Algebra, Geometry, and Number Theory Seminar

Abstracts

Michael Weymss - GW and GV invariants via the movable cone

Given any smooth threefold flopping contraction, I will give a combinatorial characterisation of which Gopakumar–Vafa (GV) curve-counting invariants are non-zero, and also explain how they change under iterated flop. To do this requires us to prescribe multiplicities to the walls in the movable cone. From this, I will draw (!), the critical locus of the associated quantum potential. There are various corollaries, including a visual proof of the Crepant Resolution Conjecture in this context. This is joint work with Navid Nabijou.

Michael Brown - Tate resolutions as noncommutative Fourier-Mukai transforms

This is joint work with Daniel Erman. The classical Bernstein-Gel'fand-Gel'fand (or BGG) correspondence gives an equivalence between the derived categories of a polynomial ring and an exterior algebra. It was shown by Eisenbud-Fløystad-Schreyer in 2003 that the BGG correspondence admits a geometric refinement that sends a sheaf on projective space to a complex of modules over an exterior algebra called a Tate resolution. The goal of this talk is to reinterpret Tate resolutions as noncommutative analogues of Fourier-Mukai transforms and to discuss some applications.

Chris Keyes - Local solubility in families of superelliptic curves

If one chooses a random integral binary degree $d$ form $f(x,z)$, what is the probability that the superelliptic curve $C: y^m = f(x,z)$ has a $p$-adic point, or better, points everywhere locally? In this talk, I will discuss joint work with Lea Beneish arxiv.org/abs/2111.04697 in which we give quantitative answers to this question in various levels of generality. For a fixed choice of $m$ and $d$, we show that for a positive proportion of forms $f(x,z)$ the curve $C$ is everywhere locally soluble, and moreover, this proportion is given by a product of the local densities. By finding bounds for these local densities, we describe the behavior of the proportion in the large $d$ limit. Finally, for superelliptic curves of the form $y^3 = f(x,z)$ for an integral binary form $f$ of degree $6$, we determine exactly the proportion that are everywhere locally soluble, in terms of explicit rational function expressions for the local densities, giving a numerical answer of 96.94%.

Shaoyun Yi - Eisenstein series from an adelic point of view

In this talk, we discuss "Hecke summation" for the classical Eisenstein series $E_k$ in an adelic setting. The connection between classical and adelic functions is made by explicit calculations of local and global intertwining operators and Whittaker functions. In the process we determine the automorphic representations generated by the $E_k$, in particular for $k=2$, where the representation is neither a pure tensor nor has finite length. We will also briefly present the analogous results for Eisenstein series of weight $2$ with level, and Eisenstein series with character. This is a joint work with Manami Roy and Ralf Schmidt.

Last semester's seminar.