Abstract: On the circle of radius R centred at the origin, consider a ``thin'' sector about the fixed line y=ax, with edges given by the lines y=(a ± ε)x , where ε= ε(R) → 0 as R→ ∞. We discuss an asymptotic count for the number of integer lattice points lying in such a sector, and moreover present results concerning the variance of such lattice points across sectors.
Abstract: I will survey what is known and conjectured about the local statistics of the zeros of the Riemann zeta function, and the relation with corresponding statistics of eigenvalues of random matrices.
Abstract: We prove that in any dimension n there exists an origin-symmetric ellipsoid of volume cn2 that contains no points of Zn other than the origin, where c>0 is a universal constant. Equivalently, there exists a lattice sphere packing in Rn whose density is at least cn2/2n . Previously known constructions of sphere packings in Rn had densities of the order of magnitude of n/2n, up to logarithmic factors. Our proof utilizes a stochastically evolving ellipsoid that accumulates at least cn2 lattice points on its boundary, while containing no lattice points in its interior except for the origin.
Abstract: A random multiplicative function is a multiplicative function α(n) such that its values on primes, (α(p)), (p=2,3,5,...), are i.i.d. random variables. Such functions serve as a simple model for the ensemble of Dirichlet characters modulo q, for instance. A basic question in the field is finding the limiting distribution of the (normalized) sum of α(n) from n=1 to n=x, possibly restricted to a subset of integers of interest. This question is currently resolved only in a few cases. We shall describe recent work where we are able to find the limiting distribution in many new instances of interest. The distribution we find is non-gaussian, in contrast to all previous works. This is joint work with Mo Dick Wong (Durham University).
Abstract: The diophantine properties of Lebesgue almost every real number (also vector and matrix) have been known since the 1930's, due to work of Khintchine and Groshev. Questions of Mahler and Sprindzhuk concern the behavior of a.e. point with respect to other natural measures, and specifically Cantor's middle thirds set. There has been intensive recent activity lately with several breakthroughs. I will survey the current state of the art.
Abstract: It is known that a modular form of weight k has about k/12 zeros in the fundamental domain (and at infinity). A natural question to ask is "Can we locate the zeros for some families of modular forms?". In 1970, F. Rankin and Swinnerton-Dyer proved the zeros of the Eisenstein series all lie on the circular part of the boundary of the fundamental domain. I will provide a brief overview of the problem and some known results, and discuss the zeros of the Miller basis of cusp forms, a natural basis for the linear space of modular forms of a given weight. As it turns out, for a fixed order of vanishing at infinity and sufficiently large weight, the finite zeros in the fundamental domain of the corresponding form in the Miller basis, all lie on the circular part of the boundary of the fundamental domain.
Abstract: Consider a random polynomial whose coefficients are independent Rademacher random variables (taking the values ±1 with equal probabilities). A central conjecture in probabilistic Galois theory predicts that such polynomials are irreducible asymptotically almost surely as their degree approaches infinity. Recent progress has shown that this conjecture follows from the Generalized Riemann Hypothesis and that the limiting infimum of the irreducibility probability is positive. In this talk, we will explore ideas from the proof of the following result: the limiting supremum of the irreducibility probability is 1, unconditionally. Specifically, we demonstrate that along special sequences of degrees, the polynomial is irreducible asymptotically almost surely. This result is based on joint work with Hokken, Kozma, and Poonen.
Abstract: A key tool in the study of random matrices drawn from the classical compact groups are the joint moments of tr(U), tr(U^2), ... , tr(U^k) (called trace moments), where U is a random matrix drawn according to the Haar probability measure on the group. These exhibit a "mock-Gaussian" behavior - they are Gaussian in a certain range, but the Gaussian regime eventually breaks down. In the Gaussian range the trace moments were computed by Diaconis-Shahshahani (in a partial range) and then Hughes-Rudnick (in the full range). In my talk I will present a recent calculation of the trace moments for the (unitary) symplectic group up to twice the Gaussian range, where non-Gaussian corrections appear. It is accomplished by exploiting the connection between hyperelliptic L-functions over finite fields and random symplectic matrices given by an equidistribution theorem of Katz-Sarnak. This allows one to reduce the problem to the evaluation of certain character sums over function fields which can be solved by existing techniques. I will also discuss an application to the linear statistics of eigenvalues of random symplectic matrices in the narrow bandwidth regime, which was first studied by Rudnick and Waxman. Based on joint work with Noam Pirani.
Abstract: Hardy's circle problem asks for the optimal bound on the remainder term in counting lattice points in a large disk. More generally, on can discuss similar questions for dilates of a fixed planar domain, and for the distribution of the suitably normalized remainder term. These questions intersects with problems in spectral theory and quantum chaos. I will survey some of what is known and what are open problems.
Abstract: The quantum Rabi model describes the interaction between a two-level atom and quantized single-mode radiation (photon). The Hamiltonian of this model is a self-adjoint operator H(g,Δ) with a discrete spectrum. The parameter g is the strength of the coupling between the photon and the atom, and 2Δ is the difference between the atomic energy levels. An eigenvalue E of H(g,Δ) is called Juddian if E+ g2 = Integer. The set of parameters for which there exists a Juddian eigenvalue is a union of a countable set of zeros of polynomial constraints. It was conjectured that the set of parameters for which there is at least one Juddian eigenvalue, is dense in the (g,Δ) plane. We proved this conjecture affirmatively. This talk will discuss the proof of the conjecture and recent results on the spectrum of the Rabi model. A key role is played by the zeros of Laguerre polynomials. Joint work with Zeev Rudnick.