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.

The scattering of electromagnetic waves by homogeneous dielectric media is characterized by a strongly singular integral equation, corresponding to the identity operator perturbed by a non-compact Green operator. Using the Kondrachov-Rellich compact imbedding and the Calderon-Zygmund theory, we prove that the Green operator is polynomially compact if the dielectric boundary is a compact smooth manifold. We then show that the electromagnetic scattering problem admits a robust solution for all non-accretive media ($\mathrm{Im}\chi\leq0$) satisfying certain geometric and topological constraints, except for the critical point $\chi=-2$, where unbounded electromagnetic enhancement may occur. Combining the polynomial compactness of the Green operator with the Arendt-Batty-Lyubich-Vu theorem in semigroup theory, we devise a non-perturbative approach to the solution of electromagnetic scattering problem, as an improvement of the Born approximation.

This work was part of the speaker's PhD thesis project completed at Harvard University.