Abstract: I will talk about a dynamical system that is associated with a random walk on a metric graph, that is, on a one-dimensional cell complex. The main difference from the often considered case is that the endpoint of a walk can be any point on an edge of a metric graph and not just one of the vertices. Let a point start its motion along the path graph from a hanging vertex at the initial moment of time. The passage time for each individual edge is fixed. At each vertex, the point selects one of the edges for further movement with some nonzero probability. Reflection occurs at the hanging vertex. Backward turns on the edges are prohibited in this model. Our aim is to analyze the asymptotic number N(T) of possible endpoints of such a random walk as the time T increases. Such a random walk can naturally arise in considering the evolution of wave packets localized in a small neighborhood of one point at the initial moment of time. Solutions to this problem, depending on the type of graph, are associated with different problems of number theory. Similar questions arise when considering the propagation of waves on hybrid manifolds. An overview of the results, which depend on the arithmetic properties of lengths, will be given as well as review of open problems.
Abstract: In this talk we present some history concerning the study of Diophantine equations over function fields. In particular, we present a general method whereby the ABC Theorem may be used to effectively solve Thue equations in this setting. We then discuss an application of this approach to determine the complete solution set to a simple quartic family of Thue equations over C(T) (Joint work with B. Faye, I. Vukusic, and V. Ziegler).
Abstract: A fundamental heuristic in analytic number theory is that the Möbius function is not correlated with other multiplicative functions unless there is a "good" reason. In suitable quantitative form this heuristic would solve many outstanding problems, such as the twin prime conjecture, Goldbach's conjecture etcetera. In this talk we discuss a version of the heuristic: Given any 1-bounded multiplicative function f(n), a deep conjecture of Elliott predicts cancellations among shifted values
∑ n ≤ x f(n+h1) ....f(n+hk) = o(x)
unless it is ``close" to a "modulated" Dirichlet character in an appropriate sense. Partial progress towards this conjecture has had numerous consequences, including solution of the Erdös discrepancy problem, progress on the (logarithmic) Chowla and Sarnak's conjectures, and many others. In this talk I will report on some new developments towards this conjecture and present several applications in number theory, ergodic theory and combinatorics. This is based on a joint work with A. Mangerel and J.Teräväinen.
Abstract: We focus the spectral structure of the quantum Rabi model (QRM), its asymmetric version (AQRM) and their covering models, the non-commutative harmonic oscillators (NCHO). The (A)QRM are widely studied as the most fundamental models of light-matter interactions both in theoretical and experimental physics. We discuss the special values of the corresponding spectral zeta functions from various number theoretic viewpoints. Particularly, for the special values of the spectral zeta function for the NCHO, we will find there are certain structures such as modular forms, elliptic curves, congruence properties which are similar to the ones possessed by Apery numbers that were used to prove the irrationality of ζ(2) and ζ(3).
Background reading: Number Theory and Quantum Physics Based on Symmetry - Themes from Quantum Optics
Abstract: We discuss different aspects of badly approximabilty. Starting with a brief review concerning badly approximable objects related to different linear and non-linear Diophantine problems we continue by introducing new theorems characterising badly approximable vectors and systems of linear forms. In particular, recently the author together with R. Akhunzhanov found certain criteria for badly approximable vectors in terms of ratios of denominators of best approximations (="convergents"). These criteria were quite recently improved by W.M. Schmidt, A. Marnat, J. Schleischitz and L. Summerer.
Abstract: The ABC conjecture of Masser-Oesterlé, 1988, states that for any t>1 there exists just finite number of triples (a, b, c) of relatively prime positive integers satisfying a+b=c, such that P=(log c)/(log Rad(abc)) > t, where the radical of an integer is the product of distinct primes dividing it, e.g. Rad(12)=6. A Belyi function is a non-constant meromorphic function on an algebraic curve possessing with at most 3 critical values. Belyi functions naturally correspond to tamely embedded graphs on surfaces, called dessins d'enfants. We provide an overview of the ABC conjecture and its power. Based on the paper "ABC allows us to count squarefrees" by Andrew Granville, IMRN,19, 1998, 991-1009, we will explain how Belyi functions imply corollaries from the ABC conjecture, if it holds. If the time permits, I will also show how to produce the triples (a, b, c) with big P from the correspondence between Belyi pairs and dessins d'enfants.
Abstract: For a reduced congruence class a mod q, it is known that there are infinitely many arbitrary long strings of consecutive primes which are all a mod q. In other words, the pattern aaa...a appears infinitely often in the sequence of increasing primes mod q. However, for any two reduced congruence classes a, b mod q, it is not known that there are infinitely many primes congruent to a mod q which are immediately followed by a prime congruent to b mod q (unless q = 3, 4, or 6, in which case the result is uninteresting). In other words, no (non-boring) non-constant pattern mod q is known to occur infinitely often. In this talk, we'll explore the same questions for the sequence of sums of two squares, in increasing order. We will show that if a,b, and c are congruence classes mod q which are sums of two squares (mod q), the patterns abc (any pattern of length 3) and aaa...abbb..b (arbitrarily many a's followed by arbitrarily many b's) each appear infinitely often in the sequence of sums of two squares mod q. Our work relies on the work of Hooley on triple correlations of sums of two squares and on McGrath's recent adaptation of Maynard's sieve to the setting of sums of two squares. Joint work with Noam Kimmel.
In 2018, Keating, Rodgers, Roditty-Gershon and Rudnick conjectured that the variance of sums of the divisor function in short intervals is described by a certain piecewise polynomial coming from a unitary matrix integral. That is to say, this conjecture ties a straightforward arithmetic problem to random matrix theory. They supported their conjecture by analogous results in the setting of polynomials over a finite field rather than in the integer setting. In this talk, we'll discuss arithmetic problems for polynomials over a finite field and their connections to matrix integrals. We will focus on variations on the divisor function problem that are instead described by symplectic or orthogonal matrix integrals. Joint work with Matilde Lalin.
Abstract: A very classical topic is to study the order of an integer A modulo a prime p, that is the least positive k so that A^k=1 mod p. Several outstanding problems are still open in this subject, for instance the typical size, the maximal size (Artin's primitive root conjecture), the minimal size, etc. An interesting extension is to take an integer n-by-n matrix A, and consider its order mod p, for primes not dividing det(A). I will discuss old and new problems in this setting.
Abstract: In the first half, we discuss and compare various statistics of sequences -- ranging from classical uniform distribution to modern fine-scale statistics (such as the pair correlation function). In the second half, we report on joint work with Christopher Lutsko in which we obtained, for the first time, a complete understanding of fine-scale statistics for sequences of slow growth. The methods of proof is based (purely) on harmonic analysis, with a heathy dose of combinatorics.
Abstract: The hyperelliptic curve y^2=f(x) has an unusual property: it admits infinitely many quadratic points (a, sqrt(f(a))). Conversely, Harris and Silverman show that curves with infinitely many quadratic points are (geometrically) hyperelliptic or bielliptic. Similarly, one may wonder which curves over number fields have an infinite collection of points of some fixed degree d. Abramovich and Harris conjectured that, analogously to the quadratic case, such curves admit degree d maps to P^1 or an elliptic curve, but this simple description turned out to be false for d>=4. I will describe recent joint work with Isabel Vogt (arXiv:2208.01067) in which we show how to reduce the general classification problem to a study of curves of low genus. As an application, we obtain a classification for d<=5. These results are obtained by analyzing discrete geometry of certain configurations of linear subspaces.
Abstract: Draw a permutation g on n letters uniformly at random. For 1 <= k < n, what is the probability that g fixes a subset of {1, ..., n} of size k? I will discuss this question, as well as analogues for matrices and polynomials over finite fields. I will in particular highlight connections between these types of problems, and problems concerning transitive actions of finite simple groups (for which no prerequisite is needed). Joint work with Sean Eberhard.
We consider the set of n by n matrices with integer elements of size at most H and obtain upper and lower bounds on the number of s-tuples of matrices satisfying various multiplicative relations, including multiplicative dependence, commutativity and bounded generation of a subgroup of GL(n,Q). Joint work with Igor Shparlinski.
We review the Gauss circle problem, and Hardy's conjecture. It is attempted to rigorously formulate the folklore heuristics behind Hardy's conjecture. Some weaker forms of the likely statement are proved to support it. Based on a joint work with Steve Lester.
I will give a survey of the theory of the "quantum cat map", a toy model for eigenfunction localization in quantum chaos. The study of this model involves several interesting problems in classical number theory, relating to Artin's primitive root conjecture, complete and incomplete exponential sums and additive combinatorics.
Abstract: The Thue-Morse sequence on the letters {a,b} starts as abbabaabbaababbabaababbaabbabaab .... where the rule is that t(0) = a, t(2n) = tn and t(2n+1) = a (resp., b) if t(n) = b (resp., a). This sequence appears in several places in mathematics, for instance as an example of an "automatic sequence", in combinatorics, in dynamics as an example of a "minimal" dynamical system, in the theory of the geodesic flow on hyperbolic surfaces (Morse), and more. In the talk, we will discuss Diophantine properties of Thue-Morse expansions. In particular, we shall explain how the Subspace Theorem, a fundamental result in Diophantine approximation, implies that the continued fraction expansion [a0; a1, a2, ...] is quadratic or transcendental when (an) begins with arbitrarily large palindromes. This establishes the transcendence of [t(0); t(1), t(2), ...] since a, abba, and every other prefix of length a power of 4 of the Thue-Morse sequence is a palindrome.