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 Lf(N) = LCM(f(1),f(2),...,f(N)), lf(N) =rad(Lf(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 Lf(N) ~ N log N if d=2and conjectured that log Lf(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 lf(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 Lf(N) and lf(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 Rd 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 at = diag (et,..., et,e-dt), let U be its expanding horospherical group, which we identify with Rd, 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.