# Seminars & Events for Minerva mini-course

##### Quantum groups and quantum cohomology

Quantum groups, which originated in mathematical physics in the study of solvable 1+1 dimensional models, are noncocommutative Hopf algebra deformations of the universal enveloping of a Lie algebra. PLEASE CLICK ON LECTURE TITLE FOR COMPLETE ABSTRACT.

##### Quantum groups and quantum cohomology - second lecture

This lecture is #2 in a series of 10 lectures: Quantum groups, which originated in mathematical physics in the study of solvable 1+1 dimensional models, are noncocommutative Hopf algebra deformations of the universal enveloping of a Lie algebra.The adjective "quantum" in quantum cohomology has a very different meaning and origin. Quantum cohomology is a commutative deformation of the algebra structure on the cohomology of an algebraic variety X that takes into account the enumerative geometry of rational curves in X. In recent years, a connection between quantum cohomology of certain very special algebraic varieties and quantum groups has been suggested by Nekrasov and Shatashvili, and from a very different angle by Bezrukavnikov and his collaborators. A large portion of these conjectures have since then been proven.

##### Quantum groups and quantum cohomology - third lecture

This lecture is #3 in a series of 10 lectures: Quantum groups, which originated in mathematical physics in the study of solvable 1+1 dimensional models, are noncocommutative Hopf algebra deformations of the universal enveloping of a Lie algebra.The adjective "quantum" in quantum cohomology has a very different meaning and origin. Quantum cohomology is a commutative deformation of the algebra structure on the cohomology of an algebraic variety X that takes into account the enumerative geometry of rational curves in X. In recent years, a connection between quantum cohomology of certain very special algebraic varieties and quantum groups has been suggested by Nekrasov and Shatashvili, and from a very different angle by Bezrukavnikov and his collaborators. A large portion of these conjectures have since then been proven.

##### Quantum groups and quantum cohomology - 4th lecture

This lecture is #4 in a series of 10 lectures: Quantum groups, which originated in mathematical physics in the study of solvable 1+1 dimensional models, are noncocommutative Hopf algebra deformations of the universal enveloping of a Lie algebra.The adjective "quantum" in quantum cohomology has a very different meaning and origin. Quantum cohomology is a commutative deformation of the algebra structure on the cohomology of an algebraic variety X that takes into account the enumerative geometry of rational curves in X. In recent years, a connection between quantum cohomology of certain very special algebraic varieties and quantum groups has been suggested by Nekrasov and Shatashvili, and from a very different angle by Bezrukavnikov and his collaborators. A large portion of these conjectures have since then been proven.

##### Stable envelopes - 5th lecture

This lecture is #5 in a series of 10 lectures: Given an algebraic symplectic variety X with an action of a torus A preserving the symplectic form one can define, under rather relaxed hypotheses, a collection of maps from the equivariant cohomology of the fixed locus X^A to the equivariant cohomology of X. These maps are indexed by a certain chamber decomposition in the Lie algebra of A and change in an interesting fashion as we cross from one chamber to the next.

##### Nakajima varieties

Nakajima varieties are very remarkable algebraic symplectic varieties

that can be associated to an arbitrary multigraph (which later in the

theory plays the role of the Dynkin diagram of a certain Lie algebra).

They include, for example, cotangent bundles of partial flag varieties,

as well as Hilbert schemes of points and more general moduli spaces

of sheaves on ADE surfaces. This talk will be an introduction to

the subject.

##### Geometric R-matrices for Nakajima varieties, I

This week, we will begin to merge the discussion of stable envelopes and Nakajima varieties to produce a geometric action of a certain Yangian on equivariant cohomology of Nakajima varieties.

##### Geometric R-matrices for Nakajima varieties, II

Part II: this week, we will begin to merge the discussion of stable envelopes and Nakajima varieties to produce a geometric action of a certain Yangian on equivariant cohomology of Nakajima varieties.

##### Rational curves in quiver varieties

We will first review the general structural features of quantum multiplication and then discuss the key points that allow to determine it for Nakajima varieties.

##### Outlook

I will explain how what we learned is applied in the enumerative geometry of curves in 3-folds and talk about various directions in which it can be generalized.