TBD
- Series
- Number Theory
- Time
- Wednesday, April 15, 2026 - 15:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Matt Olechnowicz – Concordia University – matt.olechnowicz@concordia.ca
TBD
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TBD
- Series
- Number Theory
- Time
- Wednesday, February 18, 2026 - 15:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Robert Lemke-Oliver – University of Wisconsin – lemkeoliver@wisc.edu
TBD
TBD
- Series
- Number Theory
- Time
- Wednesday, February 11, 2026 - 15:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Jesse Thorner – University of Illinois Urbana-Champaign – jat9@illinois.edu
TBD
Ultrafilters and uniformity theorems
- Series
- Number Theory
- Time
- Wednesday, December 3, 2025 - 15:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Nicole Looper – University of Illinois Chicago – nrlooper@uic.edu
Ultrafilters formalize a generalized notion of convergence based on a prescribed idea of "largeness" for subsets of the natural numbers, and underlie constructions like ultraproducts. In the study of moduli spaces, they provide a clean way to encode degenerations and to establish uniformity results that are difficult to obtain using ordinary limits. This talk will discuss applications of ultrafilters to uniformity theorems in dynamics and arithmetic geometry. After introducing local results that arise from this approach, I will sketch some of the arithmetic consequences, including uniform bounds on rational torsion points on abelian varieties. This is joint work with Jit Wu Yap
A modular framework for generalized Hurwitz class numbers
- Series
- Number Theory
- Time
- Wednesday, October 22, 2025 - 15:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Olivia Beckwith – Tulane University – obeckwith@tulane.edu
We explore the modular properties of generating functions for Hurwitz class numbers endowed with level structure. Our work is based on an inspection of the weight $\frac{1}{2}$ Maass--Eisenstein series of level $4N$ at its spectral point $s=\frac{3}{4}$, extending the work of Duke, Imamo\={g}lu and T\'{o}th in the level $4$ setting. We construct a higher level analogue of Zagier's Eisenstein series and a preimage under the $\xi_{\frac{1}{2}}$-operator. We deduce a linear relation between the mock modular generating functions of the level $1$ and level $N$ Hurwitz class numbers, giving rise to a holomorphic modular form of weight $\frac{3}{2}$ and level $4N$ for $N > 1$ odd and square-free. 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 Imamo\={g}lu. I wil lbe discussing joint work with Andreas Mono and Ngoc Trinh Le.
Numbers with close factorizations
- Series
- Number Theory
- Time
- Wednesday, October 1, 2025 - 15:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Tsz Ho Chan – Kennesaw State University – tchan4@kennesaw.edu
In this talk, we consider numbers with multiple close factorizations like $99990000 = 9999 \cdot 10000 = 9090 \cdot 11000$ and $3950100 = 1881 \cdot 2100 = 1890 \cdot 2090 = 1900 \cdot 2079$. We discuss optimal bounds on how close these factors can be relative to the size of the original numbers. It is related to the study of close lattice points on smooth curves.
Surfaces associated to zeros of automorphic L-functions
- Series
- Number Theory
- Time
- Wednesday, September 17, 2025 - 15:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Cruz Castillo – University of Illinois Urbana-Champaign – ccasti30@illinois.edu
Assuming the Riemann Hypothesis, Montgomery established results concerning the pair correlation of zeros of the Riemann zeta function. Rudnick and Sarnak extended these results for all level correlations of automorphic $L$-functions. We discover surfaces associated with the zeros of automorphic $L$-functions. In the case of pair correlation, the surface displays Gaussian behavior. For triple correlation, these structures exhibit characteristics of the Laplace and Chi-squared distributions, revealing an unexpected phase transition. This is joint work with Debmalya Basakand Alexandru Zaharescu.
Relations between rational functions and an analog of the Tits alternative
- Series
- Number Theory
- Time
- Wednesday, April 16, 2025 - 15:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Tom Tucker – Rochester University – thomas.tucker@rochester.edu
Work of Levin and Przytycki shows that if two non-special rational
functions f and g of degree $> 1 $over $\mathbb{C}$ share the same set of
preperiodic points, there are $m$, $n$, and $r$ such that $f^m g^n = f^r$.
In other words, $f$ and $g$ nearly commute. One might ask if there are
other sorts of relations non-special rational functions $f$ and $g$ over $\mathbb{C}$
might satisfy when they do not share the same set of preperiodic
points. We will present a recent proof of Beaumont that shows that
they may not, that if f and g do not share the same set of preperiodic
points, then they generate a free semi-group under composition. The
proof builds on work of Bell, Huang, Peng, and the speaker, and uses a
ping-pong lemma similar to the one used by Tits in his proof of the
Tits alternative for finitely generated linear groups.
Non-vanishing of Ceresa and Gross--Kudla--Schoen cycles
- Series
- Number Theory
- Time
- Wednesday, April 9, 2025 - 15:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Wanlin Li – Washington University – wanlin@wustl.edu
The Ceresa cycle and the Gross—Kudla—Schoen modified diagonal cycle are algebraic $1$-cycles associated to a smooth algebraic curve. They are algebraically trivial for a hyperelliptic curve and non-trivial for a very general complex curve of genus $>2$. Given an algebraic curve, it is an interesting question to study whether the Ceresa and GKS cycles associated to it are rationally or algebraically trivial. In this talk, I will discuss some methods and tools to study this problem
Bounds on Hecke Eigenvalues over Quadratic Progressions and Mass Equidistribution on Cocompact Surfaces
- Series
- Number Theory
- Time
- Wednesday, January 15, 2025 - 15:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Steven Creech – Brown University – steven_creech@brown.edu
Given a modular form $f$, one can construct a measure $\mu_f$ on the modular surface $SL(2,\mathbb{Z})\backslash\mathbb{H}$. The celebrated mass equidistribution theorem of Holowinsky and Soundararajan states that as $k\rightarrow\infty$, the measure $\mu_f$ approaches the uniform measure on the surface. Given a maximal order in a quaternion algebra which is non-split over $\mathbb{Q}$, a maximal order leads to a cocompact subgroup of $R^1\subseteq SL(2,\mathbb{Z})$ where the quotient $R^1\backslash\mathbb{H}$ is a Shimura curve. Given a Hecke form $f$ on this Shimura curve, one can construct the analogous measure $\mu_f$, and ask about the limit as $k\rightarrow\infty$. Recent work of Nelson relates this equidistribution problem for the cocompact case to obtaining bounds on sums of Hecke eigenvalues summed over quadratic progressions. In this talk, I will describe this problem in both the cocompact and non-cocompact case while highlighting how differences in algebras lead to differences in geometry. I will then state progress that I have made on bounds that correspond to square root cancellation on average for sums of Hecke eigenvalues summed over quadratic progressions when averaged over a basis of Hecke forms.