Calendar

Posted August 2, 2025
Last modified August 20, 2025

Algebra and Number Theory Seminar Questions or comments?

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?

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?

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?

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?

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?

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?

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?

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?

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.