I will review the classic paper by Atiyah and Bott titled "The Moment Map and Equivariant Cohomology". The motivational statement of that paper is that if f is a function on a symplectic manifold M whose hamiltonian flow generates a circle action, the integral of $e^&ob;if&cb;$ localizes to a certain integral over the critical locus of f. First, I will briefly tell you why I'm personally interested in this theorem, (it tells you stuff about complex Chern Simons theory,) then I will talk about how to prove it. The proof will be easy once we learn about equivariant cohomology, and how to do equivariant cohomology in a de Rahm way. 

I'll review work by McNamara and Vafa suggesting that the cobordism class of quantum gravity is trivial. Much of the talk will be spent on trying to fill this statement with at least some meaning and discuss some of the evidence that led to this speculation.

The set of closed subgroups of a topological group has a natural topology called the Chabauty topology. The topology of the space of closed subgroups of R^2 was described by Pourezza and Hubbard in 1978. It turns out to be homeomorphic to S^4, and the complement of the lattices turns out to be the suspension of the trefoil knot. I will sketch their proof, which relies on the famous Weierstrass formula

Understanding algebraic structures such as Galois extensions, quadratic forms and division algebras, can give important insights into the arithmetic of fields. In this talk, I will discuss recent work showing ways in which the arithmetic of certain fields can be partially described by topological information. I will then describe how these observations lead to arithmetic versions of the Mayer-Vietoris sequences, the Seifert–van Kampen theorem, and examples and counterexamples to local-global principles.

Cost is a $[1.\infty]$-valued measure-isomorphism invariant of aperiodic measured equivalence relations defined by Gilbert Levitt and heavily studied by Damien Gaboriau. For a large class of equivalence relations, including amenable, the cost is 1. Yoshikata Kida and Robin Tucker-Drob recently defined the notion of an inner amenable equivalence relation as an analog of inner amenability in the setting of groups. We show inner amenable equivalence relations also have cost 1. This is joint work with Robin Tucker-Trob. To get the Zoom link, sign up here: https://sites.cns.utexas.edu/utgd/utgd-online-seminar

We discover a geometric property of the space of tensors of fixed multilinear rank. Namely, we show that real tensors of any fixed multilinear rank form a minimal submanifold of the Euclidean space of tensors endowed with the Frobenius inner product. We also establish the absence of local extrema for linear functionals restricted to the submanifold of rank-one tensors, finding application in statistics.

Representations of vertex algebras of CohFT-type can be used to construct vector bundles of coinvariants and conformal blocks on moduli spaces of stable pointed curves. The name comes from the fact they define cohomological field theories. I’ll talk about the moduli space of curves, how these bundles and their classes can help us better understand it, and give some examples. This is about work with Chiara Damiolini and Nicola Tarasca.