Abstract: It is an open question whether the fractional parts of nonlinear polynomials at integers have the same fine-scale statistics as a Poisson point process. We provide explicit Diophantine conditions on the coefficients of degree 2 polynomials under which the limit of an averaged pair correlation density is consistent with the Poisson distribution, using a recent effective Ratner equidistribution result on the space of affine lattices due to Strombergsson. This is joint work with Jens Marklof.
Abstract: I will explain the general machinery behind the results on correlations of sums of two squares in polynomial rings in the large finite field limit. In particular, I will define the class of arithmetic functions for which we can determine these correlations, and I will interpret the (nontrivial) correlation factors in terms of group theory. As further applications of the general machinery I will discuss the average of sums of two squares on shifted primes, and correlations of higher divisor functions twisted by a quadratic character.
Abstract: The Markoff group of transformations is a group Γ of affine integral morphisms, which is known to act transitively on the set of all positive integer solutions to the equation x^2+y^2+z^2=xyz. The fundamental strong approximation conjecture for the Markoff equation states that for every prime p, the group Γ acts transitively on the set X(p) of non-zero solutions to the same equation over Z/pZ. Recently, Bourgain, Gamburd and Sarnak proved this conjecture for all primes outside a small exceptional set. In the current work, we study a group of permutations obtained by the action of Γ on X(p), and show that for most primes, it is the full symmetric or alternating group. We use this result to deduce that Γ acts transitively also on the set of non-zero solutions in a big class of composite moduli. If time permits, I will describe a nice interpretation of our result with respect to a natural action of Aut(F2) on pairs of generators of PSL(2,p), where F2 is the free group on 2 generators. Joint work with Chen Meiri, with help from Dan Carmon
Abstract: I will discuss moments of L-functions over function fields, and I will focus on the fourth moment in the family of quadratic Dirichlet L-functions, obtaining part of an asymptotic formula conjectured by Andrade and Keating.
Abstract: Equidistribution problems, originating from the classical works of Kronecker, Hardy and Weyl about equidistribution of sequences mod 1, are of major interest in modern number theory. We will discuss how some of those problems relate to unipotent flows and present a conjecture by Margulis, Sarnak and Shah regarding an analogue of those results for the case of the horocyclic flow over a Riemann surface. Moreover, we provide evidence towards this conjecture by bounding from above the Hausdorff dimension of the set of points which do not equidistribute.
Abstract: Multiple zeta values are usually studied in number theory, but in fact they are also closely related to moduli spaces of curves and Grothendieck-Teichmuller theory. We will explore these connections.
Abstract: In this talk, I will discuss recent results related to the moments of Dirichlet L-functions over function fields when the average is taken over monic and irreducible polynomials. If time allows I will also discuss some number field results.
Abstract: The modular function j is one of the most basic examples of a holomorphic function and arises in many mathematical contexts. Its values at quadratic imaginary arguments - known as singular moduli - generate abelian extensions of the corresponding imaginary quadratic fields. The desire to generalize this statement to the setting of real quadratic fields is at the origins of Kronecker's Jugendtraum, which later became the 12th of the problems posed by Hilbert in his celebrated address at the 1900 ICM in Paris. I will describe a conjectural framework that leads to a notion of singular moduli for real quadratic fields, relying on p-adic analysis rather than complex analysis. This is ongoing joint work with Jan Vonk.
Abstract: We study the size of the minimal gap between the first N eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio α, in comparison to the corresponding quantity for a Poissonian sequence. If α is a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size 1/N, which is essentially consistent with Poisson statistics. We also give related results for a set of α s of full measure. The proofs use a variety of ideas of an arithmetical nature, involving Diophantine approximation, the theory of continued fractions, and results in analytic number theory. The talk will give some number theoretic ingredients of the lecture that I will give in the school's Colloquium on Monday as well as present some open problems. See preprint here
Abstract: In this talk we discuss some aspects concerning the arithmetic of systems of quadratic forms. This includes the Hasse principle, failures of the Hasse principle and representation problems of integers by systems of quadratic forms.
Abstract: Let E be an elliptic curve defined over the rationals. Given an integer N, let M(N) be the number of primes p for which the reduction of the curve modulo p has precisely N points. We study the statistics of M(N) when averaged over various families of curves, showing that under certain conditions we obtain a Poisson distribution.
Abstract: Katz and Sarnak showed that the number of points on a curves in families over a field of q elements is distributed as the trace of a random matrix if we fix the genus and let q tend to infinity. There has been a lot of work recently on what happens to certain families if you fix q and let the genus tend to infinity. In particular, if you consider the family of curves C such that K(C)/K is Galois and Gal(K(C)/K) is fixed, then it has been shown for several classes of groups that the number of points is distributed as a sum of q+1 random variables. In this talk, we show this to be true if we fix Gal(K(C)/K) to be any abelian group.
Abstract: It is a classical consequence of class field theory that a prime p is of form x2+ny2 if and only if (-n) is a quadratic residue mod p and a certain integral polynomial fn has a zero mod p, see the beautiful book by Cox. Explicit class field theory tells us that fn is the minimal polynomial of the j-invariant of an elliptic curve intimately related to the Hilbert class field of an imaginary quadratic field whose discriminant equals that of the quadratic form in question. I am going to present an 'elliptic-curves-proof' of this theorem (circumventing class field theory) and obtain formulas for x and y in the spirit of Gauss, Eisenstein and others. This is a high school project by Ohad Avnery.
Abstract: I will discuss classical problems concerning the distribution of square-full numbers and their analogues over function fields. The results described are in the context of the ring Fq[T ] of polynomials over a finite field Fq of q elements, in the limit as q tends to infinity. I will also present some recent generalization of these kind of classical problems.
Abstract: We will discuss several new developments in the study of counting problems over rational function fields Fq(t), and the relationship of these problems with the geometry of moduli spaces. We will talk about work of Hast and Matei https://arxiv.org/abs/1604.02067 and Rodgers https://arxiv.org/abs/1609.02967 on sums of arithmetic functions in short intervals, and a recent result with Tran and Westerland about counting extensions of Fq(t) with bounded discriminant.
Abstract: We discuss some results concerning the local statistics of directions in certain "adelic quasicrystals". One such local statistic is the gap distribution, where we build on work by Jens Marklof and Andreas Strombergsson in the case of (affine) Euclidean lattices and quasicrystals, and by Noam Elkies and Curtis McMullen who studied fractional parts of √n. Another is the pair correlation, where we build on joint work with Jens Marklof and Ilya Vinogradov. The methods involve homogeneous dynamics and analytic number theory.
Abstract: We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz Diophantine equation in at least 4 variables. The previous best result here is by Baragar (1998) that gives an exponential rate of growth with an exponent that is not in general an integer. We give a new interpretation of this exponent of growth in terms of the unique parameter for which there exists a certain conformal measure on projective space. In a special case, in light of results of Huang and Norbury our counting result can be rephrased in terms of the simple closed curves on a special arithmetic 3 times punctured projective plane. This is joint work with Alex Gamburd and Ryan Ronan.
Abstract: Dedekind sums, arithmetic sums for SL(2,Z), are in close relation to a variety of objects ranging between geometry and number theory; the discriminant form, linking numbers of modular geodesics, signature defects of 3-dimensional Lens spaces, statistical tests for pseudo-random number generation, lattice points in tetrahedra, to name a few. Among various generalizations, there is a construction of Dedekind sums attached to each cusp of a cofinite Fuchsian group, which we call Dedekind symbols. We will focus on the distribution of values of Dedekind symbols, using the spectral theory of Kloosterman sums.
Title: Statistics of sums of two squares
Abstract: Landau proved that the mean value of b, the indicator function of sums of two squares, is $Cx/\sqrt{\log x}.$ In this talk I will discuss subtle statistical properties of b. Since those problems prove to be very difficult, we will also discuss the function field setting, in which some theorems may be proved
Contact me at: rudnick@post.tau.ac.il, Office : Schreiber 316, tel: 640-7806