**Talk at the Workshop “Complex Lagrangians, Mirror Symmetry, and Quantization”**

**Banff, October 2023**

**Title:** Frobenius structures for quantum differential equations and mirror symmetry

**ABSTRACT:** There exists a well-known connection between the Kloosterman sum in number theory, and the Bessel differential equation. This connection was explained by B. Dwork in 1974 by discovering Frobenius structures in the p-adic theory of the Bessel equation. In my talk I will speculate that this connection extends to the quantum differential equations in quantum cohomology of Nakajima varieties. As an example, I will give an explicit conjectural description of the Frobenius structures for the quantum connections of T*Gr(k,n) and also Gr(k,n). The traces of Frobenius structures are natural finite field analogs of the integral solutions of quantum differential equations known in mirror symmetry. In particular, for Gr(k,n) we arrive at the exact B-model description of quantum connection discovered by Marsh and Rietsch. Recorded talk can be found here.

**Berkeley Informal String Math Seminar.**

**Talk: Frobenius structures for quantum differential equations and mirror symm**etry

Abstract and the talk can be found here.

**Talk at Workshop Elliptic Integrable Systems, Representation Theory and Hypergeometric Function, Tokyo Japan July, 2023
**

**Title:**K-theoretic limits of elliptic stable envelope

**ABSTRACT:**In my talk I will discuss hyperplane arrangements which control the K-theoretic limits of the elliptic stable envelope. I will explain the separation of variables phenomena arising in the wall limit. This factorization phenomenon has several applications like identities between R-matrices of different quantum groups or construction of wall-crossing operators for quantum difference equations. Time permitting, I will also discuss connection with monodromy problem for quantum differential equations.

**Recorded talk can be found here:**Video

**Mini-school: Contemporary trends in integrable systems (Lisbon 2023)
Course: Geometric methods in theory of integrable spin chains
**ABSTRACT: It has been well understood that many known integrable systems appear as quantum K-theory of Nakajima quiver varieties. For example, the classical XXZ spin chain is equivalent to the quantum K-theory of cotangent bundle over Grassmannians X=T*Gr(k,n). My lectures are an informal introduction to this circle of ideas. I will explain the construction of main ingredients of the enumerative approach: vertex functions of quiver varieties, capping operators, quantum difference equations etc., and their relations to integrable systems. Time permits, I will also discuss dualities in integrable systems appearing from geometry via the 3D mirror symmetry. For references, the interested students may use arXiv:1612.08723 as an introduction to a more general theory by A. Okounkov arXiv:1512.07363.

**Recorded talks and abstracts can be fund**here

**Vertex functions, stable envelopes, 3d mirror symmetry** (joint with R.Rimanyi)

Minicourse taught at the school Representation theory and flag or quiver varieties, University of Paris, France, June 13-17, 2022**
**Recorded talks can be found here.

**Seminar Talk at M-Seminar, Kansas State University**

**Title:** Elliptic stable envelopes and symplectic duality

**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. Recorded talk.

**Seminar Talk at Berkeley Representation Theory and Mathematical Physics Seminar**

**Title: Elliptic stable envelopes and symplectic duality**

**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*P1 in details and discuss other cases in which the statement is proven. Recorded Talk.

# Elliptic stable envelopes and symplectic duality

**Seminar Talk at GPRT seminar, North Eastern University.**

**Title:** Quantum difference equations, monodromies and mirror symmetry

Recorded talk.

**Workshop on Elliptic Integrable Systems**

**Title:** Elliptic stable envelope for Hilbert scheme of points in C^2

Recorded talk

**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.