All talks are in the Forestry Auditorium, which is on the ground floor of the building Forest Resources 2

The problem session was moderated by Etnyre, Kronheimer, Ozsvath, and Schleimer.

Ian Agol, The virtual fibering conjecture and related questions
    Thurston asked a bold question of whether finite volume hyperbolic 3-manifolds might always admit a finite-sheeted cover which fibers over the circle. This talk will review some of the progress on this question, and discuss its relation to other questions including residual finiteness and subgroup separability, the behavior of Heegaard genus in finite-sheeted covers, CAT(0) cubings, the RFRS condition, and subgroups of right-angled Artin groups. In particular, hyperbolic Haken 3-manifolds with LERF fundamental group are virtually fibered. Some applications of the techniques will also be mentioned.

Jorgen Andersen, The Hitchin connection, Toeplitz operators and TQFT
    The Witten-Reshetikhin-Turaev Topological Quantum Field Theory in particular provides us with the so call quantum representations of mapping class groups. The geometric construction of these involves geometric quantization of moduli spaces, which produced a holomorphic vector bundle over Teichmüller space. This bundle supports a projectively flat connection constructed by algebraic geometric techniques by Hitchin. We will present a differential geometric construction of this connection in a generalized setting. Furthermore its relation to Toeplitz operators will be discussed. In fact we will see that the parallel transport of this connection is a Toeplitz operator, hence manifestly placing Toeplitz operators within the study of TQFT's. We will further give applications of this, like the asymptotic faithfulness of these quantum representations and asymptotic expansions of the quantum invariants. Finally we will also discuss the their application in our proof that the mapping class groups do not have Kazhdan's property T.

Denis Auroux, Fukaya categories of symmetric products and bordered Heegaard-Floer homology
    The goal of this talk will be to outline an interpretation of the work of Lipshitz-Ozsvath-Thurston on Heegaard-Floer homology for 3-manifolds with parametrized boundary in terms of the symplectic topology of symmetric products. More specifically, we will explain how to understand the algebra A(F) associated to a surface in terms of the (relative) Fukaya category of the symmetric product.

Mladen Bestvina, Recent developments in the study of Out(F_n)
    I will survey our knowledge of the topology and geometry of Out(F_n) and Outer space, particularly as they compare with mapping class groups and Teichmuller space. In the 20+ years since the definition of Outer space by Culler and Vogtmann we have succeeded (more or less) to understand the topology, and the serious work on the geometry is in its early stages.

Michel Boileau, Commensurability classes of hyperbolic knot complements without hidden symmetries
    In this talk I will report on a joint work with Steve Boyer and Geneviève Walsh. One of the main results is that the commensurability class of the complement of a hyperbolic knot without hidden symmetries contains at most two other knot complements. Moreover if the knot is periodic (i.e. there is an axis of symmetry disjoint from the knot) we can characterize when its complement is commensurable to another knot complement.

Danny Calegari, Scl, sails and surgery
    Given a group $G$ and an element $g \in [G,G]$, the {\em commutator length} of $g$, denoted $\cl(g)$, is the smallest number of commutators in $G$ whose product is $g$, and the {\em stable commutator length} of $g$ is the limit $scl(g):=\lim_{n to \infty} \cl(g^n)/n$. Commutator length in a group extends in a natural way to a pseudo-norm on the real vector space of $1$-boundaries (in group homology), and should be thought of as a kind of relative Gromov-Thurston norm. We show that the problem of computing stable commutator length in free products of abelian groups reduces to a (finite dimensional) integer programming problem. Moreover, certain families of elements in such groups (i.e. those obtained by {\em surgery} on some element in a bigger group) give rise to families of integer programming problems that are related in explicit ways. In particular one can use this to establish the existence of limit points in the range of scl in such groups, and produce elements whose stable commutator length is congruent to any rational number modulo the integers. This technology relates stable commutator length to the theory of multi-dimensional continued fractions, and Klein polyhedra, and suggests an interesting conjectural picture of scl in free groups.

Tobias Colding, Singularities of mean curvature flow
    I will discuss work on progress with Bill Minicozzi on singularities of mean curvature flow. 

Yasha Eliashberg, Computing symplectic invariants of Stein manifolds
    Two tightly related invariants of a Stein domain X of complex dimension n are its symplectic homology and linearized contact homology of its boundary. In the talk I'll explain how to compute both in terms of a symplectic handle decomposition of X, or its presentation as a symplectic Lefschetz pencil. The answer is related to Seidel's conjecture, and in the case n=2 gives a combinatorial description of the corresponding complex.
   This is a joint work with F. Bourgeois and T. Ekholm.

Ron Fintushel, Simply connected 4-manifolds with b^+ = 1
    I will discuss constructions of manifolds in this class via surgery on nullhomologous tori in standard manifolds (Santeria surgery).

Michael Freedman, The topological approach to quantum computation: an update
    In August 2000 I spoke at the AMS millennium conference about the possibility of building a quantum computer by "measuring" rather than computing the Jones polynomial. This talk describes where we are 8 1/2 years later. The leading character is still the two dimensional electron gas (2DEG) known from the Fractional Quantum Hall Effect (FQHE), but theory and experiment and have interacted. We now envision a design which does not involve the space-time braiding of quasi-particle excitations but relies instead on adaptive interferometry. I will show some data.

Stefan Friedl, Fibered 3-manifolds and symplectic 4-manifolds
    Let N be a closed 3-manifold. In 1976 Thurston showed that if N fibers over S^1, then S^1 x N is symplectic. We will show that the converse holds, i.e. if S^1 x N is symplectic, then N fibers over S^1. This is joint work with Stefano Vidussi.

Dave Gabai, Ultra large hyperbolic 3-manifolds
    

Ursula Hamenstaedt, Geometric properties of the mapping class group
    We give an overview of geometric properties of the mapping class group of a surface of finite type which can be deduced from train tracks. Special emphasis is the existence of a quasi-convex bicombing.

Shelly Harvey, Torsion in the knot concordance group
    For each sequence of polynomials P={p1(t), ... }, we define a characteristic series of groups, called the derived series localized at P. Given a knot K, such a sequence of polynomials arises naturally as the sequence of orders of the higher-order Alexander modules of K. These new series yield filtrations of the smooth knot concordance group that refine the (n)-solvable filtration. We show that each of the successive quotients of this refined filtration contains 2-torsion and elements of infinite order. More specifically, each contains a $Z^\infty$ and a ${Z/2}^\infty$. These results generalize the p(t)-primary decomposition of the algebraic knot concordance group due to Milnor, Kervaire and Levine. This is joint work with Tim Cochran and Constance Leidy.

Michael Hill, On the Non-Existence of Kervaire Invariant One Manifolds
    I will describe joint work with Hopkins and Ravenel in which we show that there are no smooth manifolds of Kervaire invariant one in dimenions greater than 126. With the exception of dimension 126, this settles the question of when there exist smooth Kervaire invariant one manifolds.

Michael Hutchings, Some extensions of the Weinstein conjecture in three dimensions
    We use the isomorphism between embedded contact homology and Seiberg-Witten Floer homology, recently established by Taubes, to obtain some improvements on the three-dimensional Weinstein conjecture. Namely, let Y be a closed oriented three-manifold with a contact form such that all Reeb orbits are nondegenerate. If Y is not a lens space, then there must be at least three Reeb orbits, and at least one non-elliptic Reeb orbit. Also, we extend the Weinstein conjecture from contact forms to stable Hamiltonian structures on three-manifolds that are not torus bundles over the circle. This is joint work with Cliff Taubes.

Bruce Kleiner, A new proof of Gromov's theorem on groups of polynomial growth
    In 1981 Gromov showed that any finitely generated group of polynomial growth contains a finite index nilpotent subgroup. This has a variety of applications, ranging from dynamics to probability theory. Gromov's proof was based in part on a beautiful rescaling argument, and the Montgomery-Zippin solution to Hilbert's fifth problem on topological groups.
   The purpose of the lecture is to describe a new, much shorter, proof of Gromov's theorem, based on harmonic maps instead of the Montgomery-Zippin theory. I will begin by reviewing the history of Gromov's theorem, and some of its applications.

Peter Kronheimer, The eigenspace decomposition of instanton homology
    Instanton homology of 3-manifolds, developed by Floer in the 1980s, can be revisited today with the benefit of the hindsight afforded by the more recently developed Seiberg-Witten and Heegaard Floer theories. What emerges is an interesting chain of ideas that relates representations of the fundamental group on the one hand to the Thurston norm on the other. Recent applications of instanton homology include a non-vanishing theorem for the Donaldson invariants of symplectic 4-manifolds, a question that had been open since the late 1980's following Donaldson's proof of the non-vanishing of his invariants for Kahler manifolds.

Mark Lackenby, Surface subgroups of arithmetic Kleinian groups
    It is a fundamental conjecture that every finitely generated Kleinian group is finite or virtually free or contains a surface subgroup. I will outline the proof of this conjecture for arithmetic Kleinian groups and, more generally, finitely generated Kleinian groups that contain a non-cyclic finite subgroup. A wide variety of techniques are required, including the theory of 3-orbifolds, the Golod-Shafarevich inequality, the solution to the geometrisation conjecture and the relationship between the spectrum of the Laplacian and the Cheeger constant of a Riemannian manifold.

Yanki Lekili, Heegaard Floer homology of broken fibrations
    We will outline a programme for identifying Perutz's Lagrangian matching invariants and Ozsvath-Szabo's Heegaard Floer invariants of three and four manifolds. In this talk, we will deal with purely Heegaard Floer theoretical side of this programme and describe an isomorphism of 3-manifold invariants for certain Spin^c structures where the groups involved can be formulated in the language of Heegaard Floer theory. As applications, we give new calculations of Heegaard Floer homology of certain classes of 3-manifolds, a characterization of Juhasz's sutured Floer homology and an outline of Floer's excision theorem in the context of Heegaard Floer homology.

Robert Lipshitz, An overview of bordered Floer theory
    Bordered Floer homology is an extension of Heegaard Floer theory to 3-manifolds with boundary. After reviewing some of the basics of Heegaard Floer theory we will outline the goals and structure of bordered Floer homology. We will then sketch a few details of the construction, in the context of some simple examples.

Dusa McDuff, Displacing Lagrangian toric fibers by probes
    Entov and Polterovich showed some time ago that the Clifford torus in the complex plane cannot be moved by a Hamiltonian isotopy to be disjoint from its original position. More generally, they showed that for many toric symplectic manifolds at least one of the Lagrangian orbits of the torus action cannot be so displaced. (These orbits are the fibers of the moment map, and so are often called fibers.)
   Recently, Fukaya, Oh, Ohta and Ono developed a version of Floer homology that is sufficiently powerful to detect the nondisplaceability of many of these fibers. In the talk I shall describe a geometric way to displace fibers using "probes". There are still many open questions here, since even in the four dimensional case there are fibers that cannot be displaced by probes but yet have vanishing Floer homology.

William Minicozzi, Mean curvature flow
    I will talk about recent joint work with Toby Colding on mean curvature flow of embedded hypersurfaces.

Yair Minsky, Hulls, hyperbolicity and rigidity in mapping class groups
    I will describe some tools for studying the coarse geometry of mapping class groups relying on the hyperbolicity of curve complexes. A particular example is a generalization of the convex hull of a finite set in a hyperbolic space. Applications include quasi-isometric rigidity and the rapid decay property. Joint work with J. Behrstock, B. Kleiner and L. Mosher.

Tom Mrowka, Instantons and the Alexander polynomial
    

Lenny Ng, Knots and the Symplectic Field Theory of cotangent bundles
    I will survey the current state of affairs in a program to extract topological information by studying the Symplectic Field Theory of cotangent bundles. This talk will focus on recent progress in understanding knot contact homology via constructions related to string topology. This is mainly joint work with Kai Cieliebak, Tobias Ekholm, and Janko Latschev.

Yi Ni, Some applications of Heegaard Floer homology to Dehn surgery
    In recent years, Heegaard Floer homology has becomes a very powerful tool in studying Dehn surgery. In this talk, we will discuss two kinds of such applications. One is to exploit the relationship between the sutured structure of the knot complement and longitudinal surgery, the other are some results about cosmetic surgeries.

Peter Ozsvath, Bimodules in bordered Floer homology
    I will focus on some calculations in bordered Floer homology. I will explain how to determine the bimodules for mapping class group, and how to use these to give a method for calculating Heegaard Floer homology HF-hat for three-manifolds. This is joint work with Robert Lipshitz and Dylan Thurston.

Alan Reid, Tau, not tau and the topology of hyperbolic 3-manifolds
    A finite volume Riemannian manifold M (or its fundamental group) is said to have Property \tau if there is a constant C>0 such that all finite sheeted covers of M have Cheeger constants bounded below by C. The Lubotzky-Sarnak Conjecture asserts that the fundamental group of a finite volume hyperbolic manifold does not have Property \tau. Property \tau, and in particular, the Lubotzky-Sarnak Conjecture have attracted a lot of attention recently because of the connections to the topology of finite sheeted covers of closed hyperbolic 3-manifolds. This talk will discuss these connections, together with work that connects this circle of ideas with the group theoretic property LERF.

Paul Seidel, Progress on exotic symplectic structures
    I will review recent progress made on construction of non-standard Liouville-type symplectic structures on various open manifolds, in particular work of McLean and Maydansky.

Dennis Sullivan, String Topology and Connections on Principal Bundles
    The free loop space is homotopy equivalent to the homotopy quotient of the based loop space acting on itself by conjugation.The free loop space also has a circle action not present in the based loop space. A principal G bundle with connection is tantamount to a homomorphism modulo conjugation of the based loop space into G.
   String topology began with the Goldman Lie bracket on linear combinations curves on surfaces which generalised the Poisson bracket of curve holonomy trace functions on representations of the surface group into G. The bracket was combined with a cobracket by Turaev to obtain a Lie bialgebra. This structure was extended by Moira Chas and the author (Abel Proceedings 2003) to the circle equivariant homology of the free loop space rel constant loops of any oriented manifold. More recently these operations were seen to be the genus zero part of a more elaborate structure with higher genus pieces (Current Developments in Mathematics 2005).This type of structure, called a quantum Lie bialgebra, was also recently defined on the contact homology of any contact manifold provided with a symplectic filling. For this structure one considers finite energy J holomorphic punctured curves in the contact manifold cross R and in the interior of the symplectic filling. (See Symplectic Field Theory in Geometry and Functional Analysis 2000 for the contact part.) Conjecturally, these two examples of quantum Lie bialgebras are related by considering the the unit cotangent sphere bundle filled in by the unit cotangent disc bundle (see Towards Relative Symplectic Field Theory by Cieliebak and Latschev).
   A closer connection between these various theories seems possible using so-called semi-infinite homology theories on the free loop space. This type of theory depends on properties emphasized in the thesis of Max Lipyanskiy (MIT 2008) of a putative action functional. One may speculate that invariants of curvature of certain connections over the manifold restricted to two discs in the manifold may define relevant actions on the component of the free loop space containing the constant loop.

Zoltan Szabo, Heegaard Floer homology and pants decompositions
    

Dylan Thurston, Pairing in bordered Heegaard Floer theory
    We introduce a technique of \emph{deforming the diagonal} for bordered Floer homology. This lets us prove the pairing theorem that reconstructs the Heegaard Floer homology for a closed 3-manifold from an invariant for two pieces. Further extensions of the theory give us actions of the mapping class group of a punctured surface on a suitable category.

Gang Tian, Hamiltonian Gromov-Witten invariants
    

Mike Usher, C^0 stability in Morse theory, Floer theory, and symplectic topology
    A wide variety of results in symplectic topology assert that some property which superficially appears to be sensitive to the derivative of a function is in fact robust under continuous perturbations. I will discuss how some old and new results in this spirit, such as lower bounds on the displacement energies of certain subsets of symplectic manifolds, can be accessed via a careful study of the filtration structure on Floer homology. On a more elementary level, the same viewpoint applied to Morse theory leads to a simple but surprising result about the zeros of closed one-forms.

Karen Vogtmann, Automorphism groups of right-angled Artin groups
    Right-angled Artin groups (RAAGs) are finitely-generated groups which are completely described by the fact that some of the generators commute; thus they interpolate between free groups at one extreme and free abelian groups at the other. RAAGs and their subgroups are the source of many important examples and counterexamples in geometric group theory.
   In this talk I will describe joint work with Ruth Charney on the outer automorphism groups of RAAGs, which can be though of analogously as interpolating between the outer automorphism groups of free groups and the general linear group GL(n,Z). We are particularly interested in determining which properties shared by Out(F_n) and GL(n,Z) are in fact true for all automorphism groups of RAAGs.

Kevin Walker, Blob homology and the contact category
     We define a chain complex B_*(C, M) (the "blob complex") associated to an n-category C and an n-manifold M. For n=1, B_*(C, S^1) is quasi-isomorphic to the Hochschild complex of the 1-category C. So in some sense blob homology is a generalization of Hochschild homology to (n>1)-categories. The degree zero homology of B_*(C, M) is isomorphic to the dual of the Hilbert space associated to M by the TQFT corresponding to C. So in another sense the blob complex is the derived category version of a TQFT. This part of the talk is joint work with Scott Morrison.
   In the second part of the talk we consider the case where C is the 3-category associated to tight contact structures on 3-manifolds. The simplest blob homology calculation in this case requires us to compute the Hochschild homology of the contact category associated to D^2 (with fixed boundary condition of k positive regions). It turns out that there is a triangle-preserving embedding of this category into the derived category of the lattice of ordered subsets of {1, ..., k-1} (ordered lexicographically, not by inclusion). This part of the talk is joint work with Alex Dugas.

Liam Watson, Surgery obstructions from Khovanov homology
    Given a manifold with torus boundary, together with an appropriate involution, we give obstructions to certain exceptional surgeries using Khovanov homology. In particular, obstructions to lens space surgeries and to finite fillings are obtained by studying the homological width of branch sets associated to surgeries on the given manifold.

Katrin Wehrheim, Topological invariants via decomposition and Lagrangian correspondences
    In joint work with Chris Woodward we provide a generally applicable blueprint for constructing topological invariants by following a 6 step program of decomposing and representing simple pieces in the symplectic category. I will explain the 6 steps using our example of Floer theoretic SU(N) - invariants for 3-manifolds Y and, depending on recent progress, generalize them to a blueprint for 4-manifold invariants using broken fibrations (similar to the work of Perutz).