Coulomb branches and Yangians

Abstract: A classical result of Jørgensen and Thurston shows that the set of volumes of finite volume complete hyperbolic 3-manifolds is a well-ordered subset of the real numbers of order type w^w; moreover, they showed that each volume can only be attained by finitely many isometry types of hyperbolic 3-manifolds. We will discuss a group-theoretic analogue of this result: If $\Gamma$ is a non-elementary hyperbolic group, then the set of exponential growth rates of $\Gamma$ is well-ordered, the order type is at least w^w, and each growth rate can only be attained by finitely many finite generating sets (up to automorphisms), and further generalizations of these results. The talk is intended to be for a wider audience. All the notions that are mentioned in the abstract will be explained. It is based on a joint work with K. Fujiwara.

Braverman, Finkelberg and Nakajima have recently given a mathematical definition of the Coulomb branches associated to certain 3-dimensional quantum field theories. They define Coulomb branches as affine algebraic varieties, and showed that many interesting varieties arise in this way.

The BFN construction also produces quantized Coulomb branches, which are non-commutative algebra. It is interesting to try to relate these non-commutative algebras with more familiar ones; one nice example that arises is the enveloping algebra of gl(n).

I'll discuss how certain quantized Coulomb branches can be described using Yangians. This means that there are explicit generators for the quantized Coulomb branch (which is otherwise rather abstractly defined), a fact which has found application in describing connections between Coulomb branches and cluster algebras. But going the other way, we may also learn more about Yangians and their modules by leveraging results from the Coulomb branch theory. In my talk, I will overview recent progress on these topics.

The classical umkehr map of Hopf assigns to a map of oriented manifolds, $f:M \to N,$ `wrong-way' homomorphisms in homology $f_!: H_*(N) \to H_*(M)$ and in cohomology $f^!:H^*(M) \to H^*(N),$ the latter a version of `integration over the fibers'. Similar wrong-way maps, sometimes known as transfer maps or Gysin maps, are defined for other generalized (co)homology theories as long as the manifolds are suitably oriented and have had many applications. While these maps are defined only for manifolds there has long been interest in extending them to singular spaces. I'll discuss joint work with Markus Banagl and Paolo Piazza in which we capitalize on recent work on the index theory of signature operators to give analytic definitions of transfer maps in K-homology for stratified spaces and relate them to topological orientations.

Mathematics

Audience: researchers in the discipline

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CRM-Regional Conference in Lie Theory

Series comments: Registration is free but mandatory:https://www.crm.umontreal.ca/act/form/inscr_lieautomne20_e.shtml

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