I will describe an approach to the problem of drawing a complete list of real quadratic fields of class number one and small regulator (known as fields of Richaud-Degert type), based on a conjectural theory of Heegner points for real quadratic fields. This is joint work with Elias Caeiro.

Joint work with Manfred Einsiedler, Amir Mohammadi, and Andreas Wieser.

Let q be an integer quadratic form in n variables, and Q a quadratic form in m variables, with m > n. Can you represent q by Q? That is to say, can you find an m × n integer matrix M such that q = Q ∘ M?

The case n = 1 reduces to the question of which integers can be represented by Q. Of course, there can be local obstructions to such a representation.

Building on work of Ellenberg and Venkatesh, and using a new effective equidistribution result, we give an effective local-to-global result about such representations once m ≥ n + 3.

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 n = d+1 , let {at } be a one parameter group of diagonal matrices in SL(n, R) which expand the first d coordinates, let U be a copy of Rd embedded in the top right corner, and let \bar ν be the pushforward of ν onto the space of lattices SL(n,R)/SL(n,Z), via the orbit map of the identity coset under U. In joint work with Khalil and Luethi, we show that the pushforward of \bar ν under a_t equidistributes as t tends to infinity. 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. The proof involves a classification of stationary measures for a certain random walk on an S-adic homogeneous space. I will discuss the main steps of the proof, emphasizing the new ideas, and briefly discuss the Diophantine applications.

In this talk I will discuss the generalization of a result of Montgomery and Vaughan regarding exponential sums with multiplicative coefficients to the setting of Weyl sums. As applications, I will present a joint equidistribution result for roots of polynomial congruences and polynomial values and some new results for mixed character sums. This is joint work with Cynthia Bortolotto and Bryce Kerr.

A set in the Euclidean plane is called an integer distance set if the distance between any pair of its points is an integer. All so-far-known integer distance sets have all but up to four of their points on a single line or circle; and it had long been suspected, going back to Erdős, that any integer distance set must be of this special form. In a recent work, joint with Marina Iliopoulou and Sarah Peluse, we developed a new approach to the problem, which enabled us to make the first progress towards confirming this suspicion. In the talk, I will discuss the study of integer distance sets, its connections with other problems, and our new developments.

In a somewhat forgotten short note of Deligne [D], he showed that some finite central extension of an arithmetic lattice in Sp(2g, ℝ) is not residually finite. A p-adic version is shown in [DGLT], and it is applied to give the first "non-approximated" groups with respect to the Frobenius norm. Following this breakthrough, the non-approximation was shown with respect to "almost all" norms [LO], but the most important cases were left open. An effort to prove that these groups are also non-sofic is described in [CL], but this is still open. We will describe all these developments, and if time permits, also an application to computer science [DDL].
References

• [D] Extensions centrales non résiduellement finies de groupes arithmétiques. Deligne, Pierre. C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 4, A203–A208.
• [DGLT] Stability, cohomology vanishing, and nonapproximable groups. De Chiffre, Marcus; Glebsky, Lev; Lubotzky, Alexander; Thom, Andreas. Forum Math. Sigma 8 (2020), Paper No. e18, 37 pp.
• [LO] Non p-norm approximated groups. Lubotzky, Alexander; Oppenheim, Izhar. J. Anal. Math. 141 (2020), no. 1, 305–321.
• [CL] Stability of homomorphisms, coverings and cocycles II: examples, applications and open problems. Chapman, Michael; Lubotzky, Alexander. Adv. Math. 463 (2025), Paper No. 110117, 38 pp.
• [DDL] Low Acceptance Agreement Tests via Bounded-Degree Symplectic HDXs. Yotam Dikstein, Irit Dinur, Alexander Lubotzky. Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 19 (2024).

The topic of my talk will be a remarkable connection between spectral properties of Taylor coefficients of entire functions and their zero-distribution on different scales. This connection works in many different instances of random and pseudo-random coefficients (among them are Besicovitch almost periodic sequences, random stationary sequences, multiplicative random sequences, arithmetic sequences of Diophantine nature, the indicator function of the square free integers, and many others). The talk is based on joint works with Jacques Benatar, Alexander Borichev and Alon Nishry.

Torsion points on subvarieties of algebraic groups are a classical theme, like in Lang's theorem regarding points on plane curves with both coordinates roots of unity. After recalling some of this history, I will discuss results and conjectures regarding covers (rather than subvarieties) of algebraic groups and their fibers over torsion points, and over points in fields generated by torsion points (e.g. cyclotomic fields). I will then put these in the context of recent research on so-called Hilbert properties of varieties, derived from Hilbert's irreducibility theorem. This includes various joint works with Lior Bary-Soroker and Sebastian Petersen.

In this talk I will discuss the hyperbolic circle problem for SL₂(ℤ). Given two points z, w that lie in the hyperbolic upper half plane, the problem is to determine the number of SL₂(ℤ) translates of w that lie in the hyperbolic disk centered at z with radius arccosh(R/2) for large R. Selberg proved that the error term in this problem is O(R^2/3). I will describe some recent work in which we improve the error term to o(R^2/3) as R tends to infinity, under the condition that z, w are Heegner points of different discriminants. This is joint work with Dimitrios Chatzakos, Giacomo Cherubini, and Morten Risager.

In joint work with Gal Binyamini and Avner Kiro we prove the analogue of Manin-Mumford for the zeros of Laguerre polynomials. The proof uses point-counting and functional transcendence ("Ax-Schanuel") but exhibits two new features. The first is in how the Laguerre zeros are put in correspondence with rational (in fact integer) points on some o-minimally definable set. This uses Gevrey asymptotics for the solutions of the Laguerre differential equation (and so definability is in the somewhat exotic o-minimal structure of Gevrey multisummable series). The Ax-Schanuel result combines the recent approach by Blasquez-Sans--Casale--Freitag--Nagloo with a classic paper of Kolchin, and formal theory of d.e.

Non-Hermitian random matrices attracted considerable interest in recent years as a tool to characterize Quantum Chaos in dissipative systems. Beyond the framework of standard Ginibre ensembles one of the general tools available for studying such matrices is the "Hermitization Trick" due to Girko. I will describe an alternative approach based on Kac-Rice formula which gives access not only to eigenvalues but also to nontrivial eigenvectors of non-Hermitian random matrices. To illustrate this approach I will consider a family of matrices interpolating between complex Ginibre and real Ginibre ensembles, which in particular allows to reveal a new scaling regime of "weak non-reality".

We prove that in any dimension n there exists an origin-symmetric ellipsoid of volume cn² that contains no points of ℤⁿ other than the origin, where c>0 is a universal constant. Equivalently, there exists a lattice sphere packing in ℝⁿ whose density is at least cn²/2ⁿ. Previously known constructions of sphere packings in ℝⁿ had densities of the order of magnitude of n/2ⁿ, up to logarithmic factors. Our proof utilizes a stochastically evolving ellipsoid that accumulates at least cn² lattice points on its boundary, while containing no lattice points in its interior except for the origin.

Does there exist a formula which takes in coefficients of a multivariable polynomial equation and outputs a solution? This question leads to some interesting problems in algebraic geometry, if the words ``formula'' and ``solution'' are suitably interpreted. I will describe some open problems and known results, as well as a recent proof of the fact that all low degree cycles on the universal hypersurface of degree d have degree kd, and are complete intersections with a subvariety of degree k. The talk is based on joint work with Yifeng Huang and Olivier Martin.

We explore the zeros of certain Poincaré 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.

In recent years there is growing interest in stability phenomena in algebraic geometry, specifically in properties of polynomial rings that are stable in the number of variables (e.g. Stillman conjecture). Problems of a similar nature were independently studied in additive combinatorics in relation to Ramsey questions in finite field geometry. A central role is played by the notion Schmidt rank/strength: high strength collections of polynomials are “psuedorandom”. We explain the notion and its special features in the different areas.

I will describe how ideas from multiplicative number theory and ergodic theory can be used to address problems of finding monochromatic solutions of quadratic equations. The talk is based on the joint works with N. Frantzikinakis and J. Moreira.

In this talk, we discuss the value distribution of Hecke eigenforms in the large weight limit. We begin with an introduction to the quantum unique ergodicity (QUE) theorem and its application to the equidistribution of zeros of Hecke eigenforms. We then turn to the study of their joint value distribution. In particular, we establish asymptotic formulas for certain low-degree mixed moments. Our approach is based on estimates of moments of L-functions.

Two well studied problems regarding random matrices (sampled from a real or complex classical group equipped with Haar measure) are:
Problem 1: The distribution of their trace, and more generally the trace of their powers.
Problem 2: The distribution of their entries, and more generally the distribution of sub-matrices.
In this talk we'll focus on random matrices over finite fields (as opposed to the complex numbers or the reals). We'll survey past works on such matrices, which traditionally focused on group-theoretic questions. We'll explain recent works and on-going works with various collaborators, tackling Problems 1 and 2 in classical groups over finite fields, as well as in certain matrix ensembles over finite fields that are not classical groups. Some applications to number theory will be given. While the questions are motivated by random matrix theory, the tools involve analytic number theory, linear algebra and representation theory.
Based on joint works with Brad Rodgers ( arxiv.org/abs/1909.03666 ); Valeriya Kovaleva ( arxiv.org/abs/2307.01344 ); Elad Zelingher; Will Sawin.

A random ensemble of cusp forms for the full modular group is introduced. For a weight-k cusp form, restricted to a compact subdomain of the modular surface, the true order of magnitude of its expected supremum is determined to be approximately √log k, in line with the conjectured bounds. In addition, the exponential concentration of the supremum around its median is established. Contrary to the compact case, the global expected supremum, attained around the cusp, grows like k^1/4. This talk is based on a joint paper with B. Huang, S. Lester and N. Yesha.

For a finite group G (generated by a single conjugacy class) Boston—Markin made an asymptotic prediction regarding the number of Galois G-extensions of the field of rational numbers ramified at a single prime number. We propose conjectures over number fields and function fields extending those of Boston—Markin, and prove those in the large finite field limit. Work in progress joint with Jordan Ellenberg aims at resolving the conjectures over function fields in the large degree limit, and work in progress joint with Ankit Sahu aims at resolving the dihedral case G = D_4 over number fields.

Let (x_n) be a sequence of integers. We examine the fluctuations in the distribution modulo 1 of the dilated sequence (α x_n) in short intervals of length S, for generic values of α. The main motivation is to compare statistics such as the number variance of these dilated sequences with those of the random (Poisson) model, thereby revealing pseudorandom behaviour. A popular example is given by x_n = p(n), where p is an integer polynomial of degree at least 2. We will discuss a recent joint work with C. Aistleitner (TU Graz), in which we establish Poissonian number variance for almost all dilations of such polynomials, uniformly throughout a large (and presumably optimal) range of S.