Geometry/Topology 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  Spring, 2007
(Please scroll down for abstracts.)




















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Schedule with abstracts







Vertex operator algebras can be regarded as generalizations of associative algebras, but have much richer structures. We study a family of vertex operator algebras which admit two copies of the affine Lie algebra actions with dual central charges, and whose top levels are identified with regular functions on the Lie groups. We discuss two constructions: one is based on the properties of intertwining operators and KnizhnikZamolodchikov equations; the other is to use the enveloping algebra of the vertex algebroid associated to the Lie group and a fixed level. We show that the two constructions yield the same vertex operator algebra. The case of integral central charges will also be discussed. 



We give a completely combinatorial definition and proof of invariance
of HeegaardFloer homology for links in the 3sphere. The definition
is based on a gridlink presentation of the link, also known as an arc
presentation. The Euler characteristic gives an apparently new method
of computing the Alexander polynomial.
Some applications include new invariants for Legendrian and transverse knots, giving new examples of knots with more than one transversal structure. This talk represents work by various subsets of Ng, Manolescu, Ozsvath, Sarkar, Szabo, and myself. 



Heegaard Floer homology is an invariant for closed threemanifolds, which also gives invariants for fourmanifolds, knot and links, and contact structures, etc. Conjecturally, Heegaard Floer homology is equivalent to the SeibergWitten theory. In this talk, we will give a combinatorial description of the hat version Heegaard Floer homology and the hat version knot Floer homology for any oriented closed threemanifolds and null homologous links. This is joint work with Sucharit Sarkar. 



In this talk I will sketch a proof of the following theorem. The fundamental class of a displaceable monotone Lagrangian submanifold vanishes. This proof uses an geometric argument and Hamiltonian Floer homology. Afterwards I will put this into a more conceptual context involving Lagrangian and Hamiltonian Floer homology and their interplay. This yields another (slightly more general) proof of the aforementioned theorem and some further corollaries concerned with topological properties of Lagrangian submanifolds. 



A convex real projective structure on a closed surface S is given by
S = Omega/Gamma, where Omega is a convex domain in R^2 contained in
RP^2, and Gamma is a discrete subgroup of PGL(3,R). There are many
such structures: the Klein model of the hyperbolic plane shows that
every hyperbolic structure on S induces a convex real projective
structure.
There is a canonical identification of a convex real projective structure on an orientable surface S of genus g > 1 and a pair consisting of a conformal structure Sigma together with a holomorphic cubic differential U on the surface. (Sigma,U) can be used to explicitly calculate the RP^2 holonomy along loops on S in various limiting cases: neck pinches (the DeligneMumford compactication of the moduli space of curves), and the case that U homothetically goes to infinity. The proofs use affine differential geometry results of ChengYau and C.P. Wang, and PDE estimates. 


(1:402:40pm, Hill 525. Note special day and time.) 
I will talk about PSL(2,C), which can be identified with the full group of orientation preserving isometries of hyperbolic 3space. The discrete subgroups of this group are called Kleinian groups, and their orbit spaces are Kleinian orbifolds. The theory of Kleinian groups changed dramatically at the beginning of 21st century. Three great conjecures have been solved (Bers' density conjecture, Marden's tameness conjecture, and Thurston's ending lamination conjecture). In spite of that, important questions remain open. One of them is the following question. When is a finitely generated subgroup of PSL(2,C) Kleinian? The exact answer is unknown even for twogenerator subgroups. We will show how geometry helps to answer this question for a special class of groups and give a taste of how complicated the structure of the parameter space of 2generator Kleinian groups is by showing a slice through this space. This is a joint work with Natalia Kopteva. 



Every genus g surface bundle admits a Heegaard splitting of genus 2g + 1 and for generic surface bundles, this is a minimal genus Heegaard splitting. For certain examples, however, the Heegaard genus is significantly lower than 2g + 1. I will describe a way to construct monodromy maps yielding surface bundles with arbitrarily high genus leaves, but admitting genus two Heegaard splittings. I will then outline a proof that all surface bundles with genus two Heegaard splittings have monodromy maps of this form. 
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Traditionally, a 6jsymbol is a certain algebraic machinery associated to a combinatorial tetrahedron with various representations attached to its faces, edges or vertices. Combining the 6jsymbols associated to the simplices of a manifold then defines an invariant of this manifold. One example is the Kashaev 6jsymbol, defined by considering the representation theory of the Weyl Hopf algebra. We will introduce a more geometric discussion of this Kashaev 6jsymbol. In particular, it is closely connected to the geometry of ideal tetrahedra in hyperbolic 3space. 
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We outline a program for determining the low volume closed and cusped hyperbolic 3manifolds. If successful it will demonstrate a close connection between low volume hyperbolic 3manifolds and hyperbolic 3manifolds with low combinatorial/topological complexity, thereby addressing the Thurston, Weeks, MatveevFomenko hyperbolic complexity conjecture. 



For any link, the Mahler measure of the Jones polynomial converges under twisting to that of a certain multivariable polynomial which depends on the number of strands we twist at each site. This is consistent with the convergence of hyperbolic volume under Dehn surgery. Twisting two strands at each site, we get the twistbracket polynomial, which generalizes the Kauffman bracket. As an application, any infinite sequence of distinct prime alternating links with cyclotomic Jones polynomials must have unbounded hyperbolic volume. This is joint work with Abhijit Champanerkar. 



Measured laminations on surfaces were introduced by Thurston as a tool to study the mapping class group and hyperbolic threemanifolds. They are obtained as limits of simple closed curves on surfaces. The first example is the set of simple closed curves on the torus which are parametrized by the rational numbers. Accordingly, the ending laminations on the torus are parametrized by the irrational numbers. Strikingly, when the genus of the surface is at least four, the space of ending lamination is connected. This, joint work with Chris Leininger, answers a question of Pete Storm. 



I will describe the ideas behind our Computational Homology project (CHomP). The goal is to be able to efficiently compute the homology of large cubical complexes and the homology of maps between large cubical complexes that arise from the numerical simulations of nonlinear systems and/or experimental data. I will explain why for a variety of applications cubical complexes are more natural than simplicial complexes and why cubical complexes lend themselves to efficient algorithms. I will also argue that the "correct" way to approximate nonlinear maps for the purposes of homological computations is through acyclic multivalued maps, not through simplicial approximations. 



Given a branched covering between closed surfaces one can easily
establish some relations, including the classical RiemannHurwitz
formula, in terms of the Euler characteristic and orientability of the
surfaces involved, the total degree, and the local degrees at the
branching points. Therefore one can view these relations as necessary
conditions for the existence of a branched covering matching a given
"combinatorial datum".
A classical problem dating back to Hurwitz asks whether these conditions are also sufficient. Thanks to the work of many authors, the problem remains open only when the base surface of the putative covering is the sphere, in which case exceptions to existence are known to occur. This talk describes joint work with Pervova, in which new infinite series both of existent coverings and of exceptions have been found, including previously unknown exceptions with the putative covering surface not being the sphere and with more than three branching points. All our series come with systematic explanations, based on three different techniques (dessins d'enfants, decomposability, graphs on surfaces) that we exploit to attack the problem, besides Hurwitz's classical technique based on permutations. If time permits some reference will also be made to: 1) applications of these results to the theory of complexity of 3manifolds, and 2) a new approach to the problem which exploits the geometry of 2orbifolds. 


(1:402:40pm. Note special time.) 
I will report on recent work motivated by the solution to the conjugacy problem in freebycyclic groups due to Bogopolski, Martino, Maslakova, and Ventura. Specifically, the goal is to construct a version of improved relative train tracks (akin to those of Bestvina, Feighn, and Handel) in an algorithmic fashion, and to use these properties to obtain generalizations of algorithmic results. 



I will describe an algorithm to compute the unit ball of the Thurston norm using normal surface theory. Applications include an algorithm to decide whether a 3manifold fibres over the circle. This is joint work with Daryl Cooper. 



There are certain families of words and word sequences (words in the generators of a twogenerator group) that arise frequently in the Teichmüller theory of hyperbolic threemanifolds and Kleinian and Fuchian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of such words and sequences: the semigroup of socalled good words of GehringMartin, the socalled killer words of GabaiMeyerhoffN.Thurston, the Farey words of KeenSeries and Minksy, the discreteness algorithm Fibonacci sequences of GilmanJiang, and parabolic dust words. We establish conenctions between these families. 