**Quantum difference equations, monodromies and mirror symmetry
(NorthEastern, Geometry, Physics, and Representation Theory Seminar, 10.1.2020**

**):**

Video, Slides

ABSTRACT:

An important enumerative invariant of a symplectic variety X is its vertex function. The vertex function is the analog of J-function in Gromov-Witten theory: it is the generating function for the numbers of rational curves in X. In representation theory these functions feature as solutions of various q-difference and differential equations associated with X, with examples including qKZ and quantum dynamical equations for quantum loop groups, Casimir connections for Yangians and other objects. In this talk I explain how these equations can be extracted from algebraic topology of symplectic dual variety X!, also known as 3D-mirror of X. This can be summarized as “identity”:

Enumerative geometry of X = algebraic topology of X!

The talk is based on work in progress with Y.Kononov: arXiv:2004.07862; arXiv:2008.06309.

# Elliptic stable envelopes and symplectic duality

(Berkeley, Representation Theory and Mathematical Physics Seminar, 05.22.2020)

ABSTRACT:

In this talk I’ll explain the following idea: “the elliptic stable envelopes of symplectic dual varieties coincide.” I’ll describe a simplest example of T^{*}P^{1} in details and discuss other cases in which the statement is proven.

**Two lectures on quantum-difference equations. ETH April 2020**

Lecture 1. Lecture notes.

ABSTRACT:

Generating functions counting quasimaps from P^{1} to quiver varieties satisfy certain q-difference equations (qde). At the same time, K-theoretic stable envelopes equip K-theories of these variates with natural action of quantum loop groups. In this talk I explain how the representation theory of these algebras can be utilized to obtain explicit description of qde’s. As an application of the general theory I consider example of a quiver variety given by the Hilbert scheme of points in complex plane. In this case we obtain explicit formula for qde in terms of the Hall algebra of elliptic curve. This formula generalizes the result of Okounkov-Pandharipande arXiv:0906.3587 to the level of equivariant K-theory. The talk is based on joint work with A. Okounkov arXiv:1602.09007.

Lecture 2. Lecture notes.

ABSTRACT:

The monodromy of the quantum difference equations can be described in purely geometric terms by the “elliptic stable envelope” introduced recently by Aganagic-Okounkov. In this talk I explain how the qde’s themselves can be reconstructed from the monodromy and certain symmetries of the elliptic stable envelopes known as 3d-mirror symmetry (symplectic duality). As an example, the qde for the Hilbert scheme of points in the complex plane discussed in the Talk 1 will be revisited in this light.

**Lecture on elliptic cohomology and stable envelopes. MIAN May 2020** (in Russian)

**Аннотация:** В докладе я постараюсь дать элементарное введение в теорию эллиптических стабильных оболочек. Я расскажу, как эллиптические оболочки используются для описания симплектической двойственности в теории представлений. Доклад основан на совместных работах с Р. Римани, А. Варченко и З. Жу arXiv:1902.03677, arXiv:1906.00134.