Abstract: We show that Comm(K) = Aut(K) = Mod(S), where K is the subgroup of the mapping class group Mod(S) generated by twists about separating simple closed curves. In particular, this verifies a conjecture of Farb. (Joint work with Dan Margalit.) |
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Abstract: This will be a warmup talk to Auroux' talks on Wed 26th and Thu 27th's. Symplectic topology in dimension four is a field where, in spite of massive recent advances, open questions abound. I will explain one of the most famous and specific of these open questions; Donaldson's point of view concerning it; and recent results of Auroux-Munoz-Presas, which hopefully bring us one step further along the way to a solution. |
Abstract: Let B_{n} be the braid group on n strands. In joint work with Bob Bell, we describe all injections of finite index subgroups of B_{n} into B_{n}. In particular, this allows us to calculate the automorphism group of the pure braid group PB_{n}. Certain elements of Aut(PB_{n}) represent new examples of elements of the abstract commensurator of B_{n}. In joint work with Chris Leininger, which builds on these ideas, we are then able to compute the abstract commensurator of B_{n} explicitly. |
Abstract: Donaldson has shown that every compact symplectic 4-manifold can be represented as a Lefschetz pencil (i.e., up to blowups, a fibration over the 2-sphere with at most nodal fibers). This leads to a description of symplectic 4-manifolds by positive factorizations in mapping class groups. In another related approach, symplectic 4-manifolds can be realized as branched covers of the complex projective plane (with a branch curve presenting cusp and node singularities). This leads to a description by factorizations in braid groups. This talk will provide an elementary introduction to these constructions, and a discussion of how they make it possible to reformulate various questions about the classification of symplectic 4-manifolds as (hard) combinatorial group theory problems. |
Abstract: The classification of symplectic 4-manifolds is closely related to that of Lefschetz fibrations and to that of plane curves with node and cusp singularities. We discuss recent results on these problems, in three main directions: (1) isotopy and regular homotopy results for plane curves; (2) surgeries along Lagrangian annuli and tori; (3) stabilization by fiber sums. We will in particular explain how classification results for fiber sums are closely related to Garside-type properties of braid groups and mapping class groups. |
Abstract: If X is a locally finite tree, its group of automorphisms G=Aut(X) is a locally compact group. A lattice in G is a discrete subgroup with cofinite Haar measure. With the right normalization of Haar measure, there is a simple combinatorial formula for the Haar measure, or ``covolume'' of a lattice. A study of these ``tree lattices'' generalizes the study of lattices in Lie groups over a nonarchimedean local field, and provides a remarkably rich theory (see the recent book by Bass-Lubotzky). One of the basic problems about a locally compact group is to classify its lattices up to commensurability. Outside the setting of linear groups, commensurability invariants have been hard to come by. We introduce two new commensurability invariants, and construct lattices realizing every possible choice of these invariants. In particular, we construct uncountably many noncommensurable lattices with any given covolume. |
Abstract:Complexes of curves are a very fruitfull tool for studying mapping class groups of surfaces. In a similar fashion, one can use complexes of embedded spheres and discs to study mapping class groups of 3-manifolds, and along the way the automorphism groups of free groups. The main theorem is a stability theorem for mapping class groups of 3-manifolds, which we also interpret in terms of automorphisms groups of free groups with boundaries (joint work with Allen Hatcher). |
Abstract:A famous theorem of Margulis asserts that any normal sub-group of a higher rank lattice is either finite or of finite index. We will discuss similar result valid in the setting of a lattice in a product of groups. Examples, apart of Lie-groups, consisting of automorphisms of trees and buildings. Our tools are ergodic theoretic. In particular we make use of the notion of the "Poisson boundary" of a group. We will not assume any familiarity with that object. The lecture will be based on a joint work with Yehuda Shalom. |
Abstract:We investigate distribution of orbits of lattices acting on homogeneous spaces. In particular, we show that for natural actions of a lattice (in a semisimple Lie group) on finite-volume spaces, dense orbits are equidistributed. We also obtain equidistribution results for infinite-volume homogeneous spaces and deduce an analog of the quantitative Oppenheim conjecture (proved by Eskin, Margulis, Mozes) for Gramm matrices evaluated at integral frames. This is a joint work with Barak Weiss. |
Abstract:The moduli space M(d) is the space of all holomorphic self-maps of the Riemann sphere, modulo the action by conjugation of the group of Mobius transformations. In this talk, I will describe the limiting dynamics at the boundary of M(d). The approach involves algebraic geometry (geometric invariant theory), hyperbolic geometry (the barycenter construction), and of course holomorphic dynamics. I won't assume any background. |
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