We show that a random walk on the mapping class group of an orientable surface gives rise to a pseudo-Anosov element with asymptotic probability one. Our methods apply to many

My research area is the topology and geometry of 3-manifolds. I was attracted to it because of the richness it acquired from W. Thurston’s revolutionary work starting in the 1970s. Thurston’s key insight was that many 3-manifolds admit homogeneous Riemannian metrics, and that one can study the topology of a 3-manifold via this geometry. This profusion of geometry has now been stunningly confirmed by Perelman’s recent proof of Thurston’s Geometrization Conjecture. As a direct result, while my work has focused on what will initially seem like purely topological problems, in fact I have used a broad range of techniques to attack them, including hyperbolic geometry, number theory, and algebraic geometry, as well as more obviously related areas such as combinatorial group theory and the theory of foliations. These connections to other fields have led me to collaborate with both number theorists and theoretical physicists, and below I’ll need to refer to both the Langlands Conjecture and the Classification of Finite Simple Groups, as well as to such topological oddities as “random 3-manifolds”. In the rest of this overview, I will outline the division of my work into broad topics, and then discuss my results in detail in later sections. As in many other areas of geometry, the study of 3-manifolds begins with codimensionone

Gregory Margulis is a mathematician of great depth and originality. Besides his celebrated results on super-rigidity and arithmeticity of irreducible lattices of higher rank semisimple Lie groups, and the solution of the Oppenheim conjecture on values of irrational indefinite quadratic forms at integral points, he has also

We construct new monomorphismsbetween mapping class groups of surfaces. The first family of examples injects the mapping class group of a closed surface into that of a different closed surface. The second family of examples are defined on mapping class groups of once-punctured surfaces and have quite curious behaviour. For instance, some pseudo-Anosov elements are mapped to multi-twists. Neither of these two types of phenomena were previously known to be possible although the constructions are elementary. 57M07; 32G15, 57R50 1

The goal of the workshop was to bring together topologists and number theorists with the intent of exploring connections between low-dimensional topology and number theory, with the special focus on topics described in the section below. We hoped that the balance between the lectures and free time as well as the intimate setting of the Banff Research Station will stimulate many informal discussions and collaborations. We were delighted to see that these objectives were fulfilled. We would like to thank BIRS staff for their hospitality and efficiency. 2. Topics of the workshop 2.1. Arithmetic Topology. Starting from the late 1960’s, B. Mazur and others observed the existence of a curious analogy between knots and prime numbers and, more generally, between knots in 3-dimensional manifolds and prime ideals in algebraic number fields, [24,25,25,30–32]. For example, the spectrum of the ring of algebraic integers in any number field has étale cohomological dimension 3 (modulo higher 2-torsion), [23]. Moreover, the étale cohomology groups of such spectra satisfy Artin-Verdier duality, which is reminiscent of the Poincare duality satisfied

We prove that every algebraic curve X/Q is birational over C to a Teichmüller curve.