Abstract:This is joint work with Ronnie Lee and Ed Miller. I shall explain how and why one "counts" the representations of fundamental groups of 3-manifolds into SU(n) by making use of various symplectic and analytic structures that representation varieties enjoy. (I will not assume any familiarity with such structures.) |
Abstract: I will describe an obstruction theory for determining which open manifolds are homeomorphic to open subsets of euclidean space. The talk should be accessible to graduate students in geometry and topology. |
Abstract:Motivated by the dynamics of billiards, we will present a classification of immersed hyperbolic planes inside the moduli space of Riemann surfaces of genus 2. Our results give a parallel, in the setting of algebraic geometry, of Ratner's theorems on unipotent flows and Lie groups. |
Abstract:The title is actually a bad pun on Beilinson's paper "coherent sheaves on P^n and problems of linear algebra". I will state some simple results about the topology of Lagrangian submanifolds in T^*S^n, and explain the algebraic part of the proof which uses some classical linear algebra. |
Abstract: We consider locally symmetric manifolds with a fixed universal covering (e.g. hyperbolic manifolds of fixed dimension), and construct for each such manifold $M$ a simplicial complex $\mathcal{R}$ whose size is proportional to the volume of $M$. When $M$ is non-compact, $\mathcal{R}$ is homotopically equivalent to $M$, while when $M$ is compact, $\mathcal{R}$ is homotopically equivalent to $M\setminus N$, where $N$ is a finite union of submanifolds of fairly smaller dimension. This reflects how the volume controls the topological structure of $M$, and yields concrete bounds for various finiteness statements which previously had no quantitative proofs. For example, it gives an explicit upper bound for the possible number of locally symmetric manifolds of volume bounded by $v>0$, and it yields an estimate for the size of a minimal presentation for the fundamental group of a manifold in terms of its volume. It also yields a number of new finiteness results. |
Abstract: We will give an overview of some results on the role of CAT(0) cubical complexes in geometric group theory. |
Abstract: This talk is about the limiting behavior of unbounded sequences in Rat_d, the space of holomorphic self-maps of CP^1 of degree d>1. In particular, we study the measures of maximal entropy and degenerations of the associated metrics of non-negative curvature on the Riemann sphere: the limits are polyhedral. The arguments rely on the study of the iterate map from Rat_d to Rat_{d^n} at the boundary. |
Abstract: Solvability of the conjugacy problem for relatively hyperbolic groups was announced by M.Gromov who introduced this class of groups. Using the definition of B.Farb of a relatively hyperbolic group, we prove this assertion and conclude that the conjugacy problem is solvable for fundamental groups of complete, finite-volume, negatively curved manifolds, and for finitely generated fully residually free groups. |
Abstract:We consider the space K(S) of complex projective structures with discrete and faithful holonomy on a surface S of negative Euler characteristic. There is a unique component of the interior of K(S) where the structures have injective developing maps. This is called the standard component; the other components are called exotic. We exhibit collections of exotic components whose closures pair-wise intersect. As a corollary we show that the closure of any exotic component is not a manifold. Joint work with Ken Bromberg. |
Abstract:Hempel has shown that the fundamental groups of knot complements are residually finite. This implies that every nontrivial knot must have a finite-sheeted, noncyclic cover. We give an explicit bound, $\Phi (c)$, such that if $K$ is a nontrivial knot in the three-sphere with a diagram with $c$ crossings then the complement of $K$ has a finite-sheeted, noncyclic cover with at most $\Phi (c)$ sheets. |
Abstract:Unlike Euclidean crystallographic groups, properly discontinuous groups of affine transformations need not be amenable. For example, a free group of rank two admits a properly discontinuous affine action on 3-space. Milnor imagined how one might construct such an action: deform a Schottky subgroup of O(2,1) inside the group of Lorentzian isometries of Minkowski space, although as he wrote in 1977, ``it seems difficult to decide whether the resulting group action is properly discontinuous.'' In 1983, Margulis, while trying to prove such groups don't exist, constructed the first examples. In his 1990 doctoral thesis, Drumm constructed explicit geometric examples from fundamental polyhedra called ``crooked planes'' and proved that every discrete subgroup which is not cocompact admits proper affine deformations. These structures seem to be intimately related to hyperbolic Riemann surfaces and their deformation theory. Jointly with Labourie and Margulis, we have given a criterion for properness of an affine deformation in terms of invariant measures for the geodesic flow. This gives an explicit description of the moduli space of proper 3-dimensional affine actions. |
Abstract: A "conifold transition" is a particular surgery on six-manifolds which arises in string theory. We will discuss work (variously in collaboration with Corti, Thomas and Yau) explaining when this surgery exists as a symplectic operation, its relation to mirror symmetry, and what light (if any) it sheds on symplectic topology. |