A family of graphs H_1,...,H_k packs into a graph G if there exist pairwise edge-disjoint copies of H_1,...,H_k in G. Gyarfas and Lehel conjectured that any family T_1, ..., T_n of trees of respective orders 1, ..., n packs into K_n. A similar conjecture of Ringel asserts that 2n copies of any trees T on n+1 vertices pack into K_{2n+1}.

In a joint work with Boettcher, Piguet, Taraz we proved a theorem about packing trees. The theorem implies asymptotic versions of the above conjectures for families of trees of bounded maximum degree. Tree-indexed random walks controlled by the nibbling method are used in the proof. In a joint work with Adamaszek, Adamaszek, Allen and Grosu, we used the nibbling method to prove the approximate version of the related Graceful Tree Labeling conjecture for trees of bounded degree.

In the talk we shall give proofs of both results.

A {\it rational elliptic curve} may be viewed as the set of solutions to an equation of the form $y2=x3+Ax+B$, where $A$ and $B$ are rational numbers. It is known that the rational points on this curve possess a natural abelian group structure, and the Mordell-Weil theorem states that this group is always finitely generated. The {\it rank} of a rational elliptic curve measures {\it how many} rational points are needed to generate all the rational points on the curve.

There is a standard conjecture---originating in work of Goldfeld and Katz-Sarnak---that states that the {\it average} rank of all elliptic curves should be 1/2; however, it has not previously been known that the average rank is even finite! In this lecture, we describe recent work that shows that the average rank is finite (in fact, we show that the average rank is bounded by 1.5).

This is joint work with Arul Shankar.

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.

Any finitely generated group can be endowed with a natural metric which is unique up to maps of bounded distortion (quasi-isometries). A fundamental question is to classify finitely generated groups up to quasi-isometry. Considered from this point of view, fundamental groups of 3-manifolds provide a rich source of examples. Surprisingly, a concise way to describe the quasi-isometric classification of 3-manifolds is in terms of a concept in computer science called "bisimulation." We will focus on describing this classification and a geometric interpretation of bisimulation. Finally, time permitting, we will provide applications to the study of Artin groups. (Joint work with Walter Neumann.)

In this talk we explain our recent proof of global stability for the Boltzmann equation (1872) with the physically important collision kernels derived by Maxwell 1867 for the full range of inverse power intermolecular potentials, $r^{-(p-1)}$ with $p > 2$ and more generally. Our solutions are perturbations of the Maxwellian equilibrium states, and they decay rapidly in time to equilibrium as predicted by Boltzmann's celebrated H-Theorem.

This equation provides a basic example where a wide range of geometric fractional derivatives occur in a physical model of the natural world. We are able to characterize these non-isotropic fractional differentiation effects precisely using in part the ``geometric Littlewood-Paley'' theory of Stein and Klainerman-Rodnianski. This is joint work with P. Gressman.

We fix F a local non archmedean field of characteristic zero, G the points over F of an algebraic reductive group defined over F and s a rational involution of G defined over F. We note H the group of fixed points of G under the action of s and X(G,s) the connected component on the neutral element of the set of complex characters of G antiinvariant under the action of s. Let P be a s-parabolic subgroup of G, id est the intersection M of P with s(P) is a s-stable Levi subgroup, we construct from a irreducible, smooth representation r of M, a rational family of distributions above the algebraic variety X(G,s), which are H-invariant linear forms on tne smooth induced representation ind(P,G; r ). Our main trick is the use of homology of groups.