Posted November 13, 2023

Last modified January 21, 2024

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Rena Chu, Duke University

Generalizations of the Schrödinger maximal operator: building arithmetic counterexamples

In 2016 Bourgain applied Gauss sums to construct a counterexample related to a decades-old question in PDEs. The story started in 1980 when Carleson asked about how "smooth" an initial data function must be to imply pointwise convergence for the solution of the linear Schrödinger equation. After progress by many authors, this was resolved by Bourgain, whose counterexample construction proved a necessary condition on the regularity, and Du and Zhang, who proved a sufficient condition. Bourgain's methods were number-theoretic, and this raised a natural question: could number-theoretic properties of other exponential sums have implications for other dispersive PDEs? We develop a flexible new method to construct counterexamples for analogues of Carleson's question. In particular, this applies the Weil bound for exponential sums, a consequence of the truth of the Riemann Hypothesis over finite fields.

Posted November 13, 2023

Last modified January 21, 2024

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Zhongkai Mi, Louisiana State University

The Lowest Discriminant Ideals of Cayley-Hamilton Hopf Algebras

Discriminant ideals for an algebra $A$ module finite over a central subring $C$ are indexed by positive integers. We study the lowest of them with nonempty zero set in Cayley Hamilton Hopf algebras whose identity fibers are basic algebras. Key results are obtained by considering actions of characters in the identity fiber on irreducible modules over maximal ideals of $C$ and actions of winding automorphisms. We apply these results to examples in group algebras of central extensions of abelian groups, big quantum Borel subalgebras at roots of unity and quantum coordinate rings at roots of unity. This is joint work with Quanshui Wu and Milen Yakimov.

Posted November 13, 2023

Last modified February 2, 2024

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Rajat Gupta, University of Texas at Tyler

On summation formulas attached to Hecke's functional equation and $p$-Herglotz functions

In this talk, we will review the work of Chandrasekharan and Narasimhan on the theory of Hecke’s functional equation (with one gamma factor) and the summation formulas of various kinds, such as the Voronoi summation formula, the Poisson summation formula, and the Abel-Plana summation formula. We will then give recent developments in this theory followed by some new results and summation formulas in the setting of Hecke’s functional equation analogous to the ones mentioned above. In particular, I will discuss these summation formulas in the case of cusp corms of weight $2k$ attached to the modular group ${\rm SL}_2(\mathbb{Z})$. Finally, I will also talk about on my recent work with Rahul Kumar on Herglotz functions and their analogues.

Posted February 5, 2024

Last modified February 14, 2024

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Hasan Saad, University of Virginia

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."

Posted November 14, 2023

Last modified February 27, 2024

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Eleanor McSpirit, University of Virginia

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.

Posted November 13, 2023

Last modified March 3, 2024

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Edmund Yik-Man Chiang, The Hong Kong University of Science and Technology

Discrete special functions: a D-modulus approach to special functions

We show that there is a holonomic D-modules (PDEs) approach to classical special functions, as such both the classical special functions and their difference analogues, some have only been found recently, can be efficiently computed by Weyl-algebraic framework. According to Truesdell, rudiments of algebraic approaches to special functions were already observed by some nineteenth century mathematicians. This algebraic method does not use well-known Lie algebra theory explicity apart from basic knowledge of solving linear PDEs. We illustrate our method with Bessel functions in this talk. We shall also explain the connection with this D-modules approach with recent advances in Nevanlinna theories for difference operators, which have its roots from discrete Painleve equations.

Posted November 14, 2023

Last modified March 26, 2024

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Micah Milinovich, University of Mississippi

Biases in the gaps between zeros of Dirichlet L-functions

We describe a family of Dirichlet L-functions that provably have unusual value distribution and experimentally have a significant and previously undetected bias in the distribution of gaps between their zeros. This has an arithmetic explanation that corresponds to the nonvanishing of a certain Gauss-type sum. We give a complete classification of the characters for when these sums are nonzero and count the number of corresponding characters. It turns out that this Gauss-type sum vanishes for 100% of primitive Dirichlet characters, so L-functions in our newly discovered family are rare (zero density set amongst primitive characters). If time allows, I will also describe some newly discovered experimental results concerning a "Chebyshev-type" bias in the gaps between the zeros of the Riemann zeta-function. This is joint work with Jonathan Bober (Bristol) and Zhenchao Ge (Waterloo).