Calendar

Posted January 11, 2025

Algebra and Number Theory Seminar Questions or comments?

Walter Bridges, University of North Texas.
The proportion of coprime fractions in number fields

The ring $\mathbb{Z}[\sqrt{-5}]$ is often one of the first examples students encounter of a ring that is not unique factorization domain. Relatedly, in the number field $\mathbb{Q}(\sqrt{-5})$, we have $$ \frac{1+\sqrt{-5}}{2}=\frac{3}{1-\sqrt{-5}}. $$ Both fractions are reduced, meaning that numerator and denominator do not share any (non-unit) factors in $\mathbb{Z}[\sqrt{-5}]$. However, neither fraction is coprime, in the sense that the numerator and denominator pair do not generate $\mathbb{Z}[\sqrt{-5}]$. In this talk, we will answer the question of how often this phenomenon occurs. That is, we compute the density, suitably defined, of the set of coprime fractions in the set of all reduced fractions in a generic number field. Our answer for $\mathbb{Q}(\sqrt{-5})$ is 80%. We will begin with a review of algebraic number theory, then discuss our notion of density in number fields. Finally, we will show that the density in question may be computed using well-known properties of Hecke L-functions. We intend this talk to be accessible to beginning graduate students.

Posted January 19, 2025

Algebra and Number Theory Seminar Questions or comments?

Asimina Hamakiotes, University of Connecticut
Abelian extensions arising from elliptic curves with complex multiplication

Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f \geq 1$. Let $E$ be an elliptic curve with complex multiplication by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j(E))$, where $j(E)$ is the $j$-invariant of $E$. Let $N\geq 2$ be an integer. The extension $\mathbb{Q}(j(E), E[N])/\mathbb{Q}(j(E))$ is usually not abelian; it is only abelian for $N=2,3$, and $4$. Let $p$ be a prime and let $n\geq 1$ be an integer. In this talk, we will classify the maximal abelian extension contained in $\mathbb{Q}(E[p^n])/\mathbb{Q}$.

Posted January 26, 2025
Last modified February 24, 2025

Algebra and Number Theory Seminar Questions or comments?

Akio Nakagawa, Kanazawa University
Hypergeometric functions over finite fields

In this talk, I will explain about Otsubo’s definition of hypergeometric functions over finite fields, and I will introduce how the confluent hypergeometric functions over finite fields are useful by showing a transformation formula for Appell–Lauricella functions over finite fields. If time allows, I will introduce my recent work on relations between hypergeometric functions and algebraic varieties.

Posted January 28, 2025
Last modified March 12, 2025

Algebra and Number Theory Seminar Questions or comments?

Eun Hye Lee, Texas Christian University
Automorphic form twisted Shintani zeta functions over number fields

In this talk, we will be exploring the analytic properties of automorphic form twisted Shintani zeta functions over number fields. I will start by stating some basic facts from classical Shintani zeta functions, and then we will take a look at the adelic analogues of them. Joint with Ramin Takloo-Bighash.

Posted March 31, 2025

Algebra and Number Theory Seminar Questions or comments?

Be'eri Greenfeld , University of Washington
Complexity and Growth of Infinite Words and Algebraic Structures

Given an infinite word (for example, 01101001$\ldots$), its complexity function counts, for each n, the number of distinct subwords of length n. A longstanding open problem is the "inverse problem": Which functions $f:\mathbb N\to \mathbb N$ arise as complexity functions of infinite words? We resolve this problem asymptotically, showing that, apart from submultiplicativity and a classical obstruction found by Morse and Hedlund in 1938, there are essentially no further restrictions. We then explore parallels and contrasts with the theory of growth of algebras, drawing on noncommutative constructions associated with symbolic dynamical systems.

Posted January 26, 2025
Last modified April 14, 2025

Algebra and Number Theory Seminar Questions or comments?

Kairi Black, Duke University
How might we generalize the Kronecker-Weber Theorem?

Hilbert's Twelfth Problem asks for a generalization of the Kronecker-Weber Theorem: cyclotomic units generate the abelian extensions of $\mathbb{Q}$, but what about for other ground fields? We consider a number field $K$ with exactly one complex embedding. In the 1970s, Stark conjectured formulas for (the absolute values of) units inside abelian extensions of $K$. We refine Stark's conjectures with a proposed formula for the units themselves, not just their absolute values.

Posted April 14, 2025

Algebra and Number Theory Seminar Questions or comments?

Qing Zhang, University of California Santa Barbara
Realizing Modular Data from Centers of Near-Group Categories

In this talk, I will discuss modular data arising from the Drinfeld centers of near-group categories. The existence of near-group categories of type $G+n$ can be established by solving a set of polynomial equations introduced by Izumi; a different set of equations, also due to Izumi, can then be used to compute the modular data of their Drinfeld centers. Smaller-rank modular categories can often be obtained from these centers via factorization and condensation. After introducing the background of this framework, I will show the existence of a near-group category of type $\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z} + 16$ and explain how the modular data of its Drinfeld center can be computed. I will then show that modular data of rank 10 can be obtained via condensation of its Drinfeld center and present an alternative realization of this data through the Drinfeld center of a fusion category of rank 4. Finally, I will discuss the modular data of the Drinfeld center of a near-group category of type $\mathbb{Z}/8\mathbb{Z} + 8$ and demonstrate that the non-pointed factor of its condensation coincides with the modular data of the quantum group category $C(g_2, 4)$. This talk is based on joint work with Zhiqiang Yu.

Posted April 16, 2025

Algebra and Number Theory Seminar Questions or comments?

Andrew Riesen, MIT
Orbifolds of Pointed Vertex Algebras

We will discuss the interplay of tensor categories $C$ with some group action $G$ and orbifolds $V^G$ of vertex operator algebras (VOAs for short). More specifically, we will show how the categorical structure of $\mathrm{TwMod}_G V$ allows one to not only simplify previous results done purely through VOA techniques but vastly extend them. One such example is the Dijkgraaf-Witten conjecture, now a theorem, which describes how the category of modules of a holomorphic orbifold should look like. Additionally, our techniques also allow us to expand the modular fusion categories known to arise from VOAs, we show that every group-theoretical fusion category comes from a VOA orbifold. This talk is based on joint work with Terry Gannon.