Abstract: Moments of L-functions has been a topic of intense research in recent years. Through the integration of random matrix theory and multiple Dirichlet series with traditional number theoretic arguments, methods for studying the moments of L-functions have been developed and, in turn, have lead to many well-posed conjectures for their behavior. In my talk, I will discuss the (integral) moments of quadratic DIrichlet L-functions evaluated at the critical point s=1/2. In particular, I will present formulas for computing the critical values for such L-functions and then compare the data for the corresponding moments to the (aforementioned) conjectured moments.

Abstract: I will explain how we calculated the coefficients of moment
polynomials for the Riemann zeta function for k = 4,5.., 13
and numerically tested them against the moment polynomial conjecture.

Abstract: Consider the problem of imaging a reflector (target) from recordings of the echoes resulting from probing the medium with waves emanating from an array of transducers (the array response matrix). We present an algorithm that selectively illuminates the edges or the interior of an extended target by choosing particular subspaces of the array response matrix. For a homogeneous background medium, we characterize these subspaces in terms of the singular functions of a space and wave number restricting operator, which are also called generalized prolate spheroidal wave functions. We discuss results indicating what can be expected from using this algorithm when the medium fluctuates around a constant background medium and the fluctuations can be modeled as a random field.

Abstract: Connections are exposed between integrable equations for spectral gap probabilities of unitary invariant ensembles of random matrices (UE) derived by different --- Tracy-Widom (TW) and Adler-Shiota-van Moerbeke (ASvM) --- methods. Simple universal relations are obtained between these probabilities and their ratios on one side, and variables of the approach using resolvent kernels of Fredholm operators on the other side. A unified description of UE is developed in terms of universal, i.e. independent of the specific probability measure, PDEs for gap probabilities, using the correspondence of TW and ASvM variables. These considerations are based on the three-term recurrence for orthogonal polynomials (OP) and one-dimensional Toda lattice (or Toda-AKNS) integrable hierarchy whose flows are the continuous transformations between different OP bases. Similar connections exist for coupled UE. The gap probabilities for one-matrix Gaussian UE (GUE) or joint gap probabilities for coupled GUE satisfy various PDEs whose number grows with the number of spectral endpoints. With the above connections serving as a guide, minimal complete sets of independent lowest order PDEs for the GUE and for the largest eigenvalues of two-matrix coupled GUE are found.

Abstract: We show that on a smooth compact Riemann surface with boundary (M0, g) the Dirichletto- Neumann map of the Schrödinger operator â g + V determines uniquely the potential V . This seemingly analytical problem turns out to have connections with ideas in symplectic geometry and differential topology. We will discuss how these geometrical features arise and the techniques we use to treat them. This is joint work with Colin Guillarmou of CNRS Nice. The speaker is partially supported by NSF Grant No. DMS-0807502 during this work.

Abstract: The problem of Electrical Impedance Tomography (EIT) with partial boundary measurements is to determine the electric conductivity inside a body from the simultaneous measurements of direct currents and voltages on a subset of its boundary. Even in the case of full boundary measurements the non-linear inverse problem is known to be exponentially ill-conditioned. Thus, any numerical method of solving the EIT problem must employ some form of regularization. We propose to regularize the problem by using sparse representations of the unknown conductivity on adaptive finite volume grids known as the optimal grids. Then the discretized partial data EIT problem can be reduced to solving the discrete inverse problems for resistor networks. Two distinct approaches implementing this strategy are presented. The first approach uses the results for the EIT problem with full boundary measurements, which rely on the use of resistor networks with circular graph topology. The optimal grids for such networks are essentially one dimensional objects, which can be computed explicitly. We solve the partial data problem by reducing it to the full data case using the theory of extremal quasiconformal (Teichmuller) mappings. The second approach is based on resistor networks with the pyramidal graph topology. Such network topology is better suited for the partial data problem, since it allows for explicit treatment of the inaccessible part of the boundary. We present a method of computing the optimal grids for the networks with general topology (including pyramidal), which is based on the sensitivity analysis of both the continuum and the discrete EIT problems. We present extensive numerical results for the two approaches. We demonstrate both the optimal grids and the reconstructions of smooth and discontinuous conductivities in a variety of domains. The numerical results show two main advantages of our approaches compared to the traditional optimization-based methods. First, the inversion based on resistor networks is orders of magnitude faster than any iterative algorithm. Second, our approaches are able to correctly reconstruct the conductivities of very high contrast, which usually present a challenge to the iterative or linearization-based inversion methods.