Informal Mathematical Physics Seminar

Schedule of talks for Fall 2015:

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

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.

organized by Igor
Krichever and Andrei
Okounkov

Mondays, 5:30, Room 507

To sign up for dinner click hereSchedule of talks for Fall 2015:

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

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.

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.

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.

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.

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).

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.

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.

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.

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