Abstract: Selberg (1940's) computed the statistics of the number of zeros of the Riemann zeta function lying in a random window. His result was much later understood to coincide with the statistics of the Gaussian Unitary ensemble of random matrix theory. For the analogous question of fluctuations energy levels of "generic" chaotic surfaces, it was conjectured that this should also be explained by random matrix theory. Rudnick (2022) proved that surfaces belonging to the moduli space of genus-g hyperbolic surfaces, exhibit smooth linear statistics, as predicted by the random matrix theory conjecture, when averaged w.r.t. Weil-Petersson measure, in the high genus limit. We show that, with high probability, these GOE fluctuations hold for a fixed surface, w.r.t. random window. An essential part of our arguments is played by Mirzakhani's integration formula, and its consequences due to Mirzakhani and Petri. Based on a joint work with Zeev Rudnick.
Abstract: Let f in Z[x] be an irreducible polynomial. One may consider the quantities L_{f}(N) = LCM(f(1),f(2),...,f(N)), l_{f}(N) =rad(L_{f}(N)) and study their rate of growth for large N. The special case f=x was already studied by Chebychev, but the case d=deg f>1 became an active area of interest only a decade ago following the work of Cilleruelo, who proved that log L_{f}(N) ~ N log N if d=2and conjectured that log L_{f}(N) ~ (d-1) N log N for d>=2, a conjecture that is still unknown for any polynomial of degree d>2. A related conjecture of Sah, namely that log l_{f}(N) ~ (d-1) Nlog N for d>=2, is also open for all polynomials of degree d>2. In my talk I will survey the progress towards these conjectures, focusing on lower bounds on L_{f}(N) and l_{f}(N). In particular I will present recent improved lower bounds for polynomials of a special form and discuss recent work on the function field analog of the problem, where stronger results are known than over the integers. Based on joint work with Sean Landsberg.
Abstract: How to detect (pseudo-)randomness of a sequence modulo one? In this talk we will try to answer this question by studying statistics for the number of points in random short intervals. We will discuss both the "local" regime, which has been extensively studied in the last couple of decades, and the "intermediate" regime which attracted less attention. In particular, we will discuss recent results for the number variance, including an ongoing work with C. Aistleitner.
Abstract: The 10X10 multiplication table has 42 distinct numbers, the 12x12 has 59, and the 60x60 has 1116. This prompts Erdos' 1955 problem on the asymptotic number A(x) of integers in a table of size √x.
Building on the works of Besicovitch, Erdos, and Tenenbaum, Ford determined the exact order of magnitude for A(x):
bounded above and below by positive constants times x/((log x)^{ δ}(log log x)^{3/2}),
where δ =1-(1+loglog(2))/log(2)=0.086...
In this talk we delve into variations of this problem, focusing on multiplication tables with rows and columns derived from arithmetic progressions.
Notably, we allow the moduli of the progressions to grow, establishing a robust lower bound in the function field setting.
While our methodology is inspired by Ford's approach, we will mention a general additive combinatorics method developed by Xu and Zhou.
To compare, the latter gives strong lower bounds for any parameters, and the former, in specific cases, yields stronger bounds, e.g., Ford's result.
Abstract: I will discuss old and new results about the distribution of zeros of various families of modular forms.
Abstract:
We introduce and motivate the problem of vanishing of L-functions associated
to quadratic Dirichlet characters at the center of the critical strip.
In particular, we discuss the conjecture of Chowla, and progress towards it conditional on the Generalized Riemann Hypothesis.
We then consider the function field analog, which has several distinctive features with connections to geometry and topology.
In particular, we present results of Li, Ellenberg, Koymans, Pagano, and others that allows one to establish a statistical version
of Chowla's conjecture in the function field setting.
Abstract: Let ν be a Bernoulli measure on a fractal in R^{d} generated by a finite collection of contracting similarities with no rotations and with rational coefficients; for instance, the usual coin tossing measure on Cantor's middle thirds set. Let a_{t} = diag (e^{t},..., e^{t},e^{-dt}), let U be its expanding horospherical group, which we identify with R^{d}, and let \bar ν be the pushforward of ν onto the space of lattices SL(d+1,R)/SL(d+1,Z), via the orbit map of the identity coset under U. In joint work in progress with Khalil and Luethi, we show that the pushforward of \bar ν under at equidistributes as t tends to infinity, as do the pushforwards under more general one parameter subgroups. This generalizes a previous result of Khalil and Luethi and implies that on a large class of rational self similar fractals, weighted badly approximately vectors are of zero measure. I will discuss these Diophantine applications and some probabilistic ideas used in the proof.
Abstract: Part of the talk is expository: I will explain how supersingular isogeny graphs can be used to construct cryptographic hash functions and survey some of the rich mathematics involved. Then, with this motivation in mind, I will discuss two recent theorems by Jonathan Love and myself. The first concerns the generation of maximal orders by elements of particular norms. The second states that maximal orders of elliptic curves are determined by their theta functions.
Abstract: We explore the zeros of certain Poincare series P(k,m) of weight k and index m for the full modular group. These are distinguished modular forms, which have played a key role in the analytic theory of modular forms. We study the zeros of P(k,m) when the weight k tends to infinity. The case where the index m is constant was considered by Rankin who showed that in this case almost all of the zeros lie on the unit arc |z|=1. In this talk we will explore the location of the zeros when the index m grows with the weight k, finding a range of different limit laws. Along the way, we also establish a version of Quantum Unique Ergodicity for some ranges.
Various Dirichlet series which arise naturally in number theory satisfy functional equations. In the early 1970's, Sato and Shintani constructed Dirichlet series satisfying functional equations by using Sato's theory of prehomogeneous vector spaces (1960's), which for suitable examples turn out to be some classical zeta functions. These zeta functions are useful for the asymptotics of arithmetic invariants. For instance Shintani (1972) used them to study class numbers of binary cubic forms, improving on results of Davenport and Heilbron obtained by the geometry of numbers. Prehomogeneous vector spaces turn out to be ubiquitous in the theory of nilpotent orbits of semisimple Lie algebras. They are also important in the geometric side of Arthur's trace formula. I will discuss the history of the subject and some recent work on the analysis of these zeta functions. No prior knowledge of the subject will be assumed. Joint work with Tobias Finis
Abstract: The original Gross-Zagier formula is a type of class number formula for a modular form of weight two and an imaginary quadratic field. Some well-known applications include many cases of the Birch and Swinnerton-Dyer conjecture, as well as an effective lower bound on the value L(1,chi), where chi is a quadratic Dirichlet character. The arithmetic Gan-Gross-Prasad conjecture predicts similar formulas for the Rankin-Selberg L-function attached to a pair (f,chi), where f is a modular form of weight 2k and chi is a Hecke character attached to an imaginary quadratic field. Shouwu Zhang proved such a formula in the case when chi is finite order. I will discuss recent work with David Lilienfeldt on the more general case where the weight of chi is less than 2k, and discuss some applications to conjectures on algebraic cycles.
Abstract: A linear recurrence sequence is a sequence {a(n):n=0,1,...} for which there are coefficients c(0),..., c(d) so that for all n, c(0)a(n)+ ... + c(d)a(n+d)=0. Examples are arithmetic progressions, geometric progressions, the Fibonacci sequence etcetera. The Kepler set of the recurrence sequence is the closure of the set of consecutive ratios a(n+1)/a(n). In the first half of the talk, we will discuss the Kepler sets over the complex numbers. In the second half of the talk, we will discuss the Kepler set over p-adic fields. Based on joint work with Daniel Berend.
Abstract: A short non-trivial solution to a homogenous linear congruence ax-by = 0 mod N with gcd(a,b,N)=1, is a solution (0,0)≠(x,y)∈Z^{2} where max{|x|, |y|}< √N . Aubry and Thue showed in the early 1900s that such a solution always exists. Short solutions are useful in various elementary number theory problems. Towards achieving the most precise and comprehensive formulation for this problem, we introduce a completely different approach presented by A. Strombergsson and A.Venkatesh (2005) when the modulus tends to infinity. Their idea was to leverage the connection between homogenous linear congruences and integer lattices and get an explicit application of equidistribution of Hecke points to address the problem of finding small solutions to linear congruences modulo primes. After outlining their arguments, we will discuss potential areas for improvements and present the main results.