Calendar

Posted November 15, 2025
Last modified January 21, 2026

Algebra and Number Theory Seminar Questions or comments?

Olivia Beckwith, Tulane University
Polyharmonic Maass forms and Hecke series for real quadratic fields

We study polyharmonic Maass forms and show that they are related to ray class extensions of real quadratic fields. In particular, we generalize work of Lagarias and Rhoades to give a basis for the space of polyharmonic Maass forms for $\Gamma(N)$. Modifying an argument of Hecke, we show that twisted traces of cycle integrals of certain depth 2 polyharmonic Maass forms are leading coefficients of Hecke $L$-series of real quadratic fields. This is ongoing joint work with Gene Kopp.

Event contact: Gene Kopp

Posted January 16, 2026
Last modified February 2, 2026

Algebra and Number Theory Seminar Questions or comments?

Koustav Mondal, Louisiana State University
Theta series and their applications

Theta series play a central role in many areas of mathematics, especially number theory. In this talk, we begin with a brief overview of two applications of theta series: point counting for congruent quadratic forms, and the evaluation of special values of $L$-functions via Ramanujan's theory of elliptic functions to alternative bases for modular forms. Along the way, we state our main results in each setting. In the second part of the talk, we return to these applications to outline the key ideas and techniques involved in the proofs, as time permits.

Event contact: Gene Kopp

Posted November 15, 2025
Last modified January 21, 2026

Algebra and Number Theory Seminar Questions or comments?

Marco Sangiovanni Vincentelli, Columbia University
An Euler system for the adjoint of a modular form

Euler systems have proven to be versatile tools for understanding Selmer groups and their connections to special values of $L$-functions. However, despite the key role they have played in making progress toward foundational conjectures in number theory, such as the Birch and Swinnerton-Dyer and Bloch–Kato Conjectures, only a handful of provably non-trivial Euler systems have been constructed to date. A significant obstacle to constructing Euler Systems lies in producing candidate Galois cohomology classes. This lecture series presents joint work with Chris Skinner that develops a method to overcome this obstacle. Using this method, we construct an Euler system for the adjoint of a modular form.

Event contact: Gene Kopp

Posted November 15, 2025
Last modified March 22, 2026

Algebra and Number Theory Seminar Questions or comments?

Kiran Kedlaya, University of California San Diego
Implementing the hypergeometric trace formula

Given parameters defining a hypergeometric motive, the trace is given by a rather explicit formula which can be written either in terms of Gauss sums (Beukers–Cohen–Mellit) or, thanks to the Gross–Koblitz formula, the Morita p-adic Gamma function (Cohen–Rodriguez Villegas–Watkins). We explain some of the process of turning this formula into an efficient algorithm "at scale", including an adaptation to compute Frobenius traces in "average polynomial time" in the sense of David Harvey's recent Arizona Winter School lectures; that part is joint with Edgar Costa and David Harvey.

Event contact: Hasan Saad and Gene Kopp

Posted March 17, 2026
Last modified March 22, 2026

Algebra and Number Theory Seminar Questions or comments?

Ian Jorquera, Colorado State University
Switching equivalence of systems of lines over finite fields

In this talk we will discuss important frame theoretic objects such at equiangular tight frames (ETFs) whose existence has important applications in fields as diverse as compressed sensing to quantum state tomography. We will then discuss some new approaches to tackling some open problems, on the existence and structure of these frame theoretic objects, by using tools from geometric algebra, and specially looking at frames over finite field vector spaces with Hermitian forms. We will then show that the switching equivalence classes of systems of lines over finite fields which are frames, often only depend on the double and triple products. This allows us to understand ETFs over finite fields in terms of their double and triple products, with a result similar to saturating the Welch bound over $\mathbb{C}$. We also show that similar to the case over $\mathbb{C}$, collections of vectors are similar to a regular simplex essentially when their triple products satisfy a certain property.

Event contact: Gene Kopp

Posted March 17, 2026
Last modified March 30, 2026

Algebra and Number Theory Seminar Questions or comments?

Shahriyar Roshan-Zamir, Tulane University
Interpolation in Weighted Projective Spaces

Over an algebraically closed field, the double point interpolation problem asks for the vector space dimension of the projective hypersurfaces of degree d singular at a given set of points. After being open for 90 years, a series of papers by J. Alexander and A. Hirschowitz in 1992--1995 settled this question in what is referred to as the Alexander-Hirschowitz theorem. In this talk, we primarily use commutative algebra to prove analogous statements in the weighted projective space, a natural generalization of the projective space. For example, we introduce an inductive procedure for weighted projective space, similar to that originally due to A. Terracini from 1915, to demonstrate an example of a weighted projective plane where the analogue of the Alexander-Hirschowitz theorem holds without exceptions and prove our example is the only such plane. Furthermore, Terracini's lemma regarding secant varieties is adapted to give an interpolation bound for an infinite family of weighted projective planes. There are no prerequisites for this talk besides some elementary knowledge of algebra.

Event contact: Gene Kopp

Posted April 20, 2026

Algebra and Number Theory Seminar Questions or comments?

Shilin Lai, University of Michigan
Euler system test vectors and relative Satake isomorphism

The construction of Euler systems often involves delicate choices of local test vectors. Using the relative Satake isomorphism, in particular the unramified Plancherel formula, we give a conceptual proof of their existence in many settings. As an example, we will treat the Gan–Gross–Prasad and Friedberg–Jacquet case uniformly. This is joint work with Li Cai and Yangyu Fan.

Event contact: Joseph DiCapua and Gene Kopp

Posted April 21, 2026

Algebra and Number Theory Seminar Questions or comments?

Mengwei Hu, Yale University
On certain Lagrangian subvarieties in minimal resolutions of Kleinian singularities

Kleinian singularities are quotients of C^2 by finite subgroups of SL_2(C). They are in bijection with the ADE Dynkin diagrams via the McKay correspondence. In this talk, I will introduce certain singular Lagrangian subvarieties in the minimal resolutions of Kleinian singularities, motivated by the geometric classification of unipotent Harish-Chandra (g,K)-modules. The irreducible components of these singular Lagrangian subvarieties are P^1's and A^1's. I will describe how they intersect with each other through the realization of Kleinian singularities as Nakajima quiver varieties. I will also discuss their connections with nilpotent K-orbits and symmetric pairs in semisimple Lie algebras.

Posted April 28, 2026

Algebra and Number Theory Seminar Questions or comments?

Caroline Semmens, University of Arkansas
TBA

Event contact: Richard Ng and Gene Kopp