For a finite 2-generated group G, one can consider the moduli of elliptic curves equipped with G-structures, which is roughly a G-Galois cover of the elliptic curve, unramified away from the origin. The resulting moduli spaces are quotients of the upper half plane by possibly noncongruence subgroups of SL(2,Z). When G is abelian, it is easy to see that such level structures are equivalent to classical congruence level structures, but in general it is difficult to classify the groups G which yield congruence level structures. In this talk I will focus on a recent joint result with Pierre Deligne, where we show that for any metabelian G, G-structures are congruence in an arithmetic sense. We do this by studying the monodromy action of the fundamental group of the moduli stack of elliptic curves (over Q) on the pro-metabelian fundamental group of a punctured elliptic curve.

In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory (also known as Hilbert's 12th Problem), and the special values of L-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field. Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points. Of these, Stark's Conjecture has special relevance toward explicit class field theory. I will describe my recent proof of the Gross-Stark conjecture, a p-adic version of Stark's Conjecture that relates the leading term of the Deligne-Ribet p-adic L-function to a determinant of p-adic logarithms of p-units in abelian extensions. Next I will state my refinement of the Gross-Stark conjecture that gives an exact formula for Gross-Stark units. I will conclude with a description of work in progress that aims to prove this conjecture and thereby give a p-adic solution to Hilbert's 12th problem.

Over the last decades, the study of "spin glasses" in physics has stimulated a large amount of theoretical activity in physics, and led to several breakthroughs. While the original puzzle of spin glass materials is still not fully solved, their theoretical analysis has created powerful techniques as well as a rich conceptual framework, to study emergent properties of strongly disordered and interacting systems. In this talk, I will use these tools and discuss how apparently unrelated complex problems such as: how to pack many objects in a given volume, how to color a graph with a given number of colors, why a liquid is turning into a glass when the temperature is lowered, and why some computational (classical and quantum) problems are hard while other are easy Â? actually (and surprisingly) do share many characteristics when looking at them through the (spin) glass.

Shnirelman discovered in the 1970s that the eigenfunctions of the Laplacian on a compact Riemannian manifold whose flow is ergodic with respect to Liouville measure exhibit an analogue of classical ergodicity at the quantum level. This phenomenon became known as "Quantum Ergodicity" and Schnirelman's proof was completed by Zelditch and Colin de Verdiere in the 1980s. Following a brief introduction to the subject, I will show that Quantum Ergodicity can also hold in systems which are essentially integrable, provided they have some arithmetic structure. Finally, in the absence of such an arithmetic structure, a very different phenomenon occurs: scarring. This talk is based on joint work with Par Kurlberg, KTH Stockholm.

Joint work with Jacob Tsimerman. Let B(g,p) denote the number of isomorphism classes of g-dimensional abelian varieties over the finite field of size p. Let A(g,p) denote the number of isomorphism classes of principally polarized g dimensional abelian varieties over the finite field of size p. We derive upper bounds for B(g,p) and lower bounds for A(g,p) for p fixed and g increasing. The extremely large gap between the lower bound for A(g,p) and the upper bound B(g,p) implies some statistically counterintuitive behavior for abelian varieties of large dimension over a fixed finite field.