Abstract: This talk reports mainly on fundamental results in diophantine approximation, and related open problems - with a focus on the infamous Littlewood conjecture. Recent results and a newly-found link to combinatorics, via Bohr sets, will be mentioned briefly.
Abstract: The asymptotic growth of log LCM(1,...,N) is N. This fact is equivalent to the prime number theorem. It is a natural question to ask "what happens if one replaces the values 1,...,N with the N consecutive values of a polynomial f(n)?". This question was answered for linear and quadratic polynomials, however, remains open for higher degree polynomials. In this talk, I will discuss the average asymptotic behavior of log LCM(f(1),...,f(N)) for a polynomial f of degree bigger than 2. Joint work with Zeev Rudnick
Abstract: We study Gaussian primes lying in narrow sectors, and show that almost all such sectors contain the expected number of primes, if the sectors are not too narrow. We will also discuss prime angles for real quadratic fields. This is joint work with Jianya Liu and Zeev Rudnick.
Abstract: We obtain an improvement and broad generalisation of a result of N. Ailon and Z. Rudnick saying that for any multiplicatively independent polynomials P_{1}(z), P_{2}(z) in C[z] there exists a polynomial F(z) in C[z] such that for any positive integer k the greatest common divisor of P_{1}^{k}-1 and P_{2}^{k}-1 divides F. Our approach is based on reducing this question to a more general question of counting intersections of level curves of complex functions. This is a joint work with I. Shparlinski.
Abstract: Given a lattice Λ and a (perhaps long) vector v in Λ, we consider two geometric quantities:
Fixing Λ and taking some infinite sequences of vectors v_{n}, we identify the asymptotic distribution of these two quantities. For example, for a.e. line L, if v_{n} is the sequence of epsilon-approximants to L then the sequence Δ(v_{n}) equidistributes according to Haar measure, and if v'_{n} is the sequence of best approximants to L then there is another measure which Δ(v'_{n}) equidistributes according to.
Joint work with Uri Shapira.
Note special time and place!
Abstract: The Lorentz gas is one of the simplest and most widely-studied models for particle transport in matter. It describes a cloud of non-interacting gas particles in an infinitely extended array of identical spherical scatterers, whose radii are small compared to their mean separation. The model was introduced by Lorentz in 1905 who, following the pioneering ideas of Maxwell and Boltzmann, postulated that its macroscopic transport properties should be governed by a linear Boltzmann equation. A rigorous derivation of the linear Boltzmann equation from the underlying particle dynamics was given, for random scatterer configurations, in three seminal papers by Gallavotti, Spohn and Boldrighini-Bunimovich-Sinai. The objective of this lecture is to develop an approach for a large class of deterministic scatterer configurations, including various types of quasicrystals. We prove the convergence of the particle dynamics to transport processes that are in general (depending on the scatterer configuration) not described by the linear Boltzmann equation. This was previously understood only in the case of the periodic Lorentz gas through work of Caglioti-Golse and Marklof-Strombergsson. Our results extend beyond the classical Lorentz gas with hard sphere scatterers, and in particular hold for general classes of spherically symmetric finite-range potentials. We employ a rescaling technique that randomises the point configuration given by the scatterers' centers. The limiting transport process is then expressed in terms of a point process that arises as the limit of the randomised point configuration under a certain volume-preserving one-parameter linear group action. (Joint work with Andreas Strombergsson, Uppsala.)
Abstract: Fundamental to random matrix theory is various factorisations of Lebesgue product measure implied by matrix change of variables. In number theory, factorisation of Siegel's invariant measure for SL(N,R) is an ingredient in Duke, Rudnick and Sarnak's asymptotic computation of the number of matrices in SL(N,Z), with a bounded norm. In this talk it will shown how factorisation of measure allows for calculations in the space of integral lattices SL(N,R)/SL(N,Z) and generalisations such as SL(N,C)/SL(N,Z[i]).
Abstract: It has been known for almost a hundred years that most polynomials with integral coefficients are irreducible and have a big Galois group. For a few dozen of years, people have been interested whether the same holds when one considers sparse families of polynomials - notably, polynomials with plus-minus 1 coefficients. In particular, "some guy on the street" conjectures that the probability for a random plus-minus 1 coefficient polynomial to be irreducible tends to 1 as the degree tends to infinity (a much earlier conjecture of Odlyzko-Poonen is about the 0-1 coefficients model) . In this talk, I will discuss these types of problems, their connection with Peres-Pementle-Rivin's result on on four random permutations and with Maynard's result on primes with missing digits.
Abstract: For a polynomial F(t,A_1,..., A_n) with coefficients in the field of p elements (p being a prime number) we study the decomposition statistics of its specializations F(t,a_1,..., a_n) with (a_1,...,a_n) restricted to lie in a subset S of F_p^n in the limit of large p and deg(F) fixed. We show that for a sufficiently large and regular subset S, e.g. a product of n intervals of length > p/c where c is a constant, the decomposition statistics is the same as for unrestricted specializations, up to a small error. This is a generalization of the well-known Polya-Vinogradov estimate of the number of quadratic residues modulo p in an interval.
Abstract: Consider an ensemble of nxn matrices constituting a finite or compact group. For instance, think of unitary matrices, orthogonal matrices, permutation matrices or GL(n,q). The Haar measure on such an ensemble (uniform measure in the finite case) gives a natural probability measure we understand pretty well. But what if we take several, independent random matrices and multiply them according to some pattern given by a fixed word, e.g., ABAB^-1? What is the measure then? I will give some motivation, review what is known and state open problems.
This is partly based on joint works with Ori Parzanchevski, Michael Magee and Liam Hanani.
Abstract: Draw a random matrix A from the unitary group U_n(C) according to the Haar measure. What is the distribution of the trace of A, and of powers of A, as n goes to infinity? Diaconis and Shahshahani have studied this question, showing that the traces, normalized appropriately, converge in distribution to i.i.d complex normal variables. We consider this question in the setting of finite fields: Drawing a matrix from U_n(F_p) or other classical group over a finite field (e.g. SL_n(F_p)), what is the distribution of traces as n goes to infinity? We show that one obtains i.i.d uniform random variables taking values in F_p. We will explain our results, and compare with the characteristic-zero situation. We will sketch the proof and suggest some open problems. Joint work with Brad Rodgers.
We present a conjecture that given a pair of multiplicatively independent integers a, b (e.g. 2 and 3), with gcd(a-1,b-1)=1, there are infinitely many k so that gcd(a^k-1,b^k-1)=1. Bugeaud, Corvaja and Zannier (2003) limit large values, showing that log[ gcd(a^k-1,b^k-1) ]=o(k). In joint work with Nir Ailon (2004), we investigated a function field analogue, showing that if f and g are nonconstant complex polynomials which are multiplicatively independent, then there exists a polynomial h such that gcd(f^k-1,g^k-1) divides h. If in addition gcd(f-1,h-1)=1, then for infinitely many k, gcd(f^k-1,g^k-1)=1. The proof uses a version due to Lang, of the Manin-Mumford conjecture about intersection of curves on group varieties with torsion points.
Abstract: An integer is called square-free if it is indivisible by the square of any integer greater than 1. Classical questions on square-free numbers include the following: - Does a given polynomial f of degree k attain infinitely many square-free values? - Must the interval [X,X+X^(1/k)] contain square-free numbers for all sufficiently large X? - Does every sufficiently large N admit a representation as the sum of a positive k-th power and a positive square-free, i.e. N = x^k + r? These questions are all still open for all but extremely small values of k; however, answers to the first two questions for all values of k are known to follow from the ABC conjecture, a result due to Granville (1998). In the talk we will discuss these questions, and show that they can all be considered as special cases of a question about square-free values of polynomials with large coefficients. We will present a conditional answer to the generalized question, assuming Vojta's version of the ABC conjecture, along with the main ingredients of the proof. Joint work with Hector Pasten.
Abstract: In Diophantine approximation we are often interested in the Lebesgue and Hausdorff measures of certain limsup sets. In 2006, Beresnevich and Velani proved a remarkable result --- the Mass Transference Principle --- which allows for the transference of Lebesgue measure theoretic statements to Hausdorff measure theoretic statements for \limsup sets arising from sequences of balls in R^k. Subsequently, they extended this Mass Transference Principle to the more general situation in which the limsup sets arise from sequences of neighbourhoods of ``approximating" planes. In this talk I will discuss a recent strengthening (joint with Victor Beresnevich, York, UK) of this latter result in which some potentially restrictive conditions have been removed from the original statement. This improvement gives rise to some very general statements which allow for the immediate transference of Lebesgue measure Khintchine--Groshev type statements to their Hausdorff measure analogues and, consequently, has some interesting applications in Diophantine approximation.
Abstract: There are beautiful and deep connections between algebraic geometry, modular forms, and certain sums arising in analytic number theory. I will describe these connections, and how they can be combined with techniques for understanding the primes. In particular, they can be used to understand prime numbers in arithmetic progressions.
Abstract: Ratner's celebrated equidistribution theorem states that the trajectory of any point in a homogeneous space under a unipotent flow is getting equidistributed with respect to some algebraic measure. In the case where the action is horospherical, one can deduce an effective equidistribution result by mixing methods, an idea that goes back to Margulis' thesis. When the homogeneous space is non-compact, one needs to impose further ``diophantine conditions'' over the base point, quantifying some recurrence rates, in order to get a quantified equidistribution result. In the talk I will discuss certain diophantine conditions, and in particular I will show how a new Margulis' type inequality for translates of horospherical orbits helps to verify such conditions, leading to a quantified equidistribution result for a large class of points, akin to the results of A. Strombergsson regarding the SL2 case.
Abstract: We give an overview of recent results about the distribution some special integers in residues classes modulo a large integer. Questions of this type were introduced by Erdos, Odlyzko and Sarkozy (1987), who considered products of two primes as a relaxation of the classical question about the distribution of primes in residue classes. Since that time, numerous variations have appeared for different sequences of integers. The types of numbers we discuss include smooth, square-free, square-full and almost primes integers. We also expose the wealth of different techniques behind these results: sieve methods, bounds of short Kloosterman sums, bounds of short character sums and many others.
We shall explain how to use Eisenstein series to give asymptotics for discrete orbits of lattices of SL(2,R) when acting on the plane. Selberg's bounds on their polynomial growth properties come in and will be used as black box. Our point of view will be of 'how' to use them. Based on work with Claire Burrin, Amos Nevo and Barak Weiss.
Abstract: We give an asymptotic formula for the number of F_q(t)-points of bounded height on the Hilbert scheme over F_q(t) which parameterises degree 2 points in the projective plane.
Contact me at: rudnick@post.tau.ac.il, Office : Schreiber 316, tel: 640-7806