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.
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.
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.
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.
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
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
Primitive integral Apollonian circle packings are fractal arrangements of tangent circles with integer curvatures. The curvatures form an orbit of a 'thin group,' a subgroup of an algebraic group having infinite index in its Zariski closure. The curvatures that appear must fall into a restricted class of residues modulo 24. The twenty-year-old local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing.
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.
Large sieve inequalities are a fundamental tool used to investigate prime numbers and exponential sums. I will explain my work that resolves a 1978 conjecture of S. Patterson (conditional on the Generalized Riemann hypothesis) concerning the bias of cubic Gauss sums. This explains a well-known numerical bias first observed by Kummer in 1846. One important byproduct of my work is a proof that
This talk is about the arithmetic of points of small canonical height relative to dynamical systems over number fields, particularly those aspects amenable to the use of equidistribution techniques. Past milestones in the subject include the proof of the Bogomolov Conjecture given by Ullmo and Zhang, and Baker-DeMarco's work on the finiteness of common preperiodic points of unicritical maps.
