Georgia Institute of Technology College of Sciences

The Cohen-Lenstra Heuristics conjecturally give the distribution of class groups of imaginary quadratic fields. Since, by class field theory, the class group is the Galois group of the maximal unramified abelian extension, we can consider the Galois group of the maximal unramified extension as a non-abelian generalization of the class group. We will explain non-abelian analogs of the Cohen-Lenstra heuristics due to Boston, Bush, and Hajir and joint work with Boston proving cases of the non-abelian conjectures in the function field analog.

We consider the following problem. Does there exist an absolute constant C such that for every natural number n, every integer 1 \leq k \leq n, every origin-symmetric convex body L in R^n, and every measure \mu with non-negative even continuous density in R^n, \mu(L) \leq C^k \max_{H \in Gr_{n-k}} \mu(L \cap H}/|L|^{k/n}, where Gr_{n-k} is the Grassmannian of (n-k)-dimensional subspaces of R^n, and |L| stands for volume? This question is an extension to arbitrary measures (in place of volume) and to sections of arbitrary codimension k of the hyperplace conjecture of Bourgain, a major open problem in convex geometry. We show that the above inequality holds for arbitrary origin-symmetric convex bodies, all k and all \mu with C \sim \sqrt{n}, and with an absolute constant C for some special class of bodies, including unconditional bodies, unit balls of subspaces of L_p, and others. We also prove that for every \lambda \in (0,1) there exists a constant C = C(\lambda) so that the above inequality holds for every natural number, every origin-symmetric convex body L in R^n, every measure \mu with continuous density and the codimension of sections k \geq \lambda n. The latter result is new even in the case of volume. The proofs are based on a stability result for generalized intersections bodies and on estimates of the outer volume ratio distance from an arbitrary convex body to the classes of generalized intersection bodies.

We consider (complex) Gaussian analytic functions on a horizontal strip, whose distribution is invariant with respect to horizontal shifts (i.e., "stationary"). Let N(T) be the number of zeroes in [0,T] x [a,b]. First, we present an extension of a result by Wiener, concerning the existence and characterization of the limit N(T)/T as T approaches infinity. Secondly, we characterize the growth of the variance of N(T). We will pose to discuss analogues of these results in a few other settings, such as zeroes of real-analytic Gaussian functions and winding of planar Gaussian functions, pointing out interesting similarities and differences. For the last part, we consider the "persistence probability" (i.e., the probability that a function has no zeroes at all in some region). Here we present results in the real setting, as even this case is yet to be understood. Based in part on joint works with Jeremiah Buckley and Ohad Feldheim.

Mathematical modeling of viruses, such as HIV, has been an extensive area of research over the past two decades. For HIV, some important factors that affect within-host dynamics include: the CTL (Cytotoxic T Lymphocyte) immune response, intra-host diversity, and heterogeneities of the infected cell lifecycle. Motivated by these factors, I consider several extensions of a standard virus model. First, I analyze a cell infection-age structured PDE model with multiple virus strains. The main result is that the single-strain equilibrium corresponding to the virus strain with maximal reproduction number is a global attractor, i.e. competitive exclusion occurs. Next, I investigate the effect of CTL immune response acting at different times in the infected-cell lifecycle based on recent studies demonstrating superior viral clearance efficacy of certain CTL clones that recognize infected cells early in their lifecycle. Interestingly, explicit inclusion of early recognition CTLs can induce oscillatory dynamics and promote coexistence of multiple distinct CTL populations. Finally, I discuss several directions of ongoing modeling work attempting to capture complex HIV-immune system interactions suggested by experimental data.