Topology/Geometry Seminar
3:30  4:30pm Tuesdays
425 Hill
Please contact Steve Ferry, Feng Luo, Saul Schleimer, or Chris Woodward if you would like to speak or suggest a speaker.
Seminar Schedule  Fall, 2006
(Please scroll down for abstracts.)







































Schedule with abstracts







This is joint work with K. Wehrheim and, in the Ainfinity setting, S. Mau. We develop functoriality of Lagrangian Floer theory for Lagrangian correspondences. Applied to the example of moduli spaces of flat SU(n)bundles, we obtain SU(n) Floer invariants for threemanifolds containing tangles (extending work by Floer and others.) As an application, we show that the symplectic mapping class group of certain representation varieties is nontrivial. 



In the 1970's Bob Edwards and Frank Quinn discovered an amazing characterization of highdimensional topological manifolds. A finite dimensional, connected locally contractible homology nmanifold is a topological manifold if and only if it has general position for 2 dimensional disks and some open subset homeomorphic to R^n. In the 1990's, Bryant, Ferry, Mio, and Weinberger produced spaces satisfying all but the last condition. A natural question is what these spaces look like. Recently, Bryant, Ferry, and Mio have proven a structure theorem for these spaces which should help in elucidating the properties of these strange spaces. 



Based on the duality between GromovWitten theory on CalabiYau threefolds and ChernSimons theory on three manifolds, M. Aganagic, A. Klemm, M. Marino, and C. Vafa proposed "the topological vertex", an algorithm on effectly computing GromovWitten invariants in all genera of any toric CalabiYau threefold. I will describe a mathematical theory of the topological vertex based on relative GromovWitten theory. This is a joint work with Jun Li, Kefeng Liu, and Jian Zhou. 



I will discuss some of my recent results and that of Ren Guo concerning the Teichmuller spaces of surfaces with boundary and their relationship to the cosine law. 



In this talk we will discuss a new quasiisometry invariant of metric spaces which we call thickness. We show that any thick metric space is not (strongly) relatively hyperbolic with respect to any nontrivial collection of subsets. The class of thick groups includes many important examples such as mapping class groups of all surfaces (except those few that are virtually free), the outer automorphism group of the free group on at least 3 generators, SL(n,Z) with n>2, and others. We shall also discuss some ways in which thick groups behave rigidly under quasiisometries. This work is joint with Cornelia Drutu and Lee Mosher. 



Knot Floer homology is a knot invariant introduced by Ozsvath and Szabo, and by Rasmussen. The Euler characteristic of knot Floer homology gives rise to the Alexander polynomial of a knot, so many properties of Alexander polynomial can be generalized to knot Floer homology. For example, if a knot is fibred, then its knot Floer homology is "monic". Ozsvath and Szabo conjectured that the converse of the previous fact is also true, namely, if the knot Floer homology is monic, then the knot is fibred. In this talk, we will discuss a proof of this conjecture, based on the works of Paolo Ghiggini and of the speaker. A corollary is that if a knot in S^3 admits a lens space surgery, then the knot is fibred. 



I plan to discuss the definition of open GromovWitten invariants for Lagrangian submanifolds that arise as the real points of a real symplectic manifold. Furthermore, I will describe a calculation of the genus zero open GromovWitten theory of the Fermat type quintic threefold and its real Lagrangian. The result fits nicely into the general framework of mirror symmetry. This calculation represents joint work with R. Pandharipande and J. Walcher 



I describe a volume function defined on the set of representations of the fundamental group of an ndimensional manifold into the group of isometries of hyperbolic nspace. This function has humour: its definition is somewhat technical and messy, but many of its properties and applications are elegant and easy to state. This talk explores some of the properties and applications and is largely based on joint work with Steve Boyer. 



We discuss an approach toward understanding of hyperbolic structures on 3manifolds by using Heegaard splittings. The goal is to use the combinatorial data in the splitting and construct a model for the hyperbolic metric. We show how this approach was applied to a family of examples and how one expects to generalize this construction. 



I will outline a classification of automorphisms (selfhomeomorphisms up to isotopy) of 3manifolds. This classification, which is analogous to the classification by NielsenThurston of automorphisms of surfaces, was developed by Leonardo Carvalho and myself. The emphasis of the outline will be on reducing surfaces, which are surfaces invariant up to isotopy used to decompose an automorphism. Following the outline, the talk will be concerned with the most difficult part of the classification, dealing with an arbitrary reducible manifold M. At this point, we must cheat a little: Given an automorphism f of M, we must first replace it by a pair of automorphisms (g, h) where g is an adjusting automorphism and h is an automorphism of the irreducible summands of M. This replacement is done using a short exact sequence of mapping class groups: from the adjusting automorphisms, to all automorphisms, to automorphisms of the irreducible summands. 



A blowup of an ideal triangulation of the interior of a compact 3manifold X is a special minimal vertex triangulation of X. Many properties are shared between ideal triangulations and their blowups; e.g., isomorphic projective solutions spaces of closed normal surfaces, efficiency of triangulations, etc. We shall give a method to construction a blowup for any given ideal triangulation and discuss applications to Dehn fillings and the construction of 3manifolds via triangulations. Time permitting we shall give a blowup of the twotetrahedra ideal triangulation of the figure eight knot complement in the 3sphere, which gives a minimal tentetrahedra triangulation of the figure eight knot exterior; and show that for any knot in the 3sphere, there is a onevertex triangulation of the 3sphere having the knot as an edge. 