Algebra, Geometry, and Number Theory Seminar

Abstracts

Will Craig - Statistical Properties of Hook Numbers

In this talk, we consider arithmetic and statistical properties of hook numbers in partitions using tools of analytic number theory and $q$-series. We consider the following three questions: (1) Is the count of the number of hooks divisible by $t$ in a partition biased towards certain arithmetic progressions? (2) How do restrictions on the parts of partitions affect the prevalence of certain hook lengths? (3) How are hooks of size $t$ distributed among self-conjugate partitions of $n$ as $n$ approaches infinity?

- - - Reserved for Colloquium (Talitha Washington)

Souvik Dey - Projective dimension of tensor product of modules

Given two non-zero finitely generated modules over a commutative Noetherian local ring, the derived tensor product has finite projective dimension if and only if so does each of the modules. This no longer remains true if "derived tensor product" is replaced by ordinary tensor-product. In this talk, we discuss several results illustrating certain hypothesis on the modules or the ring under which finiteness of projective dimension of tensor product two modules implies or is implied by the finiteness of projective dimension of the individual modules. This is based on joint work (some ongoing) with Olgur Celikbas, Toshinori Kobayashi and Hiroki Matsui. (Pretalk at 2:20.)

Sridhar Venkatesh - The Du Bois complex and some associated singularities

For a smooth variety $X$ over $\mathbb{C}$, the de Rham complex of $X$ is a powerful tool to study the geometry of $X$ because of results such as the degeneration of the Hodge-de Rham spectral sequence (when $X$ is proper). For singular varieties, it follows from the work of Deligne and Du Bois that there is a substitute called the Du Bois complex which satisfies many of the nice properties enjoyed by the de Rham complex in the smooth case. In this talk, we will discuss some classical singularities associated with this complex, namely Du Bois and rational singularities, and some recently introduced refinements, namely k-Du Bois and k-rational singularities. This is based on joint work with Wanchun Shen and Anh Duc Vo and joint work with Pat Lank. In the pretalk, I will talk about Kahler differentials: I will define the module of Kahler differentials and discuss some of its properties, particularly in the case of smooth varieties.

Hasan Saad - Distributions of points on hypergeometric varieties

In the 1960's, Birch proved that the traces of Frobenius for elliptic curves taken at random over a large finite field is modeled by the semicircular distribution (i.e. $SU(2),$ the usual Sato-Tate for non-CM elliptic curves). In this talk, we show how the theory of harmonic Maass forms and modular forms allow us to determine the limiting distribution of normalized traces of Frobenius over families of varieties. For Legendre elliptic curves, the limiting distribution is $SU(2),$ whereas for a certain family of $K3$ surfaces, the limiting distribution is $O(3).$ Since the $O(3)$ distribution has vertical asymptotes, we show how to obtain an explicit result by bounding the error. Additionally, we show how to count ``matrix'' points on these varieties and therefore determine the limiting distributions for these ``matrix points''.

Spring Break - No Seminar

Enjoy!

Hui Xue - A Selberg formula for the Shimura lift

The Shimura lift, first defined and studied by Shimura, is a family of maps that send modular forms of half-integral weight to forms of integral weight. A version of the Shimura lift was discovered much earlier by Selberg, which shows that the first Shimura lift of a Hecke eigenform times the theta function is the square of the eigenform. In this talk, we will first present a generalization of Selberg's identity, then we will discuss its relationship with the nonvanishing of central critical values of $L$-functions.

Eleanor McSpirit - Infinite Families of Quantum Modular 3-Manifold Invariants

In 1999, Lawrence and Zagier established a connection between modular forms and the Witten-Reshetikhin-Turaev invariants of 3-manifolds by constructing $q$-series whose radial limits at roots of unity recover these invariants for particular manifolds. These $q$-series gave rise to some of the first examples of quantum modular forms. Using a 3-manifold invariant recently developed Akhmechet, Johnson, and Krushkal, one can obtain infinite families of quantum modular invariants which realize the series of Lawrence and Zagier as a special case. This talk is based on joint work with Louisa Liles, and will discuss the context and motivation for such results as well as the results themselves.

Robert Dicks - The Shimura lift for the eta multiplier and generalized frobenius partitions

Recently, Ahlgren, Andersen, and Dicks proved a Shimura-type correspondence for spaces of half-integral weight cusp forms which transform with a power of the eta multiplier twisted by a Dirichlet character. This Shimura lift has arithmetic applications, such as quadratic congruences modulo arbitrary powers of a prime \ell. In this talk, we discuss this lift with a particular application in mind, that of new congruences for generalized Frobenius partitions. This is upcoming joint work with Ahlgren and Andersen.

Lola Thompson - Salem Numbers and Short Geodesics

We will discuss how Mahler measure and related concepts (e.g., Salem numbers) are connected to problems about lengths of geodesics on arithmetic hyperbolic manifolds. As a result, by solving problems using tools from number theory, we are able to answer quantitative questions in spectral geometry. This talk will build towards two goals: showing that short geodesics on arithmetic hyperbolic surfaces are rare, and showing that, on average, geodesic lengths of non-compact arithmetic hyperbolic orbifolds appear with high multiplicity. This talk is based on joint work with Mikhail Belolipetsky, Matilde Lalin, and Plinio G. P. Murillo; and with Benjamin Linowitz, D. B. McReynolds, and Paul Pollack. The 2:30 pretalk will give an introduction to Mahler measure and related concepts.

Last semester's seminar.