Algebra and Algebraic Geometry Seminar Fall 2020
The Virtual Seminar will take place on Fridays at 2:30 pm via Zoom. We will also link to relevant or interesting Zoom talks outside of the seminar.
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As a result of Covid-19, the seminar for this semester will be held virtually.
Fall 2020 Schedule
Categorical Enumerative Invariants
I will talk about recent papers with Junwu Tu, Si Li, and Kevin Costello where we give a computable definition of Costello's 2005 invariants and compute some of them. These invariants are associated to a pair (A,s) consisting of a cyclic A∞-algebra and a choice of splitting s of its non-commutative Hodge filtration. They are expected to recover classical Gromov-Witten invariants when A is obtained from the Fukaya category of a symplectic manifold, as well as extend various B-model invariants (solutions of Picard-Fuchs equations, BCOV invariants, B-model FJRW invariants) when A is obtained from the derived category of a manifold or a matrix factorization category.
Harmonic Analysis on GLn over Finite Fields
There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio: Trace(ρ(g)) / dim(ρ), for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G. Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank. Rank suggests a new organization of representations based on the very few “Small” ones. This stands in contrast to Harish-Chandra’s “philosophy of cusp forms”, which is (since the 60s) the main organization principle, and is based on the (huge collection) of “Large” representations. This talk will discuss the notion of rank for the group GLn over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks. This is joint work with Roger Howe (Yale and Texas A&M). The numerics for this work was carried with Steve Goldstein (Madison) and John Cannon (Sydney).
The Supermoduli Space of Genus Zero SUSY Curves with Ramond Punctures
Super Riemann surfaces (SUSY curves) arise in the formulation of su- perstring theory, and their moduli spaces, called supermoduli space, are the integration spaces for superstring scattering amplitudes. I will focus specifically on genus zero SUSY curves. As with ordinary curves, genus zero SUSY curves present a certain challenge, as they have an infinites- imal group of automorphisms, and so in order for the moduli problem to be representable by a Deligne-Mumford superstack, we must introduce punctures. In fact, there are two kinds of punctures on a SUSY curve of Neveu-Schwarz or Ramond type. Neveu-Schwarz punctures are entirely analogous to the marked points in ordinary moduli theory. By contrast, the Ramond punctures are more subtle and have no ordinary analog. I will give a construction of the moduli space M0,nR of genus zero SUSY curves with Ramond punctures as a Deligne-Mumford superstack by an explicit quotient presentation (rather than by an abstract existence argument).
A Rogers-Ramanujan-Slater type identity related to the Ising model
We prove three new q-series identities of the Rogers-Ramanujan-Slater type. We find a PBW basis for the Ising model as a consequence of one of these identities. If time permits it will be shown that the singular support of the Ising model is a hyper-surface (in the differential sense) on the arc space of it's associated scheme. This is joint work with G. E. Andrews and J. van Ekeren and is available online at https://arxiv.org/abs/2005.10769