I will sketch a recent approach to study the resonance spectrum of scattering Schroedinger operators, in cases where the trapped set of the corresponding classical dynamics (near some positive energy) is a fractal chaotic repeller. In that situation, we are interested in the distribution of resonances in the vicinity of the real axis, in the semiclassical limit. Our approach tends to mimic the PoincarÈ section method used to study the corresponding classical flows. Namely, we show that resonances can be defined through an implicit equation involving an appropriately defined quantized Poincare map. The subsequent study of this "quantum map" allows to recover known properties of the resonance spectrum (fractal bounds on the number of resonances in strips, spectral gap), and hopefully more.

This is joint work with J. Sjoestrand and M. Zworski.

(joint work with O. Imamoglu and A. Toth) An interesting new class of modular forms has emerged in the last several years that generalize Ramanujan's mock theta functions. The generalization is based on an observation of of Zwegers who showed that mock theta functions occur as holomorphic parts of harmonic Maass forms of weight 1/2 having singularities in cusps. Their non-holomorphic parts are directly related to actual theta functions. This realization has enhanced the study of the coefficients of the mock-theta functions, which are related to various kinds of partitions.

In this talk I will describe a new class of mock-modular forms of weight 1/2. Here the Fourier coefficients can be expressed in terms of cycle integrals of the classical modular j-function. The hope that these cycle integrals could be related to abelian extensions of real quadratic fields has not been realized. However, we are able to relate them to certain regularized Petersson inner products of weight 3/2 weakly holomorphic forms whose coefficients are traces of the values of the j-function at cm points. As an application we show that modular integrals having rational period functions are real quadratic versions of (logarithmic derivatives of) Borcherds products.

Following the brilliant insight of Riemann, that a good understanding of the distribution of prime numbers is equivalent to a good understanding of the location of zeros of pertinent L-functions, analytic number theory has traditionally centered on developing this point-of-view. The Riemann Hypothesis, that all the zeros "lie on the 1/2-line", remains unproven, so researchers have many theorems bounding some aspect of the zeros close to the 1-line. These yield the proofs of many of the key results in the subject.

Given Riemann's equivalence, and the consequent goal to prove the Riemann Hypothesis, it has long seemed "obvious" that one should approach the subject by gaining a better and better understanding of the distribution of zeros. However is this really the correct way to proceed? Is it so obvious that this is the right way to gain an improved understanding of central issues concerning the distribution of primes? Recently Soundararajan and the speaker have shown that several of the key results in the subject can be proved more succinctly, arguably more easily, without reverting to a study of zeros of L-functions, using the notion of pretentiousness.

In this talk we will explain what "pretentiousness" is, exhibit the depth of ideas we use, and show how various well-known results follow, as well as several new results.

Consider the model of a one-dimensional gas, whose particles have random initial positions and random initial velocities. Particle attract each other due to gravitation, and stick together at collisions. As time goes, the number of particles decreases while their sizes increase until there forms a giant single particle of the total mass.

The results on this process of mass aggregation are given in form of limit theorems as the number of initial particles tends to infinity. For example, the stochastic processes of the total number of particles satisfy a functional central limit theorem. I will show how this problem on the number of particles is related to positivity of integrated random walks. I will discuss the problem of finding one-sided small deviation probabilities of integrated random walks and other stochastic processes, and tell about the progress in this field.

The concept of a smooth metric measure space has recently arisen as a useful object within Riemannian geometry, for example in Perelman's formulation of Ricci flow as a gradient flow. In this setting, a key objective is to find a suitable generalization of Ricci curvature, and to understand the associated ``quasi-Einstein'' metrics. Taking two different perspectives, Lott, Villani, Sturm and Chang, Gursky and Yang have found two distinct approaches to studying smooth metric measure spaces. While the formulations are different, they both introduce an extra dimensional parameter $m$ which, in the limit $m\to\infty$, recovers the curvatures that arise in Perelman's treatment of the Ricci flow. In this way it becomes interesting to see if the two approaches are related. As the quasi-Einstein metrics of these approaches include conformally Einstein metrics, the bases of Einstein warped products, and gradient Ricci solitons, finding a relation between them might also allow us to find interesting connections between these metrics.

In this talk, I will introduce what I call ``conformally warped manifolds'' as a way to unite the approaches of Lott-Villani-Sturm and Chang-Gursky-Yang. I will discuss three results which suggest that this notion is indeed the ``best'' approach. First, I will discuss the variational problem associated to quasi-Einstein metrics, which naturally relates the Yamabe constant to Perelman's shrinker entropy. Second, I will discuss a Liouville-type theorem which illustrates the usefulness of studying the limit $m\to\infty$ as a way to overcome difficulties in the $m=\infty$ comparison theory. Third, I will discuss a compactness theorem for compact quasi-Einstein metrics analogous to Anderson's theorem for Einstein metrics, which is independent of the parameter $m$.

We prove that hyperbolic groups with a quasiconvex hierarchy are virtually subgroups of graph groups. Our focus is on "special cube complexes" which are nonpositively curved cube complexes that behave like "high dimensional graphs" and are closely related to graph groups. The main result illuminates the structure of a group by showing that it is "virtually special", and this yields the separability of the quasiconvex subgroups of the groups we study.

As an application, we resolve Baumslag's conjecture on the residual finiteness of one-relator groups with torsion. Another application shows that generic haken hyperbolic 3-manifolds have "virtually special" fundamental group. Since graph groups are residually finite rational solvable, combined with Agol's virtual fibering criterion, this proves that finite volume haken hyperbolic 3-manifolds are virtually fibered.