Abstract: Consider polynomials which take integer values on the integers (IVP). Elkies and Speyer, answering a question of Dimitrov on MathOverflow, showed that there is an exponential growth threshold for IVPs on the natural numbers. Roughly speaking, there are infinitely many IVPs with a growth rate above the threshold and only finitely many IVPs below that threshold (of arbitrary degree).
I will provide some background on this problem and will show estimates for the number of IVPs with a bounded growth rate above the threshold, for polynomials of large degree. I will also talk about a more general problem, where there is a growth condition on the integers, and the connection with logarithmic capacity.
Joint work with Avner Kiro (https://arxiv.org/abs/2508.17335).
Abstract: Let S be a finite subset of the integer lattice Zd with d>1. A lattice point x is said to be visible from the set S if, for every s in S, there is no other lattice point on the line segment joining x and s. The density of the set of points visible from S has been known since 1960, and error terms have been studied over time under conditions on S. We say that a positive integer L is exceptional if the proportion of visible points in the box [1, L]k is strictly below the density. We will present an improved upper bound for the error term, along with results on exceptional points, Schnirelmann density, and related conjectures and open questions about visible points. If time permits, we will discuss ergodic properties of the visible lattice points. Joint work with Daniel Berend and Andrew Pollington.
Abstract: Consider a system of homogeneous polynomial equations over some field K. When are we guaranteed a (nontrivial) solution in K? Brauer famously proved that for many fields of interest (e.g. Q[i]), it is enough that the number of variables be sufficiently large. Birch proved that for the rational numbers (or any number field) this also holds, as long as the polynomials have odd degree. We will survey the history of this problem and present recent work improving its quantitative and qualitative aspects. Partly based on joint work with Andrew Snowden and Tamar Ziegler.
Abstract: We examine the arithmetic properties of eigenvalues of random matrices with integer entries, focusing on the irreducibility of their characteristic polynomials and their Galois groups. Rivin, Jouve-Kowalski-Zywina, and Lubotzky-Rosenzweig previously studied characteristic polynomials arising from random walks on the Cayley graphs of Zariski-dense finitely generated subgroups of linear groups, such as SL(n,Z). Eberhard, resolving conjectures of Babai and Vu-Wood assuming ERH, analyzed discrete random matrices and established that the characteristic polynomial of a matrix with independent entries (say taking the values 0,1 with equal probabilities) is irreducible and has a large Galois group with high probability as the matrix dimension grows. Ferber, Jain, Sah, and Sawhney proved a counterpart of these results to symmetric matrices. In this talk, I will present recent results on random tridiagonal matrices where the main diagonal consists of independent Bernoulli entries, the superdiagonal and subdiagonal entries are identically one, and all other entries are zero. We show that the characteristic polynomial of such matrices is irreducible and analyze the structure of its Galois group. If time permits, we will discuss applications to the localization of the eigenstates in Anderson's 1-dim localization model. A key feature of our approach lies in combining techniques from both the above random walk framework and the above discrete matrix setting. The latter leverages the Extended Riemann Hypothesis (ERH) to reduce the problem to analyzing the distribution of eigenvalues modulo primes p. To achieve strong error bounds in these computations, we exploit the powerful mixing properties of simple groups such as PSL(2,p), a central tool in the above-mentioned random walk results.