecture, this
correspondence sends the weight filtration on the moduli of local systems to the perverse filtration on the higgs moduli. I will formulate an analogous conjecture for nodal curves with rational components, in rank one. Time permitting, I will sketch a proof.

February 1
This is a joint work with A.Kuznetsov and L.Rybnikov. We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety. We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava.

February 8
I will report the on-going joint project with Yuuya Takayama. Cherkis' bow varieties are cousins of quiver varieties, conjecturally describing moduli spaces of type A instantons on multi-Taub-NUT spaces. Our goal is to show that they are Coulomb branches of 3d N=4 framed quiver gauge theories of affine type A. This result generalizes one for unframed cases proved with Braverman and Finkelberg.

February 15, 11:40
I am going to give an overview over recent work on Pixton type formulas for the double ramification cycle, a generalization related to loci of abelian differentials with prescribed poles and zeros and a different generalization for twisted double ramification cycles over a base manifold X. This work is based on joint works with E. Clader and Pandharipande-Pixton-Zvonkine.

February 15, 5:30
Gauged linear sigma model (GLSM) is a 2d quantum field theory invented by Witten in early 90's to give a physical derivation of Landau-Ginzburg (LG)/Calabi-Yau (CY) correspondence. Since then, it has been investigated extensive in physics by Hori and others. Recently, an GLSM algebraic-geometric theory has been formulated by Fan-Jarvis-Ruan so that we can start to rigorously compute its invariants and match with physical predication. In fact, a great deal of activities are under going right now in abelian
cases where the objects of study are complete intersection of toric varieties. In this talk, we would like to look over the horizon to discuss nonabelian cases where the problems are much less understood even conjecturally. Moreover, nonabelian GLSM exhibits the new phenomenon unavailable to abelian GLSM.

February 17
The double ramification (DR) cycle is a class on the moduli space of curves that, roughly speaking, describes the locus of curves admitting a map to the projective line with specified ramification over zero and infinity.  In recent work, A. Pixton gave an explicit formula for a mixed-degree class, conjecturing that (1) its degree-g part coincides with the DR cycle, and (2) its higher-degree parts vanish.  I will discuss joint work with F. Janda giving a proof of the second of these two conjectures.

February 29
This work is in collaboration with L. Chekhov. The famous Greek astronomer Ptolemy created his well-known table of chords in order to aid his astronomical observations. This table was based on the renowned relation between the four sides and the two diagonals of a quadrilateral whose vertices lie on a common circle. In 2002, the mathematicians Fomin and Zelevinsky generalised this relation to introduce a new structure called cluster algebra. This is a set of clusters, each cluster made of n numbers called cluster variables. All clusters are obtained from some initial cluster by a sequence of transformations called mutations. Cluster algebras appear in a variety of topics, including total positivity, number theory, Teichm\”uller theory and many others.

In this talk we propose a new class of generalised cluster algebras for which the problem of quantum ordering can be solved explicitly. We start by introducing the notion of bordered cusps. This new notion arises when colliding holes in a Riemann surface. In the limit of two colliding holes, the geodesics that originally passed through the domain between colliding holes become arcs between two bordered cusps decorated by horocycles. The lengths of these arcs are $\lambda$-lengths in Thurston--Penner terminology, or cluster variables by Fomin and Zelevinsky. We then obtain new class of geodesic laminations comprising both closed curves in the interior of a Riemann surface and arcs passing between bordered cusps. We formulate the Poisson and quantum algebras of these laminations.  From the physical point of view, our construction provides an explicit coordinatization of moduli spaces of open/closed string worldsheets.

March 7
An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry, in particular of surface theory. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture and numerics.
These talk is about quadrilateral surfaces, i.e. surfaces built from planar quadrilaterals. They can be seen as discrete parametrized surfaces. Discrete curvatures as well as special classes of quadrilateral surfaces, in particular, discrete minimal surfaces are considered. Their relation to discrete integrable systems is clarified. Application in free form architecture will be demonstrated.

March 21, 11:40
Symplectic singularities were introduced by Beauville in 2000. These are especially nice singular Poisson algebraic varieties that include symplectic quotient singularities and the normalizations
of orbit closures in semisimple Lie algebras. Poisson deformations of conical symplectic singularities were studied by Namikawa who proved that they are classified by a points of a vector space. Recently I have
proved that quantizations of a conical symplectic singularities are still classified by the points of the same vector spaces. I will explain these results and then apply them to establish a version of Kirillov's orbit method for semisimple Lie algebras.

March 21, 5:30
I will introduce several universal and canonical random objects that are (at least in some sense) two dimensional or planar, along with discrete analogs of these objects. In particular, I will introduce probability measures on the space of paths, the space of trees, the space of surfaces, and the space of growth processes. I will argue that these are in some sense the most natural and symmetric probability measures on the corresponding spaces.

March 23
In this talk I will discuss some combinatorial questions related to wall-crossing functors between categories O over quantizations of symplectic resolutions with different ample cones. For suitable choices of ample cones, the wall-crossing functors are perverse in the sense of Rouquier and so give rise to bijections between the sets of simples (wall-crossing bijections). I will explain what "perverse" means in this case and
discuss some interesting special cases: the cotangent bundles of flag varieties and Hilbert schemes of points on C^2. In the former case, the wall-crossing bijections define an interesting action of the so called cactus group on the Weyl group that was not known before. In the latter case, we recover the Mullineux involution arising in the modular representation theory of symmetric groups.

March 28
Topological insulators are newly discovered materials with the defining property that any boundary cut into such crystal conducts electricity like a metal even in the presence of disorder. The main conjecture in the field is that topological insulators are classified by a certain periodic table, which I will briefly discuss. In the main part of the talk, I will present recent results where Alain Connes' non-commutative geometry program was used to full throttle to prove this conjecture for more than half of the periodic table. I will try to fit the exposition in a broader context, namely, the ongoing effort towards a constructive KK-theory, particularly constructive Kasparov products.

April 4
We propose an algebraic approach to categorification of quantum groups at a prime root of unity, with the scope of eventually categorifying Witten-Reshetikhin-Turaev three-manifold invariants. This is joint work with Mikhail Khovanov

April 11
Nekrasov and Shatashvili stated a conjecture that the operators of quantum multiplication by higher Chern classes of the tautological bundle on the Grassmannian are equal to the Baxter operators. We prove this conjecture in the K-theoretic setting using the computation of the Vertex and the quantum difference equation.
Based on a work in progress joint with A. Smirnov and A. Zeitlin

April 18
Jones polynomials and WRT invariants are well-known invariants of links in S^3. Their categorification
attracts a lot of attention now. The key numerical invariant here is the Poincare polynomial of the triply
graded Khovanov-Rozansky homology, also called HOMFLYPT homology. In spite of recent developments,
this theory remains very difficult apart from the celebrated Khovanov homology (the case of sl(2)) with
very few known formulas (only for the simplest uncolored knots). Several alternative approaches to these polynomials were suggested recently (the connections are mostly conjectural). We will discuss the direction based on DAHA, which was recently extended from torus knots to arbitrary torus iterated links (including all algebraic links). The talk will be mostly focused on the DAHA-Jones polynomial of type A_1. Based on our joint works with Ivan Cherednik.

April 20
This is joint work with D. Benson, S. Iyengar and H. Krause.  I’ll discuss the correct notion of support and its sibling, cosupport, for the stable module category (or the derived category of singularities) of a finite group scheme. As an application, I’ll describe classification of tensor ideal localizing subcategories in the stable module category in terms of supports.

April 25
DWG is a nice braid group action on the finite-dimensional representations of a semisimple group, that acts on the weights via the usual Weyl group. In their paper A.Braverman and M.Finkelberg obtain DWG using a geometric construction involving equivariant cohomology of the corresponding affine Grassmanian. On the other hand, such action appears in cohomology of cotangent bundles to the usual Grassmanians. According to the philosophy of the symplectic duality, there is a connection between certain resolutions of AG and T*Gr(k, n).
Using the construction of stable envelopes in the case of these resolutions of the affine Grassmanian, I try to describe operators from DWG on tensor products as corresponding R-matrices. In particular, since the varieties are smooth, this should give something in K-theory as well as in cohomology.

May 2
We give a manifestly rational formula for capped descendants in quantum K-theory of Nakajima varieties.