# Seminars & Events for Topology Seminar

##### Almost minimal laminations and the connectivity of ending lamination space

We show that if S is a finite type hyperbolic surface which is not the 3 or 4-holed sphere or 1-holed torus, then the Ending lamination space of S is connected, locally path connected and cyclic. Using Klarrich's theorem this implies that the boundary of a curve complex associated to any such space is connected, locally path connected and cyclic.

##### Khovanov homology, open books, and tight contact structures

I will discuss a construction modeled on Khovanov homology which associates to a surface, $S$, and a product of Dehn twists, $\Phi$, a group $Kh(S,\Phi)$. The group $Kh(S,\Phi)$ may sometimes be used to determine whether the contact structure compatible with the open book $(S,Phi)$ is tight or non-fillable. This construction generalizes the relationship between the reduced Khovanov homology of a link and the Heegaard Floer homology of its branched double cover.

##### On Simons conjecture for knots

Let $K$ be a non-trivial knot in $S3$. Simon's Conjecture asserts that $\pi_1(S3\setminus K)$ surjects only finitely many distinct knot groups. We discuss the proof of this for a class of small knots that includes 2-bridge knots.

##### On the Number of Solutions to Asymptotic Plateau Problem

We give a simple topological argument to show that the number of solutions of the asymptotic Plateau problem in hyperbolic space is generically unique. In particular, we show that the space of codimension-1 closed submanifolds of sphere at infinity, which bounds a unique absolutely area minimizing hypersurface in hyperbolic n-space, is dense in the space of all codimension-1 closed submanifolds at infinity. In dimension 3, we also prove that the set of uniqueness curves in asymptotic sphere for area minimizing planes is generic in the set of Jordan curves at infinity. We also give a nonuniqueness result by showing existence of simple closed curves in the sphere at infinity which are the asymptotic boundaries of more than one area minimizing surfaces.

##### Minimal intersection and self-intersection of curves on surfaces

Consider the set of free homotopy classes of directed closed curves on an oriented surface and denote by V the Z-module generated by this set. Goldman discovered a Lie algebra structure on this module, obtained by combining the geometric intersection of curves with the usual loop product. Later on, Turaev found a Lie coalgebra structure on the quotient of V by the one dimensional subspace generated by the trivial loop. Moreover, the Goldman Lie bracket passes to this quotient and both operations satisfy the identities of a Lie bialgebra. This Lie bialgebra has a purely combinatorial presentation.

##### Bordered Heegaard Floer homology

I will describe a construction of invariants for three-manifolds with (parameterized) boundary. The invariant associates a differential graded algebra to an oriented surface, and a (suitably generalized) module to a three-manifold whose boundary is that surface. This invaraint also enjoys a "pairing theorem" relating it with HF-hat of closed three-manifolds. I will describe the basic properties of these invariants, and also the role of certain bimodules which appear in theory. This is joint work with Robert Lipshitz and Dylan Thurston.

##### On Khovanov homology and sutured Floer homology

The relationship between Khovanov- and Heegaard Floer-type homology invariants is intriguing and still poorly-understood. In this talk, I will describe a connection between Khovanov's categorification of the reduced $n$-colored Jones polynomial and sutured Floer homology, a relative version of Heegaard Floer homology developed by Andras Juhasz. As a corollary, we will prove that Khovanov's categorification detects the unknot when $n>1$. This is joint work with Stephan Wehrli.

##### Primitive-stable representations of the free group

Automorphisms of the free group $F_n$ act on its representations into a given group $G$. When $G$ is a simple compact Lie group and $n>2$, Gelander showed that this action is ergodic. We consider the case $G=PSL(2,C)$, where the variety of (conjugacy classes of) representations has a natural invariant decomposition, up to sets of measure $0$, into discrete and dense representations. This turns out NOT to be the relevant decomposition for the dynamics of the outer automorphism group. Instead we describe a set called the "primitive-stable" representations containing discrete as well as dense representations, onwhich the action is properly discontinuous.

##### A quadratic bound on the number of boundary slopes of essential surfaces

##### Computational geometry of moduli spaces of curves

A fast algorithm for computing intersection numbers of $\psi$- and $\kappa$-classes on moduli spaces of complex algebraic curves is proposed. As a consequence, the exact large genus asymptotics of these numbers (in particular, Weil-Petersson volumes) is numerically derived.

##### Two generator subgroups of the pure braid group

A group satisfies the "Tits alternative" if every subgroup is either virtually solvable or contains a nonabelian free group. This is named after J. Tits who proved that all finitely generated linear groups enjoy this property. The Tits alternative was established for braid groups by Ivanov and McCarthy, but now also follows from linearity (due to Bigelow-Krammer). I'll discuss joint work with D. Margalit, in which we prove a strong version of the Tits alternative for the pure braid groups: every two elements of the pure braid group either commute or generate a free group. The proof uses 3-manifold topology and actions on trees.

##### Topologically invariant Chern numbers of projective varieties

In 1954 Hirzebruch asked which linear combinations of Chern numbers are topological invariants of smooth complex projective varieties. Until recently, this problem was wide open, with few non-trivial results. We give a complete solution in arbitrary dimensions. An interesting feature of this solution is how it is derived from the well known case of complex dimension two, which at first sight looks rather special and exceptional.

##### Subgroup classification in $Out(F_n)$

We prove that for every subgroup $G$ of $Out(F_n)$, one of two alternatives holds: either there is a finite index subgroup $H<G$ and a nontrivial proper free factor $A$ of $F_n$ such that each element of $H$ fixes the conjugacy class of $A$; or there is an element $g\in G$ such that no nontrivial power of $g$ fixes the conjugacy class of any nontrivial proper free factor of $F_n$. This theorem is an analogue of Ivanov's classification of subgroups of surface mapping class groups. It has application to bounded 2nd cohomology of $Out(F_n)$, by combining with results of Bestvina-Feighn and of Hamenstadt. This work is joint with Michael Handel.

##### Congruence subgroup problem for mapping class groups

I will discuss the congruence subgroup problem for mapping class groups, a problem that generalizes the classical one for arithmetic groups. I will discuss an unpublished proof by W. Thurston for an affirmative answer to this problem for genus zero mapping class groups. Time permitting, I will discuss the current state of this problem.

##### Recurrence of random paths and counting closed geodesics in strata

We discuss the problem of counting closed geodesics in a stratum of the moduli space of Abelian(quadratic) differentials. This is joint work with Alex Eskin and Kasra Rafi.

##### Bordered Floer homology: bimodules and computations

We will review the structure of bordered Floer homology, including how it depends on the parametrization of the boundary. We will then discuss how to compute it, and consequently another algorithm for computing HF-hat. This is work in progress with Peter Ozsvath and Dylan Thurston.

##### Existence and rigidity of pseudo-Anosov flows transverse to R-covered foliations

Pseudo-Anosov flows are extremely common in three manifolds and they are very useful. How many pseudo-Anosov flows are there in a manifold up to topological conjugacy? We analyse this question in the context of flows transverse to a given foliation F. We prove that if F is R-covered (leaf space in the universal cover is the real numbers) then there are at most two pseudo-Anosov flows transverse to F. In addition if there are two, then the manifold is hyperbolic and the the foliation F blows down to a foliation topologically conjugate to the stable foliation of a particular type of an Anosov flow. The results use the topological theory of pseudo-Anosov flows, the universal circle for foliations and the geometric theory of R-covered foliations. We also discuss the existence of transverse pseudo-Anosov flows in this setting.

##### Heegaard Floer homology and pants decompositions

##### Annulus open book decompositions and the self linking number

We introduce a construction of an immersed surface for a null-homologous braid in an annulus open book decomposition. This is hinted by the so called Bennequin surface for a braid in $R3$. By resolving the singularities of the immersed surface, we obtain an embedded Seifert surface for the braid. Then we compute a self-linking number formula using this embedded surface and observe that the Bennequin inequality is satisfied if and only the contact structure is tight. We also prove that our self-linking formula is invariant (changes by 2) under a positive (negative) braid stabilization which preserves (changes) the transverse knot class.

##### On the geometry of space-time

In the relativistic point of view, the geometry of space should evolve with time, in a manner directed by Einstein equations. I will briefly summarize two interesting aspects with open questions: