A series of talks at the Ohio State University given by some really cool people! The talks will be held on the 13th of April, 2023, and are open to everyone. All talks will be held in the Mathematics Tower (MW) at OSU.
Title: Symplectic involutions and cohomology of Kummer-type fourfolds
Abstract: The middle cohomology of hyperkahler fourfolds of Kummer type was studied by Hassett and Tschinkel, who showed that a large portion is generated by cycle classes of fixed-point loci of symplectic involutions. In recent joint work with K. Honigs, we study certain symplectic fourfolds over arbitrary fields. We have extended the results of Hassett and Tschinkel and characterized the Galois action on the cohomology. This has natural consequences for derived equivalences between certain generalized Kummer fourfolds, a result which uses as input joint work with K. Honigs and J. Voight on the n-torsion in abelian surfaces versus their duals.
Title: Fourier Interpolation and the Weil Representation
Abstract: In 2017, Radchenko-Viazovska proved a remarkable interpolation result for even Schwartz functions on the real line: such a function is entirely determined by its values and those of its Fourier transform at square roots of integers. We give a new proof of this result, exploiting the fact that Schwartz functions are the underlying vector space of the Weil representation 𝑊. This allows us to deduce the interpolation result from the computation of the cohomology of a certain congruence subgroup of 𝑆𝐿2(ℤ) with values in 𝑊. This is joint work in progress with Akshay Venkatesh.
Title: Nonabelian Cohen—Lenstra heuristics in the presence of roots of unity
Abstract: The nonabelian Cohen—Lenstra program studies the distribution of the Galois group of maximal unramified extension of K as K varies in a family of global fields. In this talk, we will discuss some new developments for this type of question. We will introduce a cohomological invariant of a Galois extension of F_q. We show that by keeping track of this invariant we can generalize the nonabelian Cohen—Lenstra Heuristics given by Liu, Wood, and Zureick-Brown to cover the case when the base field contains extra roots of unity; moreover, we show that the new conjecture is a nonabelian generalization of the work by Lipnowski, Tsimerman, and Sawin. We will prove the conjecture with a large q limit, and discuss how to make a similar conjecture for number fields.
Title: Counting fields generated by points on plane curves
Abstract: For a smooth projective curve C/Q, how many field extensions of Q -- of given degree and bounded discriminant --- arise from adjoining a point of C(\overline{Q})$? Can we further count the number of such extensions with a specified Galois group? Asymptotic lower bounds for these quantities have been found for elliptic curves by Lemke Oliver and Thorne, for hyperelliptic curves by Keyes, and for superelliptic curves by Beneish and Keyes. We discuss similar asymptotic lower bounds that hold for all smooth plane curves C.
Title: Algebraic and arithmetic properties of curves via Galois cohomology
Abstract: A lot of the algebraic and arithmetic information of a curve is contained in its interaction with the Galois group. This draws inspiration from topology, where given a family of curves over a base B, the fundamental group of B acts on the cohomology of the fiber. As an arithmetic analogue, given an algebraic curve C defined over a non-algebraically closed field K, the absolute Galois group of K acts on the etale cohomology of the geometric fiber and this action gives rise to various Galois cohomology classes. In this talk, we discuss how to use these classes to detect algebraic and arithmetic properties of the curve, such as the non-existence of rational points (following Grothendieck's section conjecture) and whether the curve is hyperelliptic.