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Expander graphs in pure and applied mathematics
 Bull. Amer. Math. Soc. (N.S
"... Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number th ..."
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Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry and more. This expository article describes their constructions and various applications in pure and applied mathematics. This paper is based on notes prepared for the Colloquium Lectures at the
Random walks on the mapping class group
"... We show that a random walk on the mapping class group of an orientable surface gives rise to a pseudoAnosov element with asymptotic probability one. Our methods apply to many ..."
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Cited by 8 (1 self)
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We show that a random walk on the mapping class group of an orientable surface gives rise to a pseudoAnosov element with asymptotic probability one. Our methods apply to many
Random Heegaard splittings
 J. Topology
"... Consider a random walk on the mapping class group, and let wn be the location of the random walk at time n. A random Heegaard splitting M(wn) is a 3manifold obtained by using wn as the gluing map between two handlebodies. We show that the joint distribution of (wn, w −1 n) is asymptotically indepen ..."
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Consider a random walk on the mapping class group, and let wn be the location of the random walk at time n. A random Heegaard splitting M(wn) is a 3manifold obtained by using wn as the gluing map between two handlebodies. We show that the joint distribution of (wn, w −1 n) is asymptotically independent, and converges to the product of the harmonic and reflected harmonic measures defined by the random walk. We use this to show that the translation length of wn acting on the curve complex, and the distance between the disc sets of M(wn) in the curve complex, grows linearly in n. In particular, this implies that a random
A Summary of the Work of Gregory Margulis
, 2008
"... Gregory Margulis is a mathematician of great depth and originality. Besides his celebrated results on superrigidity 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 inte ..."
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Gregory Margulis is a mathematician of great depth and originality. Besides his celebrated results on superrigidity 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
Injections of mapping class groups.
"... 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 oncepunctured surfaces and have quit ..."
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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 oncepunctured surfaces and have quite curious behaviour. For instance, some pseudoAnosov elements are mapped to multitwists. Neither of these two types of phenomena were previously known to be possible although the constructions are elementary. 57M07; 32G15, 57R50 1
RESEARCH SUMMARY
"... My research area is the topology and geometry of 3manifolds. 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 3manifolds admit homogeneous Riemannian metrics, and that one can study the topol ..."
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My research area is the topology and geometry of 3manifolds. 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 3manifolds admit homogeneous Riemannian metrics, and that one can study the topology of a 3manifold 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 3manifolds”. 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 3manifolds begins with codimensionone
REPORT ON BIRS WORKSHOP 07W5052 ”LOWDIMENSIONAL TOPOLOGY AND NUMBER THEORY”
"... The goal of the workshop was to bring together topologists and number theorists with the intent of exploring connections between lowdimensional 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 a ..."
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The goal of the workshop was to bring together topologists and number theorists with the intent of exploring connections between lowdimensional 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 3dimensional 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 2torsion), [23]. Moreover, the étale cohomology groups of such spectra satisfy ArtinVerdier duality, which is reminiscent of the Poincare duality satisfied
Every curve is a Teichmüller curve 1 Every curve is a Teichmüller curve
, 2009
"... We prove that every algebraic curve X/Q is birational over C to a Teichmüller curve. ..."
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We prove that every algebraic curve X/Q is birational over C to a Teichmüller curve.
• Lie groups: � Compact Lie groups. � Noncompact simple Lie groups.
, 2009
"... and representations ..."
SIEVE IN DISCRETE GROUPS, ESPECIALLY SPARSE
"... Abstract. We survey the recent applications and developments of sieve methods related to discrete groups, especially in the case of infinite index subgroups of arithmetic groups. 1. ..."
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Abstract. We survey the recent applications and developments of sieve methods related to discrete groups, especially in the case of infinite index subgroups of arithmetic groups. 1.