Calendar
Posted August 2, 2025
Last modified August 20, 2025
Algebra and Number Theory Seminar Questions or comments?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
Joseph DiCapua, Louisiana State University
Lubin–Tate Formal Group Laws
In this expository talk, we introduce Lubin–Tate formal group laws. The torsion points of a Lubin–Tate formal group law are defined, and we discuss the endomorphism ring of such a formal group law. Certain torsion points are used to define Coleman's trace operator, an important tool in Iwasawa theory. We briefly mention how Lubin–Tate formal group laws are used in one construction of the maximal abelian extension of a finite extension of the $p$-adics.
Posted August 2, 2025
Last modified August 20, 2025
Algebra and Number Theory Seminar Questions or comments?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
Joseph DiCapua, Louisiana State University
Parametrization of Formal Norm Compatible Sequences
We give a classification of power series parametrizing Lubin–Tate trace compatible sequences. This proof answers a question posed in the literature by Berger and Fourquaux. Lubin–Tate trace compatible sequences are a generalization of norm compatible sequences, which arise in Iwasawa theory and local class field theory. The result we prove generalizes the interpolation theorem proved by Coleman in the classical norm compatible sequence case. We also, jointly with Victor Kolyvagin, give a method for finding such series explicitly in certain special cases.
Posted August 2, 2025
Last modified September 10, 2025
Algebra and Number Theory Seminar Questions or comments?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
Hang Xue, The University of Arizona
Fourier–Jacobi periods on unitary groups
We explain what Fourier–Jacobi periods on unitary groups are and prove the global Gan–Gross–Prasad conjecture about them. We also give an application to the Tate conjecture of product of unitary Shimura varieties.
Posted August 2, 2025
Last modified September 10, 2025
Algebra and Number Theory Seminar Questions or comments?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
Andreas Mono, Vanderbilt University
A modular framework for generalized Hurwitz class numbers
We discover a neat linear relation between the mock modular generating functions of the level $1$ and level $N$ Hurwitz class numbers. This relation gives rise to a holomorphic modular form of weight $\frac{3}{2}$ and level $4N$ for $N > 1$ odd and square-free. This follows from a more general inspection of the weight $\frac{1}{2}$ Maass–Eisenstein series of level $4N$v at its spectral point $s = \frac{3}{4}$. This idea goes back to Duke, Imamoğlu and Tóth in level $4$ and relies on the theory of so-called sesquiharmonic Maass forms. Furthermore, we connect the aforementioned results to a regularized Siegel theta lift as well as a regularized Kudla–Millson theta lift for odd prime levels, which builds on earlier work by Bruinier, Funke and Imamoğlu. This is joint work with Olivia Beckwith. We conclude by presenting the situation in higher weights as well, which is sole work.
Posted September 23, 2025
Algebra and Number Theory Seminar Questions or comments?
2:00 pm Lockett 233 or click here to attend on Zoom
Liang Chang, Nankai University
Modular data of non-semisimple modular categories
Modular tensor categories are semisimple tensor categories with nondegenerated braiding, which have many applications in low dimensional topology and topological physics. Recently, the notion of modularity is extended to non-semisimple tensor category. In this talk, we will talk about the work to extend the well-understood theory of semisimple modular categories, such as the SL(2, Z)-representation and rank finiteness, to the non-semisimple case by using representations of factorizable ribbon Hopf algebras.
Posted September 30, 2025
Algebra and Number Theory Seminar Questions or comments?
2:00 pm Lockett 233 or click here to attend on Zoom
Jianqi Liu, University of Pennsylvania
Modular invariance of intertwining operators from the factorization theorem of conformal blocks
The notion of conformal blocks on stable curves defined by modules over a vertex operator algebra (VOA) generalizes the WZNW-conformal blocks defined by modules over affine Lie algebras. Recent advances in the theory of VOA-conformal blocks have shed new light on the representation theory of VOAs. In particular, the factorization theorem and the vector bundle property of the sheaves of VOA-conformal blocks lead to the modular invariance of intertwining operators for strongly rational VOAs. In this talk, I will explain the proof of this theorem and present a short proof of the associativity of fusion rings for VOAs. This talk is based on joint work with Xu Gao.
Posted August 2, 2025
Last modified October 8, 2025
Algebra and Number Theory Seminar Questions or comments?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
Kenz Kallal, Princeton University
Algebraic theory of indefinite theta functions
Jacobi's theta function $\Theta(q) := 1 + 2q + 2q^4 + 2q^9 + \cdots $, and more generally the theta functions associated to positive-definite quadratic forms, have the property that they are modular forms of half-integral weight. The usual proof of this fact is completely analytic in nature, using the Poisson summation formula. However, $\Theta$ was originally invented by Fourier (Théorie analytique de la chaleur, 1822) for the purpose of studying the diffusion of heat on a uniform circle-shaped material: it is the fundamental solution to the heat equation on a circle. By algebraically characterizing the heat equation as a specific flat connection on a certain bundle on a modular curve, we produce a completely algebraic technique for proving modularity of theta functions. More specifically, we produce a refinement of the algebraic theory of theta functions due to Moret-Bailly, Faltings–Chai, and Candelori. As a consequence of the algebraic nature of our theory and the fact that it applies to indefinite quadratic forms / non-ample line bundles (which the prior algebraic theory does not), we also generalize the Kudla–Millson analytic theory of theta functions for indefinite quadratic forms to the case of torsion coefficients. This is joint work in progress with Akshay Venkatesh.
Posted September 2, 2025
Algebra and Number Theory Seminar Questions or comments?
2:00 pm Lockett 233 or click here to attend on Zoom
Guanyu Li, Cornell University
Derived Commuting Schemes, Representation Homology, and Cohomology of Lie Algebras
The commuting schemes of an algebraic group or a Lie algebra play important roles in many areas of mathematics. They can be viewed as special cases of representation schemes, which are often highly singular. Derived algebraic geometry provides tools to remedy the deficiency. In particular, the derived representation scheme, together with its associated algebraic invariant known as representation homology, offers deeper insights into the structure of representation schemes. While the representation homology of reductive groups and reductive Lie algebras has been studied in the literature, it is natural to ask about the behavior of these objects and their relationships in the non-reductive setting. In this talk, I will discuss the derived commuting scheme of a maximal unipotent subgroup of a semisimple group scheme, as well as the derived commuting scheme of its Lie algebra. First, the higher structure of the derived commuting scheme detects whether the underlying commuting scheme is a complete intersection. Unlike the reductive case, the derived commuting scheme of a unipotent subgroup is equivalent to that of its Lie algebra. Using an analogue of the trace map, most of the homology classes can be explained in terms of the classical cohomology of a maximal nilpotent Lie algebra, described via the root system of the semisimple Lie algebra. This could be interpreted that the singularities of the commuting scheme of a maximal nilpotent subalgebra are largely determined by root system data. If time permits, I will also discuss a possible nilpotent analogue of the Macdonald identity, together with an interpretation in terms of representation homology.
Posted September 9, 2025
Algebra and Number Theory Seminar Questions or comments?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
Kalani Thalagoda, Tulane University
A summation formula for Hurwitz class numbers
The Hurwitz class numbers, $H(n)$, count ${\rm SL}(2,\mathbb{Z})$-classes of binary quadratic forms inversely weighted by stabilizer size. They are famously connected to the sum of three squares problem and to class numbers of imaginary quadratic fields. The work of Zagier in 1975 showed that their generating functions are related to a weight $3/2$ Harmonic Maass form. In this talk, I will discuss a summation formula for mock modular forms of moderate growth, with an emphasis on its application to Hurwitz class numbers. This is joint work with Olivia Beckwith, Nicholas Diamantis, Rajat Gupta, and Larry Rolen.
Posted October 26, 2025
Last modified November 1, 2025
Algebra and Number Theory Seminar Questions or comments?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
Che-Wei Hsu, National Taiwan University
Hypergeometric Evaluations of L-values and Harmonic Maass Forms
In their earlier work, Bruinier, Ono, and Rhoades asked for an explicit construction of good harmonic Maass forms associated with CM newforms. Building on Ramanujan's theories of elliptic functions to alternative bases, we express $L$-values of certain weakly holomorphic cusp forms in terms of hypergeometric functions. As an application, we construct harmonic Maass forms with explicitly computable Fourier coefficients that are good for specific Hecke eigenforms including certain CM cusp forms.
In this talk, I will briefly review the basic notions of harmonic Maass forms and then present the ongoing joint work with Jia-Wei Guo, Fang-Ting Tu, and Yifan Yang.
Posted November 15, 2025
Algebra and Number Theory Seminar Questions or comments?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
Jiaqi Hou, Louisiana State University
Restriction bounds for Maass forms
I will talk about the analytic problem of bounding Hecke–Maass forms. From the general theory of bounding Laplace eigenfunctions on Riemannian manifolds, one obtains local bounds for many different kinds of norms, and these bounds are believed to be far from optimal if the manifold is negatively curved. I will discuss how Hecke–Maass forms on arithmetic hyperbolic 3-folds behave along totally geodesic surfaces and present an improved L^2 bound by the method of arithmetic amplification.
Posted November 15, 2025
Last modified December 1, 2025
Algebra and Number Theory Seminar Questions or comments?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
Esme Rosen, Louisiana State University
Hypergeometric Motives and Modular Forms
We will discuss hypergeometric motives and their different aspects, which include classical hypergeometric functions and hypergeometric functions over finite fields. Using the Explicit Hypergeometric Modularity Method recently introduced by Allen, Grove, Long, and Tu, we also explain the relationship between certain special hypergeometric motives and modular forms.
Posted November 15, 2025
Last modified January 21, 2026
Algebra and Number Theory Seminar Questions or comments?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
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?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
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?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
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?
1:20 pm – 2:20 pm Lockett 233 or click here to attend on Zoom
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?
2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
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?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
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?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
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?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
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
Last modified May 8, 2026
Algebra and Number Theory Seminar Questions or comments?
2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom
Caroline Semmens, University of Arkansas
Isogeny-Torsion Graphs of Some Quadratic Number Fields
An elliptic curve over a number field $K$ is a smooth projective curve $E$ with a point defined over $K$. By the Mordell–Weil theorem, the $K$-rational points on elliptic curves form a finitely generated abelian group. Isogenies of elliptic curves are maps between elliptic curves which are also group homomorphisms. We can organize isogeny classes into graphs and label the vertices with the torsion structure of corresponding elliptic curves. These graphs are called isogeny-torsion graphs, and in 2021, Chiloyan and Lozano-Robledo classified all isogeny torsion graphs over $\mathbb{Q}$. In this talk, we explore progress on this classification question over other number fields, using work done by Banwait, Najman, and Padurariu extending Mazur’s theorem. Of particular interest is the quadratic field $K = \mathbb{Q}(\sqrt{213})$. This talk is based on joint work with Clayton Boothe, Michael Logal, and Lance E Miller.
Event contact: Richard Ng and Gene Kopp