UCSD Number Theory Seminar (Math 209)

Thursdays 2-3pm PST, APM 6402 and virtually

We will run a "hybrid" seminar this quarter, where we meet in the seminar room and simultaneously over Zoom. Register via this link to get the Zoom link via email.

Most talks will be preceded by a "pre-talk" for graduate students and postdocs only, starting 40 minutes before the announced time for the main talk and lasting about 30 minutes. Pre-talks are also available via Zoom.

Don't forget to register for Math 209 if you are a UCSD graduate student. Continued department financial support for this seminar is contingent on maintaining sufficient enrollment.

To subscribe to our weekly seminar announcement, or to join the number theory group's Zulip discussion server for additional announcements, please contact the organizers.

As of spring 2020, this site is dynamically generated from researchseminars.org, which see for other seminars worldwide (or for this seminar listed in your local timezone).

Winter Quarter 2023

For previous quarters' schedules, click here.

January 12

Arghya Sadhukhan (Maryland)
Understanding the dimension of some (union of) affine Deligne-Lusztig varieties via the quantum Bruhat graph

The study of affine Deligne-Lusztig varieties (ADLVs) $X_w(b)$ and their certain union $X(\mu,b)$ has been crucial in understanding mod-$p$ reduction of Shimura varieties; for instance, precise information about the connected components of ADLVs (in the hyperspecial level) has proved to be useful in Kisin's proof of the Langlands-Rapoport conjecture. On the other hand, first introduced in the context of enumerative geometry to describe the quantum cohomology ring of complex flag varieties, quantum Bruhat graphs have found recent applications in solving certain problems on the ADLVs. I will survey such developments and report on my work surrounding a dimension formula for $X(\mu,b)$ in the quasi-split case, as well as some partial description of the dimension and top-dimensional irreducible components in the non quasi-split case.

January 19

Longke Tang (Princeton)
Prismatic Poincaré duality

Prismatic cohomology is a new p-adic cohomology theory introduced by Bhatt and Scholze that specializes to various well-known cohomology theories such as étale, de Rham and crystalline. I will roughly recall the properties of this cohomology and explain how to prove its Poincaré duality.

January 26

Nolan Wallach (UC San Diego)
The Whittaker Inversion Theorem and some applications

The Whittaker Plancherel theorem appeared as Chapter 15 in my two volume book, Real Reductive Groups. It was meant to be an application of Harish-Chandra’s Plancherel Theorem. As it turns out, there are serious gaps in the proof given in the books. At the same time as I was doing my research on the subject, Harish-Chandra was also working on it. His approach was very different from mine and appears as part of Volume 5 of his collected works; which consists of three pieces of research by Harish-Chandra that were incomplete at his death and organized and edited by Gangolli and Varadarajan. Unfortunately, it also does not contain a proof of the theorem. There was a complication in the proof of this result that caused substantial difficulties which had to do with the image of the analog of Harish-Chandra’s method of descent. In this lecture I will explain how one can complete the proof using a recent result of Raphael Beuzzart-Plessis. I will also give an application of the result to the Fourier transforms of automorphic functions at cusps. (This seminar will be given remotely, but there will still be a live audience in the lecture room.)

February 2

Gilyoung Cheong (UC Irvine)
Polynomial equations for matrices over integers modulo a prime power and the cokernel of a random matrix

Over a commutative ring of finite cardinality, how many $n \times n$ matrices satisfy a polynomial equation? In this talk, I will explain how to solve this question when the ring is given by integers modulo a prime power and the polynomial is square-free modulo the prime. Then I will discuss how this question is related to the distribution of the cokernel of a random matrix and the Cohen--Lenstra heuristics. This is joint work with Yunqi Liang and Michael Strand, as a result of a summer undergraduate research I mentored.

February 9

Simon Marshall (Wisconsin)
Large values of eigenfunctions on hyperbolic manifolds

It is a folklore conjecture that the sup norm of a Laplace eigenfunction on a compact hyperbolic surface grows more slowly than any positive power of the eigenvalue. In dimensions three and higher, this was shown to be false by Iwaniec-Sarnak and Donnelly. I will present joint work with Farrell Brumley that strengthens these results, and extends them to locally symmetric spaces associated to $\mathrm{SO}(p,q)$.

February 16

Daniel Vallieres (CSU Chico/UC San Diego)
Iwasawa theory and graph theory

Analogies between number theory and graph theory have been studied for quite some times now. During the past few years, it has been observed in particular that there is an analogy between classical Iwasawa theory and some phenomena in graph theory. In this talk, we will explain this analogy and present some of the results that have been obtained so far in this area.

February 23

Sameera Vemulapalli (Princeton)
Counting low degree number fields with almost prescribed successive minima

The successive minima of an order in a degree n number field are n real numbers encoding information about the Euclidean structure of the order. How many orders in degree n number fields are there with almost prescribed successive minima, fixed Galois group, and bounded discriminant? In this talk, I will address this question for n = 3, 4, 5. The answers, appropriately interpreted, turn out to be piecewise linear functions on certain convex bodies. If time permits, I will also discuss function field analogues of this problem.

March 2

Yujie Xu (MIT)
Hecke algebras for p-adic groups and the explicit Local Langlands Correspondence for G_2

I will talk about my recent joint work with Aubert where we prove the Local Langlands Conjecture for G_2 (explicitly). This uses our earlier results on Hecke algebras attached to Bernstein components of reductive p-adic groups, as well as an expected property on cuspidal support, along with a list of characterizing properties. In particular, we obtain "mixed" L-packets containing F-singular supercuspidals and non-supercuspidals.

March 9

Jef Laga (Cambridge)
Arithmetic statistics via graded Lie algebras

I will explain how various results in arithmetic statistics by Bhargava, Gross, Shankar and others on 2-Selmer groups of Jacobians of (hyper)elliptic curves can be organised and reproved using the theory of graded Lie algebras, following earlier work of Thorne. This gives a uniform proof of these results and yields new theorems for certain families of non-hyperelliptic curves. I will also mention some applications to rational points on certain families of curves. The talk will involve a mixture of representation theory, number theory and algebraic geometry and I will assume no familiarity with arithmetic statistics.

March 16

Xu Gao (UC Santa Cruz)
$p$-adic representations and simplicial distance in Bruhat-Tits buildings

$p$-adic representations are important objects in number theory, and stable lattices serve as a connection between the study of ordinary and modular representations. These stable lattices can be understood as stable vertices in Bruhat-Tits buildings. From this viewpoint, the study of fixed point sets in these buildings can aid research on $p$-adic representations. The simplicial distance holds an important role as it connects the combinatorics of lattices and the geometry of root systems. In particularly, the fixed-point sets of Moy-Prasad subgroups are precisely the simplicial balls. In this talk, I'll explain those findings and compute their simplicial volume under certain conditions.