Abstract: Let f(x) in Z[x] be a monic irreducible polynomial of degree greater than one. In 1964, Hooley showed that the sequence u/n, where f(u) = 0 mod n, ordered in the obvious way, is uniformly distributed modulo one. In my talk I shall explain the number theoretic motivation for the study of this problem and in particular discuss the joint distribution of the roots of a pair of monic irreducible polynomials of degree greater than one f(x), g(x) in Z[x].
Abstract: Given an algebraic variety over the rationals, I will discuss the joint distribution of the number of prime factors of each of the coordinates as we vary an integral point. Under suitable assumptions and with the appropriate normalisation, we establish a multivariate central limit theorem, thus generalising the celebrated Erdos-Kac theorem. This is based on joint work with Daniel Loughran and Efthymios Sofos.
Abstract. We investigate the approximation rate of a typical element of the Cantor set by dyadic rationals. This is a manifestation of the times two times three phenomenon, and is joint work with Demi Allen from Bristol and Han Yu from Cambridge.
Abstract: Suppose that we choose arbitrarily a subset of residue classes modulo each prime, and use them with the Chinese Remainder Theorem to define subsets of residue classes modulo all squarefree moduli. Under extremely general conditions, it follows that the fractional parts of these sets become equidistributed modulo 1 for almost all moduli. The talk will discuss the precise statement of this general principle, as well as some generalizations and applications.(Joint work with K. Soundararajan)
Abstract: How random is a uniformly distributed sequence? Fine-scale statistics provide an answer to this question. Our focus is on the pair and triple correlation statistics of sequences on the unit circle. In particular, we report on recent progress concerning the fractional parts of nα (joint work with Nadav Yesha) and α n2 (joint work with Aled Walker), where α is a fixed positive number.
Abstract: Under certain congruence conditions, the elliptic curves defined over the complex numbers with complex multiplication (CM) by a given order can be reduced to supersingular curves (SSC) defined over a finite field of prime characteristic. The (finite) set of isomorphism classes of SSC curves carries a natural probability measure. It was shown by Philippe Michel via progress on the subconvexity problem that the reductions of CM curves equidistribute among the SSC curves when the discriminant of the order diverges along the congruence conditions. We will describe a proof of equidistribution in the product of the simultaneous reductions with respect to several distinct primes of CM curves of a given order using a recent classification of joinings for certain diagonalizable actions by Einsiedler and Lindenstrauss. This is joint work with Menny Aka, Philippe Michel, and Andreas Wieser.
Abstract: We give an introduction to our recent work with Etienne Le Masson, Joe Thomas and Cliff Gilmore on spatial delocalisation of Eigenfunctions of the Laplacian for random surfaces of large genus. In particular we describe the L^p norms of Eigenfunctions in terms of purely geometric conditions of hyperbolic surfaces, which are shown to be almost surely satisfied in large genus in the Weil-Petersson model for random surfaces. The work is motivated by analogous large random graph results by Bauerschmidt, Knowles and Yau and the delocalisation of cusp forms on arithmetic surfaces of large level (related to Quantum Unique Ergodocity).
Abstract: The disjointness conjecture of Sarnak states that the Mobius function is disjoint with dynamical systems of zero entropy. In this talk I will describe how to establish this conjecture for a class of skew products. This is joint work with Wen Huang and Ke Wang.
Abstract: Recently substantial progress has been made in the study of 2-parts of class groups of quadratic number fields, most notably by Alexander Smith. In this talk we give an introduction to the topic. We start with a classical result due to Gauss known as genus theory, which describes the 2-torsion of the class group. We will then give a description of the 4-torsion and 8-torsion of the class group. Finally we sketch how one can apply these techniques to improve the current lower bounds on the number of squarefree integers d such that the negative Pell equation x2 - dy2 = -1 is soluble in integers x and y. This last part is joint work Stephanie Chan, Djorjdo Milovic and Carlo Pagano.
Abstract: On a compact hyperbolic surface, the Laplacian has a spectral gap between 0 and the next smallest eigenvalue if and only if the surface is connected. The size of the spectral gap measures how "highly connected" the surface is. We study the spectral gap of a random covering space of a fixed surface, and show that for every ε>0 , with high probability as the degree of the cover tends to ∞, the smallest new eigenvalue is at least 3/16-ε. The number 3/16 is, mysteriously, the same spectral gap that Selberg obtained for congruence modular curves. Our main tool is a new method to analyze random permutations "sampled by surface groups". I intend to give some background to the result and discuss some ideas from the proof. This is based on joint works with Michael Magee and Frederic Naud.
Delivered remotely on zoom
Abstract: We study the distribution of lattice points lying on expanding circles in the hyperbolic plane. The angles of lattice points arising from the orbit of the modular group PSL(2,Z), and lying on hyperbolic circles centered at i, are shown to be equidistributed for generic radii (among the ones that contain points). We also show that angles fail to equidistribute on a thin set of exceptional radii, even in the presence of growing multiplicity. Surprisingly, the distribution of angles on hyperbolic circles turns out to be related to the angular distribution of euclidean lattice points lying on circles in the plane, along a thin subsequence of radii. This is joint work with D. Chatzakos, S. Lester and I. Wigman.
Abstract: The squarefree integers are divisible by no square of a prime. It is well-known that they have positive density within the integers, namely 6/pi^2. We consider the number of squarefree integers in a random interval of size H: # {n in [x,x+H] : n squarefree}, where x is a random number between 1 and X. The variance of this quantity has been studied by R. R. Hall (1982). Using Hardy-Littlewood type results for squarefrees, Hall obtained asymptotics for this quantity in the range H< X2/9, and Keating and Rudnick recently conjectured that his result persists for the entire range H < X. We make progress on this conjecture by relating the problem to statistics of the Riemann zeta function. We will show how, on RH, one can verify the conjecture for H up to X1/2. If time permits, we will discuss how one can get by without RH. Joint work with Maks Radziwill and Brad Rodgers.
A relatively recent conjecture due to Cilleruelo states that for an irreducible nonlinear polynomial f with integer coefficients of degree d > 1, the least common multiple L(N) of the sequence f(1), f(2), ..., f(N) has asymptotic growth L(N) ~(d-1)N log(N) as N goes to infinity. I will discuss the background and status of this conjecture.
Abstract: A number field is monogenic if its ring of integers is generated by a single element. It is conjectured that for any degree d > 2, the proportion of degree d number fields which are monogenic is 0. There are local obstructions that force this proportion to be < 100%, but beyond this very little is known. I will discuss work with Alpoge and Bhargava showing that a positive proportion of cubic fields (d = 3) have no local obstructions and yet are still not monogenic. This uses new results on ranks of Selmer groups of elliptic curves in twist families.
Abstract: It is well known (and an easy exercise) that for a prime q, the reduction mod q map from SL(2,Z) to SL(2,Z/qZ) is surjective. This is also true for SL(N,Z), and is called the strong approximation theorem. In this talk we discuss a quantitative version: Given a matrix in SL(2,Z/qZ), what is the size of the smallest lift to SL(2,Z)? Sarnak showed that almost every matrix in SL(2,Z /qZ) ) can be lifted to a matrix in SL(2, Z) ), where every coordinate is bounded by q3/2+o(1) . The exponent 3/2 is optimal since the number of matrices in SL(2,Z) with coordinates bounded by T is asymptotic to T2. Conjecturally, a similar optimal lifting of almost every element should hold for every sequence of principal congruence subgroups of arithmetic groups, for example when SL(2) is replaced with SL(N). We prove that a solution to this problem follows from a proper generalization of the conjectures of Sarnak and Xue on limit multiplicity, whose aim is to approximate the Generalized Ramanujan Conjecture. Unconditionally, we prove an optimal lifting theorem in the context of the action of SL(3,Z) on the projective plane. Based on joint works with Konstantin Golubev and Hagai Lavner.
Abstract: We obtain new upper bounds on the minimal density of lattice coverings of Rn by dilates of a convex body K. We also obtain bounds on the probability (with respect to the natural Haar-Siegel measure on the space of lattices) that a randomly chosen lattice L satisfies L + K = Rn. As a step in the proof, we utilize and strengthen results on the discrete Kakeya problem. This is joint work with Oded Regev and Barak Weiss.
Abstract: The object we will talk about is a polynomial whose coefficients are integers chosen at independently at random according to some distribution. For example, the coefficients are chosen uniformly from some fixed interval, or are chosen from the cubes in the intervals. The common intuition is that random polynomials should be irreducible with his probability and should have the large Galois group with high probability. When the coefficients are chosen from 0,1 (with the condition the the free coefficient is 1) this is known as the Odlyzko-Poonen conjecture. When the coefficients are chosen from 1 to 210, this was proven by LBS and Kozma. Assuming the General Riemann Hypothesis, Breuillard and Varju solved the O-P conjecture. In this talk I will present new results (with Kozma and Koukoulopoulos) that allow very general distributions, and in particular significantly improve the unconditional state of the art. The starting point of the new results is a variant of the LBS-Kozma method, and from there one needs to develop function field versions of recent results on the distribution of primes numbers By Bourgain and by Maynard.
Abstract: There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio: Trace(r(g)) / dim(r), for an irreducible representation r of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G. In 2015, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank. Rank suggests a new organization of representations based on the very few "Small" ones. This stands in contrast to Harish-Chandra's "philosophy of cusp forms", which since the 60's is the main organizational principle, and is based on the (huge collection) of "Large" representations. This talk will discuss the notion of rank for the group GL(n) over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks.
This is joint work with Roger Howe (Yale and Texas A&M). The numerics for this work was carried by Steve Goldstein (Madison).
Abstract: In a recent letter Sarnak proved the following Optimal Strong Approximation result for the arithmetic group SL(2,Z); for any integer q, almost any element of the finite group SL(2,Z/qZ) can be lifted to an element in SL(2,Z), which is congruent to it modulo q and whose coefficients are bounded by q3/2 (the smallest possible exponent). This raises the natural question of whether similar Optimal Strong Approximation results hold for other arithmetic groups G(Z), or S-arithmetic groups G(Z[1/S]), where G is an algebraic group. The Ramanujan Conjectures are central open problems in the modern number theory. The Ramanujan Conjectures are known to have many applications; for instance, they imply Optimal Strong Approximation. Unfortunately, for most algebraic groups G, the Ramanujan Conjectures are either unknown (e.g. SL(2)) or, even worse, are incorrect (e.g. U(3) and Sp(4)). To overcome this obstacle, Sarnak introduced back in the 90's the Density Hypotheses, which are meant to serve as a replacement of the Ramanujan Conjectures. In this talk I will elaborate on the Ramanujan Conjectures, their possible failures, and Sarnak's Density Hypotheses. Finally, I will describe how to use recent results coming from the Langlands program to prove the Density Hypotheses for all classical definite (i.e. compact at infinity) algebraic groups. As an application of this we get Optimal Strong Approximation for the S-arithmetic subgroups of classical definite groups.
Abstract: We discuss a solution of the function field analog of the twin prime conjecture, saying that over a finite field, there are infinitely many monic irreducible polynomials with any fixed difference. This is based on a joint work with Will Sawin, where special properties of the Mobius functions in positive characteristic are used to make the problem amenable to geometric methods.
Abstract: I will discuss a quantitative version of Hilbert's irreducibility theorem for function fields: If f(T1,...,Tn,X) is an irreducible polynomial with coefficients in the field Fq(u) of rational functions over a finite field Fq, then the proportion of tuples (t1,...,tn) of polynomials in Fq[u] of degree d such that f(t1,...,tn,X) is reducible is O(dq-d/2). The proof combines the function field Large Sieve (developed by Hsu) with a clever but elementary approach to handle the inseparable case. Based on joint work with Lior Bary-Soroker.