Years ago, Zeev Rudnick defined the Poisson generic real numbers as those where the number of occurrences of the long strings in the initial segments of their fractional expansions in some base follow the Poisson distribution. Peres and Weiss proved that almost all real numbers, with respect to Lebesgue measure, are Poisson generic. They also showed that Poisson genericity implies Borel normality, but the two notions do not coincide, witnessed by the famous Champernowne constant. I will discuss these results and present a construction of a Poisson generic real for a fixed parameter lambda.
Abstract: Hilbert's irreducibility theorem may be formulated as the statement the rational points on the line cannot be covered by rational points coming from finitely many covers. When one wants to replace the line, by more complicated varieties, such as elliptic curves or algebraic groups, such a naive statement fails, due to the existences of isogenies. It turns out the correct generalization is for ramified covers. In the talk, we will discuss some recent progress on these problems.
Abstract: For a subgroup of the modular group, we ask: which integers occur as the trace of an element of the subgroup? For the modular group itself, every integer occurs. The question is particularly interesting for "thin" groups, which are certain subgroups of infinite index. We use circle method to prove a local-global theorem on the set of traces, when the subgroup contains a parabolic element. This yields fine information on the length spectrum (the set of lengths of closed geodesics) of the hyperbolic surface associated to this group. This is joint work with Alex Kontorovich.
Abstract: In 1927, E. Artin proposed a conjecture for the natural density of primes p for which g is a primitive root mod p. By observing numerical deviations from Artin's originally predicted asymptotic, Derrick and Emma Lehmer (1957) identified the need for an additional correction factor; leading to a modified conjecture which was eventually proved to be correct by Hooley (1967) under the assumption of GRH. An appropriate analogue of Artin's primitive root conjecture may also be formulated over a global function field, where Bilharz provided a proof that is correct under the assumption that g is a "geometric" element. In this talk we discuss the correction factor that emerges when one removes the assumption that g is geometric; thereby completing the proof of Artin's primitive root conjecture for arbitrary function fields in one variable over a finite field.
Abstract: Robin boundary conditions are used in heat conductance theory to interpolate between a perfectly insulating boundary, described by Neumann boundary conditions, and a temperature fixing boundary, described by Dirichlet boundary conditions. The Neumann and Dirichlet spectrum of a square are just sums of two squares, hence have a direct arithmetic significance. The Robin spectrum is more mysterious and until now the number theory behind it was not explored. We study the statistics and the arithmetic properties of the Robin spectrum of a rectangle. Based on a joint work with Zeev Rudnick.
I will discuss some recent work, joint with Alon Nishry and Brad Rodgers, concerning the distribution of Dirichlet and trigonometric polynomials generated by multiplicative coefficients f(n). In the first part of the talk we will explore some old and new results for deterministic sequences f(n) (Mobius, Legendre symbol,...), stopping along our journey to marvel at a variety of wild and thorny conjectures. The second half of the talk will be devoted to Steinhaus random multiplicative coefficients f(n)=X(n).
Abstract: The prime number theorem can be stated as saying that the logarithm of the least common multiple (LCM) of the first N integers is asymptotically equal to N, as was known to Chebyshev. Motivated by this formulation, we look at a generalization - the least common multiple of polynomial sequences. The case of a polynomial in one variable was first studied by Cilleruelo in 2011, who determined the asymptotics of the quadratic case, and has since been explored by various other researchers. The conclusion is that for an irreducible polynomial F(x) of degree d at least 2, the logarithm of the LCM of F(1),...., F(N) grows roughly as N log(N), though we still do not know the asymptotics, conjectured to be (d-1)N log(N). In this talk we consider polynomials in two variables. We discover that already in the quadratic case, there is a range of asymptotic behaviours. We show that for "generic" quadratic polynomials, the growth of log LCM of the values of F(x,y) up to N has order of magnitude N loglog N /(log N)1/2 (which as we shall explain, is the answer for a suitable random model), but for certain degenerate cases such as (x+y)2 or x2 + y2, the answers are different.
Abstract: An outstanding conjecture in quantum chaos is that the statistics of the energy levels of "generic" chaotic systems with time reversal symmetry are described by those of the Gaussian Orthogonal Ensemble (GOE) in Random Matrix Theory. Conjectural examples are the eigenvalues of the Laplacian on a "generic" hyperbolic surface. This conjecture has proved to be extremely difficult, with no single case being proved, the closest case being some results for the Riemann zeros which seem to have similar statistics, those of the Gaussian Unitary Ensemble. It has long been desired to improve the situation by averaging over a suitable ensemble of chaotic systems. I will describe a version of such ensemble averaging on the moduli space of compact hyperbolic surfaces, equipped with the Weil-Petersson measure, using the pioneering work of Maryam Mirzakhani. For a suitable quantity, we obtain a small confirmation of GOE statistics. See preprint.
I will survey some old and relatively new results about the decimal (base 10) expansion of 1/p, where p is a prime. The subject is very classical, with links to key issues in number theory several of which are still open. I will also briefly discuss corresponding problems for polynomials over a finite field.
delivered in person in Schreiber 309.
Abstract: The equation x2 + 1 = 0 mod p has solutions whenever p = 2 or 4n+1. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. That the roots of the former equation are equidistributed is a famous theorem of Duke, Friedlander and Iwaniec from 1995. We examine what happens to the distribution when one adds a restriction on the primes which has to do with the angle in the plane formed by their corresponding representation as a sum of squares. This simple arithmetic question has a solution which involves multiple disciplines of number theory, but the talk does not assume any previous background.
delivered in person in Schreiber 309.
Abstract: In 1963 Christopher Hooley showed that the roots of a quadratic congruence mod m, appropriately normalized and averaged, are uniformly distributed mod 1. In this lecture, which is based joint work with Matthew Welsh (Bristol), we will study pseudo-randomness properties of the roots on finer scales and prove for instance that the pair correlation density converges to an intriguing limit. A key step in our approach is to translate the problem to convergence of certain geodesic random line processes in the hyperbolic plane.
Abstract: We will sketch the proof of a breakthrough result (from around 2005) by Bourgain, Chang, Glibichuk, and Konyagin who proved that there is cancellation in exponential sums formed by summing exp(2 pi h/p) for h ranging over elements in a "small" multiplicative subgroup H of the finite field Z/pZ. The result was discussed in the first talk of the semester, for showing that the digits of 1/p are uniformly distributed if the period is not too small. The proof uses ideas from additive combinatorics, in particular the "sum-product theorem" and the Balog-Gowers-Szemeredi theorem (roughly, subsets of Z/pZ with "large additive energy" must contain "large" subsets S with property that the sumset S+S is "small").
delivered in person in Schreiber 309.
Abstract: Given a prime p, the generators of the multiplicative group of the integers modulo p are called primitive roots. In 1930 Vinogradov conjectured that the smallest generator, the least primitive root, is smaller than any power of p. This talk will be a general introduction to the subject. I will discuss the classic results of Vinogradov and Burgess towards this conjecture and describe some more recent improvements for primes such that p-1 does not have small odd prime factors.
Abstract: The aim of this talk is to describe typical compact hyperbolic surfaces: results will be stated for most surfaces rather than every single one of them. In order to motivate this idea, I will first present examples introduced in literature as limiting cases of famous theorems, and argue that they might be seen as "atypical". This will allow us to appreciate the contrast with a fast-growing family of new results in both geometry and spectral theory, which are established with probability close to one in various settings, while being false for these atypical surfaces. In particular, I will discuss results on the distribution of eigenvalues and the geometry of long geodesics, as well as ongoing research on spectral gaps.
Abstract: Chebyshev famously observed empirically that more often than not, there are more primes of the form 3 mod 4 up to x than primes of the form 1 mod 4. This was confirmed theoretically much later by Rubinstein and Sarnak in a logarithmic density sense. Our understanding of this is conditional on the generalized Riemann Hypothesis as well as Linear Independence of the zeros of L-functions.
We investigate similar questions for sums of two squares in arithmetic progressions. We find a significantly stronger bias than in primes, which happens for almost all integers in a natural density sense. Because the bias is more pronounced, we do not need to assume Linear Independence of zeros, only a Chowla-type Conjecture on non-vanishing of L-functions at 1/2. The bias is stronger because it arises from a multiplicative contribution of squares as opposed to additive contribution (as in the case of primes).
To illustrate, we have under GRH that the number of sums of two squares up to x that are 1 mod 3 is greater than those that are 2 mod 3 for all but o(x) integers.
Abstract: In this talk I will describe the distribution of lattice points lying on circles. A striking result of Katai and Kornyei shows that along a density one subsequence of admissible radii the angles of lattice points lying on circles are uniformly distributed in the limit as the radius tends to infinity. Their result goes further, proving that uniform distribution persists even at very small scales, meaning that the angles are uniformly distributed within quickly shrinking arcs. A more refined problem is to understand how the lattice points are spaced together at the local scale, e.g. given a circle containing N lattice points determine the number of gaps between consecutive angles of size less than 1/N. I will discuss some recent joint work with Par Kurlberg in which we compute the nearest neighbor spacing of the angles along a density one subsequence of admissible radii.
Lior Bary Soroker (TAU). Distribution of rational points on elliptic curves.
Hilbert's irreducibility theorem may be formulated as the statement the rational points on the line cannot be covered by rational points coming from finitely many covers.
When one wants to replace the line, by more complicated varieties, such as elliptic curves or algebraic groups, such a naive statement fails, due to the existences of isogenies. It turns out the correct generalization is for ramified covers.
In the talk, we will discuss some recent progress on these problems.