Older abstracts
Fall 2023
Date: Sep. 21.
Place: LL 902 (LC)
Speaker: Patrick McFaddin
Title: Arithmetic tori, toric varieties, and K-theory
Abstract: Complex algebraic tori (and their compactifications) play a tremendously important role in algebra and algebraic geometry. One reason for their ubiquity is that compactifications of tori can be encoded using combinatorial data in the associated lattices of characters and cocharacters. Over base fields which are not algebraically closed, the study of (compactifications of) tori becomes much more subtle, and Galois actions on these associated lattices must be considered. In this talk, I will discuss tori and toric varieties defined over an arbitrary base field, together with a few examples. I will then give an overview of work of Merkurjev and Panin who studied these objects using a certain non-commutative category of K-motives. Lastly, I will discuss partial progress toward determining the Galois-lattice structure of the Grothendieck group of a toric variety.
Date: Oct. 5.
Place: LL 902 (LC)
Speaker: David Swinarski
Title: The worst destabilizing 1-parameter subgroup for cuspidal curves
Abstract: Kempf proved that when a point is unstable in the sense of geometric invariant theory, there is a "worst" destabilizing 1-parameter subgroup $\lambda$. It is natural to ask: what are the worst destabilizing 1-ps for the unstable points in familiar GIT problems, such as those used to construct the moduli space of curves $M_g$? We first consider Chow points of rational curves with one unibranch singular point. Then the problem can be restated as an explicit problem in convex geometry (finding the proximum of a polyhedral cone to a point outside it). We present some observations based on computer experiments, and prove the answer for cusps $y^2 = x^{2r+1}$ of order $r$. This is joint work with Joshua Jackson (Sheffield).
Date: Oct 12.
Place: LL 902 (LC)
Speaker: Sebastian Bozlee
Title: A stratification of moduli of arbitrarily singular curves
Abstract: What does the space of all algebraic curves look like? One way to answer the question is to break up this gigantic, unruly space into nice pieces, called strata, and then to describe those.
For nodal (i.e. not particularly singular) curves such strata have long been constructed and are used throughout the theory. In this talk, I will describe how to construct a similar stratification of moduli of arbitrarily singular curves, indexed by combinatorial data generalizing dual graphs. The key idea is to combine the geometry of moduli of smooth curves with moduli of subalgebras.
Date: Oct 19.
Place: LL 902 (LC)
Speaker: Cris Poor
Title: Formal series of Jacobi forms
Abstract: Do all formal series of Jacobi forms invariant under a certain involution converge? We discuss recent progress on this open problem. The Fourier-Jacobi expansions of paramodular forms, necessarily convergent, are examples of series of Jacobi forms invariant under an involution. The open problem is the converse: do any such formal series converge and thereby define paramodular forms? Evidence for this converse is given by the success of computer programs that rigorously determine specific vector spaces of paramodular forms. We reduce the open problem to the behavior of a vector space under the action of a Hecke algebra.
Date: Nov. 9.
Place: LL 902 (LC)
Speaker: Giancarlo Castellano
Title: Special values of Rankin–Selberg $L$-functions for $\GL_n \times \GL_m$ over totally real fields
Abstract: In recent years, considerable advancements have been made in the arithmetic of automorphic $L$-functions. Ever since the seminal work of Raghuram–Shahidi (2008), progress has been especially fruitful in relating special values of Rankin–Selberg automorphic $L$-functions for $\GL_n \times \GL_{n-1}$ to representation-theoretic invariants of cohomological automorphic representations known as \emph{Whittaker–Betti periods}. The case where both representations are cuspidal was settled in Raghuram (2016) for arbitrary number fields; work of Grobner–Harris (2016), Grobner (2018), Grobner–Lin (2021) and Li–Liu–Sun (2021) progressively extended those results to the case where the representation of $\GL_{n-1}$ is allowed to be isobaric.
In contrast, a lot less seems to be known about special values of Rankin–Selberg automorphic $L$-functions for $\GL_n \times \GL_m$ with $1 \le m < n - 1$. Among the most prominent results we have, on the one hand, a theorem of Grobner–Sachdeva over CM-fields (with a parity restriction on $m$ and $n$), and on the other, results of Harder–Raghuram and Raghuram about \emph{quotients} of special $L$-values (the base field being assumed totally real and totally imaginary, respectively) where such quotients are brought into connection with quantities known as \emph{relative periods}.
In this talk, I will relate the main outcomes of my PhD work. On the one hand, I obtained a result on special values of Rankin–Selberg $L$-functions for $\GL_n \times \GL_m$ over a totally real field, with $n$ odd and $m < n$ even, by a method similar to the one used in the work of Grobner–Sachdeva. As a consequence, one obtains a corollary on quotients of such special values, which, when compared and combined with Harder–Raghuram's results, yields a relation between Whittaker periods, on the one hand, and Harder–Raghuram relative periods on the other. In the special case $m = n - 1$, the latter result is to be compared with earlier work of Raghuram about relative periods being equal to quotients of Whittaker periods of opposite signatures, up to an explicit power of the imaginary unit $\mathrm{i}$.
Date: Nov. 16.
Place: LL 902 (LC)
Speaker: Han-Bom Moon
Title: Derived category of moduli space of vector bundles
Abstract: The derived category of moduli spaces of vector bundles on a curve is expected to be decomposed into the derived categories of symmetric products of the base curve. I will briefly explain the statement and how representation theory is involved in the proof of this statement.
Date: Nov. 30.
Place: LL 902 (LC)
Speaker: Tamar Blanks
Title: Witt invariants of Weyl groups
Abstract: The trace form is a simple and powerful tool in the study of fields and field extensions. For a finite-dimensional extension $K/F$, it is the symmetric bilinear form over $F$ sending $(x,y) \in K \times K$ to the trace $Tr_{K/F}(xy)$. In generalizing this notion, one can ask: given a particular type of algebraic structure (such as field extensions), what kind of quadratic form data is available? What invariants exist, if we want the invariant to be valued in the Witt ring of quadratic forms?
More precisely, for a smooth linear algebraic group $G$ over a field $k$, a Witt invariant of $G$ is a natural transformation from $H^1(-,G)$ to $W(-)$, that is, a rule that assigns an element of the Witt ring to each $G$-torsor in a way that is compatible with extension of scalars. In 2003, J.-P. Serre showed that when $G=S_n$, all Witt invariants come from the trace form. I will discuss recent work extending Serre's result from $S_n$ to other Weyl groups, and place this work in the broader context of cohomological invariants. I will then outline some related challenges and open questions in this area.