Mondays, 5:30, Room 507To sign up for dinner click here
|Sept 7||no seminar||Labor day|
|Sept 14||Eric Rains||Noncommutative geometry and special
|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
||Symplectic duality and the affine
|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
||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)
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.
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