Informal Mathematical Physics Seminar

organized by Igor Krichever and Andrei Okounkov

Mondays, 5:30, Room 507

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Schedule of talks for Fall 2015:

Sept 7 no seminar Labor day
Sept 14 Eric Rains Noncommutative geometry and special functions
Sept 21 Alexander Shapiro Towards cluster structure on quantum groups
Sept 28 Josh Sussan A categorical braid group action at a prime root of unity
Oct 5 Yi Sun Traces of intertwiners for quantum affine sl_2 and Felder-Varchenko functions
Oct 12
Alexander Braverman
Symplectic duality and the affine Grassmannian
Oct 19 Dmitry Zakharov Bounded, non-vanishing solutions of KdV and a new class of potentials of the Schroedinger operator
Oct 26 Paul Zinn-Justin From conormal bundles of Schubert varieties to loop models
Nov 2
Timo Seppalainen 1st Minerva lecture, note special time: 4:00-5:30
Use this form to sign up for dinner after the lecture
Nov 9 Anton Zeitlin Decorated Super-Teichmueller Theory
Nov 16 Noah Arbesfeld
Chern classes of tautological bundles on Hilbert schemes via Virasoro intertwiners
Nov 23 Georg Oberdieck Quantum Cohomology of the Hilbert scheme of points of a K3 surface
Nov 30 seminar staff
preparation for the next week's talk
Dec 7 Mina Aganagic Two mathematical applications of (little) string theory

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.


September 14
Many of the "special" functions arising in mathematical physics (as well as many other applications) either come from solutions of special linear ODEs or describe natural flows in spaces of linear ODEs.  In either case, one is led to the problem of understanding families of ODEs with specified singularity structure; for the latter, one also wants to understand when there are simple relations between the solutions of two such equations.  The existing methods for addressing these questions begin to falter when one generalizes to difference or q-difference (or elliptic difference!) equations; I'll give an informal introduction to a new approach via noncommutative geometry, which not only gives a number of new results about moduli spaces of difference equations, but also a number of new results in noncommutative algebraic geometry.

September 21
A Poisson-Lie group G with its standard Poisson structure admits a family of cluster coordinates with the defining property of having log-canonical Poisson brackets:
{X_i, X_j} = a_{ij} X_i X_j
On the level of quantum groups, these coordinates become a family of q-commuting generators
X_i X_j = q^{a_{ij}} X_j X_i
for (a localization of) the quantized algebra O_q[G] of functions on G.

Showing that a localization of the quantum group U_q(g) is isomorphic to certain quantum torus algebra (i.e. an algebra with q-commuting generators) is a much desired property known as the quantum Gelfand-Kirillov conjecture. In a joint work with Gus Schrader, we have constructed an embedding of the quantum group U_q(g) into a quantum torus algebra naturally defined from the quantum double Bruhat cell O_q[G^{w_0,w_0}]. Our construction is motivated by Poisson geometry of the Grothendieck-Springer resolution and is closely related to the global sections functor of the quantum Beilinson-Bernstein theorem. I will explain our work, outline a few other ways to obtain such "cluster" structure on U_q(g), and discuss its applications to representation theory.

September 28
Khovanov introduced the subject of Hopfological algebra as an approach to categorifying the Witten–Reshetikhin–Turaev 3-manifold invariant. We review the categorification of the upper half of small quantum sl(2) at a prime root of unity due to Khovanov and Qi. Then we explain how this gives rise to a categorification of a braid group action at a prime root of unity. This is joint work with You Qi.

October 5

This talk concerns two approaches for studying a family of special functions occurring in the study of the q-Knizhnik-Zamolodchikov-Bernard (q-KZB) equation.  The philosophy of KZ-type equations predicts that it admits solutions via (1) traces of intertwining operators between representations of quantum affine algebras produced by Etingof-Schiffmann-Varchenko and (2) certain theta hypergeometric integrals we term Felder-Varchenko functions.  In a series of papers in the early 2000's, Etingof-Varchenko conjectured that these families of solutions are related by a simple renormalization; in the trigonometric limit, they proved such a link and used it to study these functions.

In recent work, I resolve the first case of the Etingof-Varchenko conjecture by showing that the traces of quantum affine sl_2-intertwiners of Etingof-Schiffmann-Varchenko valued in the 3-dimensional evaluation representation converge in a certain region of parameters and give a representation-theoretic construction of Felder-Varchenko functions.  I will explain the two constructions of solutions, the methods used to relate them, and connections to affine Macdonald theory and the Felder-Varchenko conjecture on the q-KZB heat operator and corresponding SL(3,Z)-action.

This talk is based on the preprint arXiv:1508.03918.

October 12

The purpose of the talk will be two-fold: in the fist part I am going to review the phenomenon of symplectic duality, making a special emphasis on the categorical aspects of it as well as on its relation to the notion of Higgs and Coulomb branch of the moduli space of vacua in 3-dimensional quantum field theory. In the 2nd part I am going to briefly talk about a recent mathematical construction of the above-mentioned Coulomb branch via the affine Grassmannian (joint work with Finkelberg and Nakajima), which gives rise to a (rather long) list of symplectically dual pairs. If time permits, I will try to explain some ideas how this construction might be used for proving the categorical version of symplectic duality.

October 19

The KdV hierarchy is an infinite collection of commuting isospectral deformations of the one-dimensional Schr\”odinger operator, and its spectral theory is thus intimately related to the initial-value problem for KdV. For two classes of initial data, the spectral theory is well understood, and the initial value problem can be considered solved. A potential rapidly vanishing at infinity can be reconstructed from its spectral data by using the inverse spectral transform (IVT), and the spectral data evolves linearly with KdV. An important class of such potentials are the Bargmann potentials, or soliton solutions of KdV. The spectrum of a periodic potential consists of an infinite sequence of bands separated by spectral gaps. For a dense collection of potentials, there are only finitely many gaps, the eigenfunction is identified as a section of a line bundle over a corresponding hyperelliptic curve, and the KdV evolution is linear on the Jacobian of the curve. 

It has long been known that finite-gap potentials should be obtainable as limits of Bargmann potentials, but a precise description of such a limit was not known. We reformulate the IVT by studying the singularities of the eigenfunctions of the corresponding Schr\”odinger operator, which gives us some additional freedom for describing the Bargmann potentials. Replacing the isolated singularities with cuts on the spectral plane, we obtain a new Riemann—Hilbert problem whose solutions describe potentials of the Schr\”odinger operator that are non-vanishing at infinity, but are not periodic, and can be thought of as a one-dimensional soliton gas. This RH problem can be studied numerically, and we also study the spectra of the corresponding Sch\”odinger operators.

Joint work with Sergey Dyachenko and Vladimir Zakharov.

October 26

In this work in collaboration with A. Knutson, we investigate the correspondence between algebraic geometry and quantum integrable systems -- a subject in which great progress has been made recently thanks to the work of Maulik and Okounkov -- from the point of view of Grobner degenerations. The latter is very combinatorial in nature and works equally well for cohomology and K-theory. Following Knutson and Miller, I shall recall the simplest framework in which one can develop this approach, namely (matrix) Schubert varieties and Schubert and Grothendieck polynomials. After that, I shall formulate a broad extension of these results which will naturally lead us to loop models on general lattices: first noncrossing loops (Temperley--Lieb model), then, if time allows, crossing loops (Brauer model).

November 9

I will talk about the construction of the generalization of Penner's coordinates on the decorated super-Teichmueller space of a surface with $s\ge 1$ punctures, which is a principal bundle over the corresponding super-Teichmueller space. We will discuss all necessary ingredients e.g. super-version of the Ptolemy transformations, combinatorial approach to the description of the spin structures on punctured surfaces as well as the even Ptolemy-invariant 2-form, which is the generalization of the Weil-Petersson 2-form. Based on the preprint arXiv:1509.06302.

November 16

We explain how to obtain formulas for multiplication by Chern classes of tautological bundles on the Hilbert scheme of points on a general surface S from intertwiners for highest weight representations of a Virasoro-like algebra. We present the proof when S=C^2, which is due to Maulik and Okounkov, and then extend the result to general S.

November 23

The quantum cohomology of the Hilbert scheme of points of the complex plane, or more general toric symplectic resolutions have been studied in work of Okounkov-Pandharipande, Maulik-Okounkov and others.
On the other hand, not much is known about the related case of projective symplectic resolutions, such as the Hilbert scheme of points of a K3 surface.

In this talk I will present a conjecture for quantum multiplication with divisor classes in the quantum comology ring of Hilb(K3) in terms of explicit relations with the Nakajima basis.

December 7

I will describe the relation between q-deformed conformal blocks of W-algebras and generating functions of K-theoretic instanton invariants for quiver gauge theories. The W-algebra and the corresponding quiver gauge theory are both associated with the same simply laced Lie algebra of ADE type. The relation can be stated and proven without recourse to physics. Yet, in discovering it, a six dimensional string theory ("the little string”) played an important role. The second, related, application of the little string theory yields a generalization of the (quantum) geometric Langlands correspondence for ADE groups. The result can once more be phrased  in purely mathematical terms. This time it involves quantum K-theory of ADE quiver varieties and the elliptic stable envelopes.

Seminar arxiv:  Spring 2015 Fall 2014 Spring 2014 Fall 2013 Spring 2013 Fall 2012 Spring 2012