# The Joint Los Angeles Topology Seminar

### 2019-2020

 UCLA MS 6627 2020-04-27 16:00-18:00 Matthew Hedden (Michigan State University) Title TBD Abstract TBD. Morgan Weiler (Rice University) Title TBD Abstract TBD. USC Room TBD 2020-03-30 Time TBD Andrew Hanlon (Stony Brook) Title TBD Abstract TBD. Paul Wedrich (Max Planck/Bonn/MSRI) Title TBD Abstract TBD. Caltech Linde 310 2020-03-09 16:00-18:00 Yongbin Ruan (Zhejiang University) BCOV axioms of Gromov-Witten theory of Calabi-Yau 3-fold One of biggest and most difficult problems in the subject of Gromov-Witten theory is to compute higher genus Gromov-Witten invariants of compact Calabi-Yau 3-fold such as the quintic 3-folds. There have been a collection of remarkable axioms/conjectures from physics (BCOV B-model) regarding the universal structure or axioms of higher genus Gromov-Witten theory of Calabi-Yau 3-folds. In the talk, I will first explain 4 BCOV axioms explicitly for the quintic 3-folds. Then, I will outline a solution for 3+1/2 of them. Josh Greene (Boston College) On loops intersecting at most once How many simple closed curves can you draw on the closed surface of genus g in such a way that no two are isotopic and no two intersect in more than k points? It is known how to draw a collection in which the number of curves grows as a polynomial in g of degree k + 1, and conjecturally, this is the best possible. I will describe a proof of an upper bound that matches this function up to a factor of log(g). It involves hyperbolic geometry, covering spaces, and probabilistic combinatorics. UCLA Geology 3656 2019-11-04 16:30-18:30 Nate Bottman (USC) Functoriality for the Fukaya category and a compactified moduli space of pointed vertical lines in C^2 A Lagrangian correspondence between symplectic manifolds induces a functor between their respective Fukaya categories. I will begin by introducing this construction, along with a family of abstract polytopes called 2-associahedra (introduced in math/1709.00119), which control the coherences among this collection of functors. Next, I will describe new joint work with Alexei Oblomkov (math/1910.02037), in which we construct a compactification of the moduli space of configurations of pointed vertical lines in $\mathbb{C}^2$ modulo affine transformations $(x,y) \mapsto (ax+b,ay+c)$. These spaces are proper complex varieties with toric lci singularities, which are equipped with forgetful maps to $\overline{M}_{0,r}$. Our work yields a smooth structure on the 2-associahedra, thus completing one of the last remaining steps toward a complete functoriality structure for the Fukaya category. Peter Smillie (Caltech) Hyperbolic planes in Minkowski 3-space Can you parametrize the space of isometric embeddings of the hyperbolic plane into Minkowski 3-space? I'll give a partial result and conjectural answer, in terms of, equivalently, domains of dependence, measured laminations, or lower semicontinuous functions on the circle. Using the Gauss map and its inverse, I'll then interpret this result in terms of harmonic maps to the hyperbolic plane. Finally, I'll restrict to the case where the isometric embedding is invariant under a group action, and describe connections to Teichmuller space. This is all joint work with Francesco Bonsante and Andrea Seppi.

### 2018-2019

 Caltech Linde 310 2019-04-29 16:00-18:00 Claudius Zibrowius (University of British Columbia) Khovanov homology and the Fukaya category of the 3-punctured disc This talk will focus on a classification result for complexes over a certain quiver algebra and its consequences for Khovanov homology of 4-ended tangles. In particular, I will introduce a family of immersed curve invariants for pointed 4-ended tangles, whose intersection theory computes reduced Khovanov homology. This is joint work in progress with Artem Kotelskiy and Liam Watson, which was inspired by recent work of Matthew Hedden, Christopher Herald, Matthew Hogancamp and Paul Kirk. Nathan Dowlin (Dartmouth) A spectral sequence from Khovanov homology to knot Floer homology Khovanov homology and knot Floer homology are two knot invariants which are defined using very different techniques, with Khovanov homology having its roots in representation theory and knot Floer homology in symplectic geometry. However, they seem to contain a lot of the same topological data about knots. Rasmussen conjectured that this similarity stems from a spectral sequence from Khovanov homology to knot Floer homology. In this talk I will give a construction of this spectral sequence. The construction utilizes a recently defined knot homology theory HFK_2 which provides a framework in which the two theories can be related. USC KAP 414 2019-04-22 16:30-18:30 David Ayala (Montana State U.) Factorization homology: sigma-models as state-sum TQFTs Roughly, factorization homology pairs an n-category and an n-manifold to produce a chain complex. Factorization homology is to state-sum TQFTs as singular homology is to simplicial homology: the former is manifestly well-defined (i.e. independent of auxiliary choices), continuous (i.e. carries a continuous action of diffeomorphisms), and functorial; the latter is easier to compute. Examples of n-categories to input into this pairing arise, through deformation theory, from perturbative sigma-models. For such n-categories, this state sum expression agrees with the observables of the sigma-model this is a form of Poincar duality, which yields some surprising dualities among TQFTs. A host of familiar TQFTs are instances of factorization homology; many others are speculatively so. The first part of this talk will tour through some essential definitions in whats described above. The second part of the talk will focus on familiar instances of factorization homology, highlighting the Poincare/Koszul duality result. The last part of the talk will speculate on more such instances. Francisco Arana Herrera (Stanford) Counting square-tiled surfaces with prescribed real and imaginary foliations Let X be a closed, connected, hyperbolic surface of genus 2. Is it more likely for a simple closed geodesic on X to be separating or non-separating? How much more likely? In her thesis, Mirzakhani gave very precise answers to these questions. One can ask analogous questions for square-tiled surfaces of genus 2 with one horizontal cylinder. Is it more likely for such a square-tiled surface to have separating or non-separating horizontal core curve? How much more likely? Recently, Delecroix, Goujard, Zograf, and Zorich gave very precise answers to these questions. Surprisingly enough, their answers were exactly the same as the ones in Mirzakhanis work. In this talk we explore the connections between these counting problems, showing they are related by more than just an accidental coincidence. UCLA MS 6221 2019-04-15 16:00-18:00 Peter Lambert-Cole (Georgia Tech) Bridge trisections and the Thom conjecture The classical degree-genus formula computes the genus of a nonsingular algebraic curve in the complex projective plane. The well-known Thom conjecture posits that this is a lower bound on the genus of smoothly embedded, oriented and connected surface in CP2. The conjecture was first proved twenty-five years ago by Kronheimer and Mrowka, using Seiberg-Witten invariants. In this talk, we will describe a new proof of the conjecture that combines contact geometry with the novel theory of bridge trisections of knotted surfaces. Notably, the proof completely avoids any gauge theory or pseudoholomorphic curve techniques. James Conway (UC Berkeley) Classifying contact structures on hyperbolic 3-manifolds Two of the most basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. In dimension 3, these questions have been answered for large classes of manifolds, but with a notable absence of hyperbolic manifolds. In this talk, we will see a new classification of contact structures on an family of hyperbolic 3-manifolds arising from Dehn surgery on the figure-eight knot, and see how it suggests some structural results about tight contact structures. This is joint work with Hyunki Min. USC KAP 2018-11-12 16:30-18:30 Peter Samuelson (UC Riverside) The Hall algebra of the Fukaya category of a surface The Hall algebra of an abelian (or triangulated) category has a basis given by isomorphism classes of objects, and the product "counts extensions" ("counts distinguished triangles"). This construction has been important in representation theory, e.g. it gives a conceptual construction of quantum groups. We will discuss a conjectural description of the Hall algebra of the Fukaya category of a surface (using the version defined by Haiden, Katzarkov, and Kontsevich). We also discuss a connection to the skein algebra of the surface. (This is joint work with B. Cooper.) Sherry Gong (UCLA) egarding the computation of singular instanton homology for links We discuss some computations arising from the spectral sequence constructed by Kronheimer and Mrowka relating the Khovanov homology of a link to its singular instanton homology. Caltech Linde 310 2018-11-05 16:00-18:00 Chris Gerig (Harvard) SW=Gr Whenever the Seiberg-Witten (SW) invariants of a 4-manifold X are defined, there exist certain 2-forms on X which are symplectic away from some circles. When there are no circles, i.e. X is symplectic, Taubes' SW=Gr'' theorem asserts that the SW invariants are equal to well-defined counts of J-holomorphic curves (Taubes' Gromov invariants). In this talk I will describe an extension of Taubes' theorem to non-symplectic X: there are well-defined counts of J-holomorphic curves in the complement of these circles, which recover the SW invariants. This Gromov invariant'' interpretation was originally conjectured by Taubes in 1995. Biji Wong (CIRGET Montreal) A Floer homology invariant for 3-orbifolds via bordered Floer theory Using bordered Floer theory, we construct an invariant for 3-orbifolds with singular set a knot that generalizes the hat flavor of Heegaard Floer homology. We show that for a large class of 3-orbifolds the orbifold invariant behaves like HF-hat in that the orbifold invariant, together with a relative Z_2-grading, categorifies the order of H_1^orb. When the 3-orbifold arises as Dehn surgery on an integer-framed knot in S^3, we use the {-1,0,1}-valued knot invariant epsilon to determine the relationship between the orbifold invariant and HF-hat of the 3-manifold underlying the 3-orbifold. UCLA MS 6627 2018-10-15 16:00-18:00 Lei Chen (Caltech) Section problems In this talk, I will discuss a direction of study in topology: Section problems. There are many variations of the problem: Nielsen realization problems, sections of a surface bundle, sections of a bundle with special property (e.g. nowhere zero vector field). I will discuss some techniques including homology, Thurston-Nielsen classification and dynamics. Also I will share many open problems. Some of the results are joint work with Nick Salter. Lisa Piccirillo (UT Austin) The Conway knot is not slice Surgery-theoretic classifications fail for 4-manifolds because many 4-manifolds have second homology classes not representable by smoothly embedded spheres. Knot traces are the prototypical example of 4-manifolds with such classes. Ill give a flexible technique for constructing pairs of distinct knots with diffeomorphic traces. Using this construction, I will show that there are knot traces where the minimal genus smooth surface generating second homology is not the obvious one, resolving question 1.41 on the 1978 Kirby problem list. I will also use this construction to show that Conway knot does not bound a smooth disk in the four ball, which completes the classification of slice knots under 13 crossings and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.

### 2017-2018

 UCLA MS 6627 2018-04-23 16:00-18:00 Allison Moore (UC Davis) Distance one lens space fillings and band surgery Band surgery is an operation that transforms a link into a new link. When the operation is compatible with orientations on the links involved, it is called coherent band surgery, otherwise it is called non-coherent. We will look at the behavior of the signature of a knot under non-coherent band surgery, and also classify all band surgery operations from the trefoil knot to the $T(2, n)$ torus knots and links. This classification is by way of a related three-manifold problem that we solve by studying the Heegaard Floer d-invariants under integral surgery along knots in the lens space $L(3,1)$. If time permits, I will mention some motivation for the the study of band surgery on knots from a DNA topology perspective. Parts of this project are joint work with Lidman and Vazquez. Danny Ruberman (Brandeis) Seiberg-Witten invariants of 4-dimensional homology circles Most applications of gauge theory in 4-dimensional topology are concerned with simply-connected manifolds with non-trivial second homology. I will discuss the opposite situation, first describing a Seiberg-Witten invariant for manifolds with first homology = Z and vanishing second homology; this invariant has an unusual index-theoretic correction term. I will discuss recent work with Jianfeng Lin and Nikolai Saveliev giving a new formula for this invariant in terms of monopole homology, and some calculations and applications. Caltech E-Bridge 201 2018-04-02 16:00-18:00 Yongbin Ruan (University of Michigan)) The structure of higher genus Gromov-Witten invariants of quintic 3-fold The computation of higher genus Gromov-Witten invariants of quintic 3--fold (or compact Calabi-Yau manifold in general) has been a focal point of research of geometry and physics for more than twenty years. A series of deep conjectures have been proposed via mirror symmetry for the specific solutions as well as structures of its generating functions. Building on our initial success for a proof of genus two conjecture formula of BCOV, we present a proof of two conjectures regarding the structure of the theory. The first one is Yamaguchi-Yau's conjecture that its generating function is a polynomial of five generators and the other one is the famous holomorphic anomaly equation which governs the dependence on four out of five generators. This is a joint work with Shuai Guo and Felix Janda. Li-Sheng Tseng (UC Irvine) Symplectic geometry as topology of odd sphere bundles We will motivate the consideration of odd-dimensional sphere bundles over symplectic manifolds where the Euler class of the fiber bundles is given by powers of the symplectic structure. The topological invariants of these odd sphere bundles are directly related to the symplectic invariants of the base manifold. We will describe how we can use such a relation to reinterpret symplectic invariants as topological invariants of the higher dimensional odd sphere bundles, and also, how topological methods to study the odd sphere bundles can point to new methods to study symplectic geometry. This talk is based on a joint work with Hiro Tanaka. Caltech E-Bridge 201 2017-12-04 16:00-18:00 Zhouli Xu (MIT) Smooth structures, stable homotopy groups of spheres and motivic homotopy theory Following Kervaire-Milnor, Browder and Hill-Hopkins-Ravenel, Guozhen Wang and I showed that the 61-sphere has a unique smooth structure and is the last odd dimensional case: $S^1, S^3, S^5$ and $S^{61}$ are the only odd dimensional spheres with a unique smooth structure. The proof is a computation of stable homotopy groups of spheres. We introduce a method that computes differentials in the Adams spectral sequence by comparing with differentials in the Atiyah-Hirzebruch spectral sequence for real projective spectra through Kahn-Priddy theorem. I will also discuss recent progress of computing stable stems using motivic homotopy theory with Dan Isaksen and Guozhen Wang. Raphael Zentner (University of Regensburg) Irreducible SL(2,C)-representations of integer homology 3-spheres We prove that the splicing of any two non-trivial knots in the 3-sphere admits an irreducible SU(2)-representation of its fundamental group. This uses instanton gauge theory, and in particular a non-vanishing result of Kronheimer-Mrowka and some new results that we establish for holonomy perturbations of the ASD equation. Using a result of Boileau, Rubinstein and Wang (which builds on the geometrization theorem of 3-manifolds), it follows that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C). USC KAP 414 2017-11-20 15:45-18:00 Daniel Alvarez-Gavela (Stanford) The simplification of singularities of Lagrangian and Legendrian fronts The envelope of light rays reflected or refracted by a curved surface is called a caustic and generically has semi-cubical cusp singularities at isolated points. In generic families depending on one real parameter the cusps of the caustic will be born or die in pairs. At such an instance of birth/death the caustic traces a swallowtail singularity. This bifurcation is also known as the Legendrian Reidemeister I move. For families depending on more parameters or for front projections of higher dimensional Legendrians (or Lagrangians), the generic caustic singularities become more complicated. As the dimension increases the situation quickly becomes intractable and there is no explicit understanding or classification possible in the general case. In this lecture we will present a full h-principle (C^0-close, relative, parametric) for the simplification of higher singularities of caustics into superpostions of the familiar semi-cubical cusp. As a corollary we will obtain a Reidemeister type theorem for families of Legendrian knots in the standard contact Euclidean 3-space which depend on an arbitrary number of parameters. We will also explain the relation to Nadler's program for the arborealization of singularities of Lagrangian skeleta and give several other potential applications of the h-principle to symplectic and contact topology. Ciprian Manolescu (UCLA) A sheaf-theoretic model for SL(2,C) Floer homology I will explain the construction of a new homology theory for three-manifolds, defined using perverse sheaves on the SL(2,C) character variety. Our invariant is a model for an SL(2,C) version of Floers instanton homology. I will present a few explicit computations for Brieskorn spheres, and discuss the connection to the Kapustin-Witten equations and Khovanov homology. This is joint work with Mohammed Abouzaid. UCLA MS 6627 2017-11-06 16:00-18:00 Sheel Ganatra (USC) Liouville sectors and localizing Fukaya categories We introduce a new class of Liouville manifolds-with-boundary, called Liouville sectors, and show they have well-behaved, covariantly functorial Fukaya/Floer theories. Stein manifolds frequently admit coverings by Liouville sectors, which can then be used to study the Fukaya category of the total space. Our first main result in this setup is a local criterion for generating (global) Fukaya categories. One of our goals, using this framework, is to obtain a combinatorial presentation of the Fukaya category of any Stein manifold. This is joint work with John Pardon and Vivek Shende. Nathan Dunfield (UIUC) An SL(2, R) Casson-Lin invariant and applications When M is the exterior of a knot K in the 3-sphere, Lin showed that the signature of K can be viewed as a Casson-style signed count of the SU(2) representations of pi_1(M) where the meridian has trace 0. This was later generalized to the fact that signature function of K on the unit circle counts SU(2) representations as a function of the trace of the meridan. I will define the SL(2, R) analog of these Casson-Lin invariants, and explain how it interacts with the original SU(2) version via a new kind of smooth resolution of the real points of certain SL(2, C) character varieties in which both kinds of representations live. I will use the new invariant to study left-orderability of Dehn fillings on M using the translation extension locus I introduced with Marc Culler, and also give a new proof of a recent theorem of Gordon's on parabolic SL(2, R) representations of two-bridge knot groups. This is joint work with Jake Rasmussen (Cambridge).

### 2016-2017

 Caltech Sloan 151 2017-04-17 16:00-18:00 Steven Frankel (Yale University) Calegari's conjecture for quasigeodesic flows We will discuss two kinds of flows on 3-manifolds: quasigeodesic and pseudo-Anosov. Quasigeodesic flows are defined by a tangent condition, that each flowline is coarsely comparable to a geodesic. In contrast, pseudo-Anosov flows are defined by a transverse condition, where the flow contracts and expands the manifold in different directions. When the ambient manifold is hyperbolic, there is a surprising relationship between these apparently disparate classes of flows. We will show that a quasigeodesic flow on a closed hyperbolic 3-manifold has a coarsely contracting-expanding transverse structure, a generalization of the strict transverse contraction-expansion of a pseudo-Anosov flow. This behavior can be seen "at infinity," in terms of a pair of laminar decompositions of a circle, which we use to proof Calegari's conjecture: every quasigeodesic flow on a closed hyperbolic 3-manifold can be deformed into a pseudo-Anosov flow. Duncan McCoy (UT Austin) Characterizing slopes for torus knots We say that p/q is a characterizing slope for a knot K in the 3-sphere if the oriented homeomorphism type of p/q-surgery is sufficient to determine the knot K uniquely. I will discuss the problem of determining which slopes are characterizing for torus knots, paying particular attention to non-integer slopes. This problem is related to the question of which knots in the 3-sphere have Seifert fibered surgeries. USC KAP 245 2017-04-10 16:30-18:30 Julien Paupert (Arizona State) Rank 1 deformations of non-cocompact hyperbolic lattices Let X be a negatively curved symmetric space and Gamma a noncocompact lattice in Isom(X). We show that small, parabolic-preserving deformations of Gamma into the isometry group of any negatively curved symmetric space containing X remain discrete and faithful (the cocompact case is due to Guichard). This applies in particular to a version of Johnson-Millson bending deformations, providing for all n infnitely many noncocompact lattices in SO(n,1) which admit discrete and faithful deformations into SU(n,1). We also produce deformations of the figure-8 knot group into SU(3,1), not of bending type, to which the result applies.This is joint work with Sam Ballas and Pierre Will. Oleg Lazarev (Stanford University) Contact manifolds with flexible fillings In this talk, I will show that all flexible Weinstein fillings of a given contact manifold have isomorphic integral cohomology. As an application, in dimension at least 5 any almost contact class that has an almost Weinstein filling has infinitely many exotic contact structures. Using similar methods, I will also construct the first known infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic. These results are proven by studying Reeb chords of loose Legendrians and positive symplectic homology. UCLA MS 6627 2017-03-13 16:00-18:00 Mark Hughes (Brigham Young University) Neural networks and knot theory In recent years neural networks have received a great deal of attention due to their remarkable ability to detect subtle and very complex patterns in large data sets. They have become an important machine learning tool and have been used extensively in many fields, including computer vision, fraud detection, artificial intelligence, and financial modeling. Knots in 3-space and their associated invariants provide a rich data set (with many unanswered questions) on which to apply these techniques. In this talk I will describe neural networks, and outline how they can be applied to the study of knots in 3-space. Indeed, these networks can be applied to answer a number of algebraic and geometric problems involving knots and their invariants. I will also outline how neural networks can be used together with techniques from reinforcement learning to construct explicit examples of slice and ribbon surfaces for certain knots. John Etnyre (Georgia Tech) Embeddings of contact manifolds I will discuss recent results concerning embeddings and isotopies of one contact manifold into another. Such embeddings should be thought of as generalizations of transverse knots in 3-dimensional contact manifolds (where they have been instrumental in the development of our understanding of contact geometry). I will mainly focus on embeddings of contact 3-manifolds into contact 5-manifolds. In this talk I will discuss joint work with Ryo Furukawa aimed at using braiding techniques to study contact embeddings. Braided embeddings give an explicit way to represent some (maybe all) smooth embeddings and should be useful in computing various invariants. If time permits I will also discuss other methods for embedding and constructions one may perform on contact submanifolds. UCLA MS 5127 2016-11-07 16:00-18:00 Burak Ozbagci (Koc University) Fillings of unit cotangent bundles of nonorientable surfaces We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism. (This is a joint work with Youlin Li.) Matt Hogancamp (USC) Categorical diagonalization and link homology I will discuss joint work with Ben Elias in which we introduce the notion of a diagonalizable functor and give a categorical analogue of the usual minimal polynomial condition for diagonalizability. As our main application we prove that the Rouquier complex associated to the full-twist braid acts diagonalizably on the category of Soergel bimodues. This has important consequences for the triply graded Khovanov-Rozansky link homology, which I will explain. I will conclude by discussing connections with some recent, very exciting work of Gorsky-Negut-Rasmussen, which suggests that categorical diagonalization is the key to understanding a deep (conjectural) connection between Khovanov-Rozansky homology and Hilbert schemes. USC KAP 245 2016-10-31 16:30-18:30 Tian Yang (Stanford University) Volume conjectures for Reshetikhin-Turaev and Turaev-Viro invariants In a joint work with Qingtao Chen we conjecture that, at the root of unity exp(2πi/r) instead of the root exp(πi/r) usually considered, the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic 3-manifold grow exponentially, with growth rates respectively connected to the hyperbolic and complex volume of the manifold. This reveals an asymptotic behavior of the relevant quantum invariants that is different from that of Witten's invariants (which grow polynomially by the Asymptotic Expansion Conjecture), and may indicate a geometric interpretation of the Reshetikhin-Turaev invariants that is different the SU(2) Chern-Simons gauge theory. Recent progress toward these conjectures will be summarized, including joint work with Renaud Detcherry and Effie Kalfagianni. Kasra Rafi (University of Toronto and MSRI) Caltech Sloan 151 2016-10-17 16:00-18:00 Hongbin Sun (UC Berkeley) NonLERFness of arithmetic hyperbolic manifold groups We will show that, for "almost" all arithmetic hyperbolic manifolds with dimension >3, their fundamental groups are not LERF. The main ingredient in the proof is a study of certain graph of groups with hyperbolic 3-manifold groups being the vertex groups. We will also show that a compact irreducible 3-manifold with empty or tori boundary does not support a geometric structure if and only if its fundamental group is not LERF. Sucharit Sarkar (UCLA) Equivariant Floer homology Given a Lie group G acting on a symplectic manifold preserving a pair of Lagrangians setwise, I will describe a construction of G-equivariant Lagrangian Floer homology. This does not require G-equivariant transversality, which allows the construction to be flexible. Time permitting, I will talk about applying this for the O(2)-action on Seidel-Smith's symplectic Khovanov homology. This is joint with Kristen Hendricks and Robert Lipshitz.

### 2015-2016

 USC KAP 414 2016-03-21 16:30-18:30 Nicolas Tholozan (Univ. Luxembourg) Compact quotients of pseudo-Riemannian hyperbolic spaces A pseudo-Riemannian manifold is a manifold where each tangent space is endowed with a quadratic form that is non-degenerate, but not necessarily positive definite. A typical example is the hyperbolic space H(p,q), which is a pseudo-Riemannian manifold of signature (p,q) and constant negative sectional curvature. It is homogeneous, as it admits a transitive isometric action of the Lie group SO(p,q+1). A long standing question is to determine for which values of (p,q) one can find a discrete subgroup of SO(p,q+1) acting properly discontinuously and cocompactly on H(p,q). In this talk I will show that there is no such action when p is odd and q >0. The proof relies on a computation of the volume of the corresponding quotient manifold. The proof also implies that, when p is even, this volume is essentially rational. I will discuss in more details the case of H(2,1) (the 3-dimensional anti-de Sitter space), for which compact quotients exist and have been described by work of Kulkarni-Raymond and Kassel. Peter Samuelson (University of Iowa) The Homfly skein and elliptic Hall algebras The Homfly skein relations from knot theory can be used to associate an algebra to each (topological) surface. The Hall algebra construction associates an algebra to each smooth (algebraic) curve over a finite field. Using work of Burban and Schiffmann, we show that the skein algebra of the torus is isomorphic to the Hall algebra of an elliptic curve. If time permits we discuss a third (categorical) construction of the same algebra. (Joint with Morton and Licata.) UCLA MS 5127 2016-02-29 16:15-18:30 Eugene Gorsky (UC Davis) Heegaard Floer homology of some L-space links A link is called an L-space link if all sufficiently large surgeries along it are L-spaces. It is well known that the Heegaard Floer homology of L-space knots have rank 0 or 1 at each Alexander grading. However, for L-space links with many components the homology usually has bigger ranks and a rich structure. I will describe the homology for algebraic and cable links, following joint works with Jen Hom and Andras Nemethi. In particular, for algebraic links I will construct explicit topological spaces with homology isomorphic to link Floer homology. Sheel Ganatra (Stanford University) Automatically generating Fukaya categories and computing quantum cohomology Suppose one has determined the Floer theory algebra of a finite non-empty collection of Lagrangians in a Calabi-Yau manifold. I will explain that, if the resulting algebra satisfies a finiteness condition called homological smoothness, then the collection automatically split-generates the Fukaya category. In addition, the Hochschild invariants of the algebra (and hence of the whole Fukaya category) are automatically isomorphic to the quantum cohomology ring. This result immediately extends to the setting of monotone/non-Calabi-Yau symplectic manifolds, under an additional hypothesis on the rank of the algebra???s 0th Hochschild cohomology. The proofs make large use of joint work with Perutz and Sheridan, which in turn is part of a further story about recovering Gromov-Witten invariants from the Fukaya category. Caltech Sloan 153 2016-02-08 16:00-18:00 Anna Wienhard (University of Heidelberg) Maximal representations and projective structures on iterated sphere bundles The Toledo number is a numerical invariant associated to representations of fundamental groups of surfaces into Lie groups of Hermitian type. Maximal representations are those representations for which the Toledo number is maximal. They form connected components of the representation variety. In the case when the Lie group is SL(2,R)= Sp(2,R) they correspond precisely to holonomy representations of hyperbolic structures. Maximal representations into the symplectic group Sp(2n,R) generalize this situation with a lot of new features appearing. I will describe some of these new features and explain how maximal representations arise as homonym representations of projective structures on iterated sphere bundles over surfaces. Shicheng Wang (Peking University) Chern--Simons theory, surface separability, representation volumes, and dominations of 3-manifolds The talk will start with mapping degree sets and simplicial volumes. We then discuss recent results on virtual representation volumes and on virtual dominations of 3-manifolds, as well as their relations. Time permitted, we may end with the high dimensional applications of representation volumes. This is joint work with P. Derbez, Y. Liu and H. Sun. UCLA MS 6229 2015-11-30 16:00-18:00 Ailsa Keating (Columbia University) Higher-dimensional Dehn twists and symplectic mapping class groups Given a Lagrangian sphere S in a symplectic manifold M of any dimension, one can associate to it a symplectomorphism of M, the Dehn twist about S. This generalises the classical two-dimensional notion. These higher-dimensional Dehn twists naturally give elements of the symplectic mapping class group of M, i.e. $\pi_0 (Symp (M))$. The goal of the talk is to present parallels between properties of Dehn twists in dimension 2 and in higher dimensions, with an emphasis on relations in the mapping class group. Hiro Lee Tanaka (Harvard University) Factorization homology and topological field theories This is joint work with David Ayala and John Francis. Factorization homology is a way to construct invariants of manifolds out of some algebraic data. Examples so far include singular homology, intersection homology, Bartlett's spin net formalism for Turaev-Viro invariants, Reshetikhin-Turaev invariants for framed knots, and Salvatore's non-Abelian Poincare Duality. It has also been used by Ayala-Francis to prove the cobordism hypothesis. In this talk we'll give some basic examples and prove some classification results akin to Brown Representability. Caltech Sloan 151 2015-11-16 16:00-18:00 Mike Hill (UCLA) A higher-height lift of Rohlin's Theorem: on \eta^3 Rohlin's theorem on the signature of Spin 4-manifolds can be restated in terms of the connection between real and complex K-theory given by homotopy fixed points. This comes from a bordism result about Real manifolds versus unoriented manifolds, which in turn, comes from a C_2-equivariant story . I'll describe a surprising analogue of this for larger cyclic 2 groups, showing that the element eta cubed is never detected! In particular, for any bordism theory orienting these generalizations of Real manifolds, the three torus is always a boundary. Joshua Greene (Boston College) Definite surfaces and alternating links I will describe a characterization of alternating links in terms intrinsic to the link complement and derive some consequences of it, including new proofs of some of Tait's conjectures. USC KAP 245 2015-10-19 16:30-18:30 Jeff Danciger (UT Austin) Convex projective structures on non-hyperbolic three-manifolds We discuss a program underway to determine which closed three-manifolds admit convex real projective structures and its implications in the search for low-dimensional matrix representations of three-manifold groups. While every hyperbolic structure is a convex projective structure, examples of convex projective structures on non-hyperbolic three-manifolds were found only recently by Benoist. We produce a large source of new examples, including the doubles of many hyperbolic knot and link complements. The strategy is to suitably deform cusped hyperbolic three-manifolds and then (convexly) glue them together. Joint work with Sam Ballas and Gye-Seon Lee. Faramarz Vafaee (Caltech) L-spaces and rationally fibered knots The main focus of the talk will be on proving fiberedness results for knots in L-spaces with either L-space or S1 x S2 surgeries. Recall that an L-space is defined to be a rational homology three-sphere with the same Heegaard Floer homology as a lens space. We prove that knots in L-spaces with S1 x S2 surgeries are Floer simple and fibered. Moreover, the induced contact structure on the ambient manifold is tight. We also prove that a knot K in an L-space Y with a non-trivial L-space surgery is fibered provided that the orthogonal complement of K with respect to the linking form of Y vanishes. This generalizes the result of Boileau-Boyer-Cebanu-Walsh, in which they assume the knot is primitive. This work is joint with Yi Ni.

### 2014-2015

 UCLA MS 6627 2015-04-06 16:00-18:00 Steven Sivek (Princeton University) Augmentations of Legendrian knots and constructible sheaves Given a Legendrian knot in R^3, Shende-Treumann-Zaslow defined a category of constructible sheaves on the plane with singular support controlled by the front projection of the knot. They conjectured that this is equivalent to a category determined by the Legendrian contact homology of the knot, namely Bourgeois-Chantraine's augmentation category. Although this conjecture is false, it does hold if one replaces the augmentation category with a closely related variant. In this talk, I will describe this category and some of its properties and outline the proof of equivalence. This is joint work with Lenny Ng, Dan Rutherford, Vivek Shende, and Eric Zaslow. Hirofumi Sasahira (Nagoya University) Spin structures on Seiberg-Witten moduli spaces We will prove that under a certain condition the moduli space of solutions to the Seiberg-Witten equations on a 4-manifold has a canonical spin structure. The spin bordism class of the moduli space is a differential topological invariant of the 4-manifold. We will show that this invariant is nontrivial for the connected sum of some symplectic 4-manifolds. UCLA MS 5127 2014-11-17 16:00-18:00 David Rose (USC) Annular Khovanov homology via trace decategorification We'll review work of the speaker, joint with Lauda and Queffelec, relating Khovanov(-Rozansky) homology to categorified quantum sl_m via categorical skew Howe duality. We'll then discuss work in progress (joint with Queffelec) showing how to obtain annular Khovanov homology from this "skew Howe 2-functor" via trace decategorification. This provides a conceptual basis for this invariant, and in particular explains the recent discovery of Grigsby-Licata-Wehrli that the annular Khovanov homology of a link carries an action of sl_2. Our framework extends to give the first construction of sl_n annular Khovanov-Rozansky homology (which carries an action of sl_n), and should lead to a proof of a conjecture of Auroux-Grigsby-Wehrli relating annular Khovanov homology to the Hochschild homology of endomorphism algebras in category O. Liam Watson (University of Glasgow) A categorified view of the Alexander invariant Alexander invariants are classical objects in low-dimensional topology stemming from a natural module structure on the homology of the universal abelian cover. This is the natural setting in which to define the Alexander polynomial of a knot, for example, and given that this polynomial arises as graded Euler characteristic in knot Floer homology, it is natural to ask if there is a Floer-theoretic counterpart to the Alexander invariant. There is: This talk will describe a TQFT due to Donaldson, explain how it is categorified by bordered Heegaard Floer homology, and from this place the Alexander invariant in a Heegaard Floer setting. This is joint work with Jen Hom and Tye Lidman. Caltech Sloan 151 2014-11-17 16:00-18:00 Boris Coskunuzer (Koc University and MIT) Minimal Surfaces with Arbitrary Topology in H^2xR In this talk, we show that any open orientable surface can be embedded in H^2xR as a complete area minimizing surface. Furthermore, we will discuss the asymptotic Plateau problem in H^2xR, and give a fairly complete solution. Ina Petkova (Rice University) Combinatorial tangle Floer homology In joint work with Vera Vertesi, we extend the functoriality in Heegaard Floer homology by defining a Heegaard Floer invariant for tangles which satisfies a nice gluing formula. We will discuss the construction of this combinatorial invariant for tangles in S^3, D^3, and I x S^2. The special case of S^3 gives back a stabilized version of knot Floer homology. USC KAP 414 2014-11-03 16:00-18:00 Anna Wienhard (Heidelberg and Caltech) Anosov representations and proper actions When M is a Riemannian manifold, a discrete subgroup of isometries acts properly on M. This is not true for semi-Riemannian manifolds. For a homogeneous space there is criterion, due to Benoist and Kobayashi, which describes when the action of a discrete subgroup of isometries is proper. In this talk I will explain a connection between Anosov representations and proper actions on homogeneous spaces, which relies on a new characterization of Anosov representations. As an application, for a fixed convex cocompact subgroup G' of a Lie group G of rank one, one gets a precise description of the set of proper actions of G' on the group G by left and right multiplication. This is joint work with Francois Gueritaud, Olivier Guichard, and Fanny Kassel. Jeremy Toulisse (University of Luxembourg) Minimal maps between hyperbolic surfaces, and anti-de Sitter geometry Around 1990, Geoff Mess discovered deep connections between 3-dimensional anti-de Sitter (AdS) geometry and the theory of hyperbolic surfaces. These ideas were further expanded by Schoen, Labourie, Schlenker, Krasnov and others to establish an equivalence between minimal Lagrangian diffeomorphisms between hyperbolic surfaces and maximal surfaces in AdS space-time. We will explain this connection, and extend it to manifolds with conical singularities.

### 2011-2012

 Caltech Sloan 257 2012-04-13 16:00-18:00 Juan Souto (University of British Columbia) Metrics on the sphere with large volume and spectral gap I will explain how to construct, for d>2, metrics on the d-dimensional sphere with bounded geometry, arbitrarily large volume, and spectral gap bounded from below away from 0. As a consequence we obtain that there are hyperbolic knot complements M_i whose volume tends to infinity and whose Cheeger constant is larger than some epsilon>0 for all i. This is joint work with Marc Lackenby. Nick Sheridan (MIT) Homological Mirror Symmetry for a Calabi-Yau hypersurface in projective space We prove homological mirror symmetry for a smooth Calabi-Yau hypersurface in projective space. In the one-dimensional case, this is the elliptic curve, and our result is related to that of Polishchuk-Zaslow; in the two-dimensional case, it is the K3 quartic surface, and our result reproduces that of Seidel; and in the three-dimensional case, it is the quintic three-fold. After stating the result carefully, we will describe some of the techniques used in its proof, and draw lots of pictures in the one-dimensional case. Caltech Sloan 257 2012-03-02 16:00-18:00 David Futer (Temple University) The Jones polynomial and surfaces far from fibers This talk explores relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses, we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We also show that certain coefficients of the Jones and colored Jones polynomials measure how far this surface is from being a fiber in the knot complement. This is joint work with Effie Kalfagianni and Jessica Purcell. Sucharit Sarkar (Clay Math Institute / Columbia) A Khovanov homotopy type We will start by describing Khovanov's categorification of the Jones polynomial from a cube of resolutions of a link diagram. We will then introduce the notion of a framed flow category, as defined by Cohen, Jones and Segal. We will see how a cube of resolutions produces a framed flow category for the Khovanov chain complex, and how the framed flow category produces a space whose reduced cohomology is the Khovanov homology. We will show that the stable homotopy type of the space is a link invariant. Time permitting, we will show that the space is often non-trivial, i.e., not a wedge sum of Moore spaces. This work is joint with Robert Lipshitz. UCLA MS 6229 2012-02-13 16:00-18:00 Selman Akbulut (Michigan State) Exotic smooth structures on 4-manifolds I will discuss corks and plugs (and possibly anti-corks) which are useful tools for understanding exotic smooth manifolds. A natural puzzle is to find the corks and plugs of a given small exotic manifold, such as the Dolgachev surface and the Akhmedov-Park's exotic CP^2 # 2(-CP^2), whose handle-body pictures I will describe. Vladimir Markovic (Caltech) Virtual geometry of Riemann surfaces and 3-manifolds I will discuss my work with J. Kahn about the Ehrenpreis conjecture and the surface subgroup theorem for hyperbolic 3-manifolds. USC KAP 148 2012-01-23 15:30-17:30 Nathan Dunfield (UIUC) Twisted Alexander polynomials of hyperbolic knots I will discuss a twisted Alexander polynomial naturally associated to a hyperbolic knot in the 3-sphere via a lift of its holonomy representation to SL(2, C). It is an unambiguous symmetric Laurent polynomial whose coefficients lie in a number field coming from the hyperbolic geometry. The polynomial can be defined as the Reidmeister torsion of a certain acyclic chain complex, namely the first homology of the knot exterior with coefficients twisted by the holonomy representation tensored with the abelianization map. This polynomial contains much topological information, for instance about the simplest surface bounded by the knot. I will present computations showing that for all 313,209 hyperbolic knots in S^3 with at most 15 crossings it in fact gives perfect such information, in contrast with a related polynomial coming from the adjoint representation of SL(2, C) on it's Lie algebra. This is joint work with Stefan Friedl and Nicholas Jackson. Sergei Gukov (Caltech) Mirror symmetry for colored knot homology It is known that knot homologies admit a physical description as spaces of open BPS states. We study operators and algebras acting on these spaces. This leads to a very rich story, which involves wall crossing phenomena, algebras of closed BPS states acting on spaces of open BPS states, and deformations of Landau-Ginzburg models. One important application to knot homologies is the existence of "colored differentials" that relate homological invariants of knots colored by different representations. Based on this structure, we formulate a list of properties of the colored HOMFLY homology that categorifies the colored HOMFLY polynomial. By calculating the colored HOMFLY homology for symmetric and anti-symmetric representations, we find a remarkable "mirror symmetry" between these triply-graded theories. This talk is based on http://arxiv.org/abs/1112.0030. USC KAP 245 2011-11-28 15:30-17:30 Aaron Lauda (USC) Towards odd Khovanov homology via odd categorified quantum groups Khovanov homology is a categorification of the Jones polynomial that paved the way for other categorifications of quantum link invariants. The theory of categorified quantum groups provides a representation theoretic explanation of these homological link invariants via the work of Webster and others. Surprisingly, the categorification of the Jones polynomial is not unique. Ozsvath, Rasmussen, and Szabo introduced an "odd" analog of Khovanov homology that also categorifies the Jones polynomial, and the even and odd categorification are not equivalent. In this talk I will explain joint work with Alexander Ellis and Mikhail Khovanov that aims to develop odd analogs of categorified quantum groups to give a representation theoretic explanation of odd Khovanov homology. Vera Vertesi (MIT) Transverse invariants in Heegaard Floer homology Using the language of knot Floer homology recently two invariants were defined for Legendrian knots. One in the standard contact 3-sphere defined by Ozsvath, Szabo and Thurston in thecombinatorial settings of knot Floer homology, one by Lisca, Ozsvath, Stipsicz and Szabo in knot Floer homology for a general contact 3--manifold. Both of them naturally generalizes to transverse knots. In this talk I will give a characterization of the transverse invariant, similar to the one given by Ozsvath and Szabo for the contact invariant. Namely for transverse braids both transverse invariants are given as the bottommost elements with respect to the filtration of knot Floer homology given by the axis. The above characterization allows us to prove that the two invariants are the same in the standard contact 3--sphere. This is a joint work with J. Baldwin and D.S. Vela-Vick. UCLA MS 6229 2011-10-24 16:00-18:00 Kristen Hendricks (Columbia) A rank inequality for the knot Floer homology of branched double covers Given a knot K in the three sphere, we compare the knot Floer homology of (S^3, K) with the knot Floer homology of (Sigma(K), K), where Sigma(K) is the double branched cover of the three-sphere over K. By studying an involution on the symmetric product of a Heegaard surface for (Sigma(K), K) whose fixed set is a symmetric product of a Heegaard surface for (S^3, K), and applying recent work of Seidel and Smith, we produce an analog of the classical Smith inequality for cohomology for knot Floer homology. To wit, we show that the rank of the knot Floer homology of (S^3,K) is less than or equal to the rank of the knot Floer homology of (Sigma(K), K). Matt Hedden (Michigan State) The Khovanov module and unlink detection I'll discuss a module structure on Khovanov homology and prove that it detects unlinks. The prove uses Kronheimer and Mrowka's result that Khovanov homology detects unknots, a refinement of Ozsvath and Szabo's spectral sequence from Khovanov homology to the Heegaard Floer homology of the branched double cover of a link, and a theorem which shows that Heegaard Floer homology detects S^1xS^2 summands in the prime decomposition of a 3-manifold. This is joint work with Yi Ni. Caltech Sloan 257 2011-09-30 16:00-18:00 Steven Sivek (Harvard) Monopole Floer homology and Legendrian knots I will define invariants of Legendrian knots using Kronheimer and Mrowka's construction of monopole Floer homology for sutured manifolds. These invariants have several interesting properties: their behavior under stabilization and contact surgery suggests that they are closely related to the Lisca-Ozsvath-Stipsicz-Szabo invariant in knot Floer homology, and they are functorial with respect to Lagrangian concordance. As an application, I will construct many examples of non-loose knots in overtwisted contact manifolds. Qian Yin (University of Michigan / UCLA) Lattes Maps and Combinatorial Expansion A Lattes map is a rational map that is obtained from a finite quotient of a conformal torus endomorphism.Thurston maps are branched covering maps over the 2-sphere with finite post-critical sets. We characterize Lattes maps by their combinatorial expansion behavior, and deduce new necessary and sufficient conditions for a Thurston map to be topologically conjugate to a Lattes map. In the Sullivan dictionary, this characterization corresponds to Hamenstadt's entropy rigidity theorem.

### 2010-2011

 UCLA MS 6229 2011-02-28 16:00-18:00 Ko Honda (USC) HF=ECH via open book decompositions The goal of this talk is to sketch a proof of the equivalence of Heegaard Floer homology (due to Ozsvath-Szabo) and embedded contact homology (due to Hutchings). This is joint work with Vincent Colin and Paolo Ghiggini. Thomas Kragh (MIT) Nearby Lagrangians and Fibered Spectra I will start by giving basic definitions, state the nearby Lagrangian conjecture and describe some previous results on this. I will then discuss fibered (ring)-spectra and how to use these on the nearby Lagrangian problem. I will sketch how to prove that any nearby Lagrangian is up to a finite covering space a homology equivalence. Combining this with results by Abouzaid we arrive at the conclusion that any nearby Lagrangian is a homotopy equivalence. USC KAP 156 2011-02-07 16:00-18:00 Rumen Zarev (Columbia) Bordered Floer homology and the contact category Bordered Heegaard Floer homology is a verison of Heegaard Floer homology for 3-manifolds with boundary, developed by Lipshitz, Ozsvath, and Thurston. A key component of the theory is a DG-algebra A(F) associated to a parametrized surface F. After describing the basics of bordered Floer homology, I will discuss how the homology of A(F) can be naturally identified with a full subcategory of the category of contact structures on Fx[0,1], with convex boundary conditions. Caltech Sloan 257 2011-01-14 16:00-18:00 Tian-Jun Li (Minnesota) Symplectic 4-manifolds with Kodaira dimension zero Symplectic 4-manifolds with negative Kodaira dimension have been classified up to symplectomorphisms. In the next case, Kodaira dimension zero, there is a speculation that such a manifold is diffeomorphic to K3, Enriques surface or a torus bundle over torus. I will discuss what is known towards this conjectured smooth classification. Matthew Day (Caltech) A Birman exact sequence for automorphism groups of free groups The classical Birman exact sequence relates the mapping class group of a surface with punctures to the mapping class group of the same surface with some of the punctures patched over. We describe a similar sequence for automorphism groups of free groups. Instead of mapping class groups of surfaces with punctures, we consider groups of automorphisms of a free group that fix the conjugacy classes of some of the basis elements. Instead of filling in punctures, we induce automorphisms on a free group of lower rank by deleting basis elements. Our results concern the kernel of this map: the kernel is finitely generated and we give an explicit recursive presentation for it. Further, the kernel is not finitely presentable (aside from a few trivial cases). We show this by building explicit infinite-rank subgroups of its second homology and cohomology. This is joint work with Andrew Putman. Caltech Sloan 257 2010-12-03 16:00-18:00 Joshua Greene (Columbia) Mutation and alternating links I will discuss the proof and consequences of the following result. Suppose that D_1 and D_2 are reduced alternating diagrams for a pair of links whose branched double-covers have isomorphic Heegaard Floer homology groups. Then the spaces are diffeomorphic, and moreover D_1 and D_2 are related by a sequence of Conway mutations. Andrew Putman (Rice) Equivariant homological stability for congruence subgroups An important structural feature of the kth homology group of SL_n(Z) is that it is independent of n once n is sufficiently large. This property is called "homological stability" for SL_n(Z). Congruence subgroups of SL_n(Z) do not satisfy homological stability; however, I will discuss a theorem that says that they do satisfy a certain equivariant version of homological stability. UCLA MS 6229 2010-11-01 16:00-18:00 Allison Gilmore (Columbia) An algebraic proof of invariance for knot Floer homology We investigate the algebraic structure of knot Floer homology in the context of categorification. Ozsvath and Szabo gave the first completely algebraic description of knot Floer homology via a cube of resolutions construction. Starting with a braid diagram for a knot, one singularizes or smooths each crossing, then associates an algebra to each resulting singular braid. These can be arranged into a chain complex that computes knot Floer homology. Using this construction, we give a fully algebraic proof of invariance for knot Floer homology that avoids any mention of holomorphic disks or grid diagrams. We close with an alternative description of knot Floer homology in terms of Soergel bimodules that suggests a close relationship with HOMFLY-PT homology. John Baldwin (Princeton) A combinatorial spanning tree model for knot Floer homology I'll describe an ongoing project with Adam Levine to iterate Manolescu's exact triangle in knot Floer homology. This iteration results in a spectral sequence which converges to (a stabilized version of) knot Floer homology. With coefficients in a Novikov ring, the E_2 term of this spectral sequence is roughly generated by Kauffman states of the knot or, equivalently, by spanning trees of its black graph. One can describe the d_2 differential combinatorially and can prove that this spectral sequence collapses at E_3. Therefore (E_2,d_2) provides a combinatorial chain complex for delta-graded knot Floer homology. USC KAP 245 2010-09-27 15:30-17:30 Anthony Licata (Stanford) Heisenberg algebras, Hilbert schemes, and categorification Representations of Heisenberg algebras appear naturally in many mathematical contexts. In geometry, one notable appearance is in the work of Nakajima and Grojnowski relating Heisenberg algebras and Hilbert schemes of points on a complex surface. We will describe a categorification of the Nakajima-Grojnowski construction. This is joint work with Sabin Cautis. Joan Licata (Stanford) Legendrian contact homology for Seifert fibered spaces In this talk, I'll focus on Seifert fibered spaces whose fiber structure is realized by the Reeb orbits of an appropriate contact form. A Legendrian knot in such a manifold is described by a specially labeled Lagrangian diagram, and from this data one can compute both the "classical" invariants for Legendrian knots in rational homology three-spheres and also a new invariant which takes the form of a differential graded algebra. This work is joint with J. Sabloff.

### 2009-2010

 USC KAP 245 2010-05-03 15:30-17:30 Vera Vertesi (MSRI) Legendrian and transverse classification of twist knots In 1997 Chekanov gave the first example of a knot type whose Legendrian representations are not distinguishable using only the classical invariants: the 5_2 knot. Epstein, Fuchs and Meyer extended his result by showing that there are at least n different Legendrian representations of the (2n+1)2 knot with maximal Thurston-Bennequin number. The aim of this talk to give a complete classification of Legendrian representations of twist knots. In particular the (2n+1)_2 knot has exactly ceil(n^2/2) Legendrian representations with maximal Thurston-Bennequin number. This is a joint work (in progress) with John Etnyre and Lenhard Ng. Daniel Krasner (Columbia and MSRI) Graphical calculus of Soergel bimodules in Khovanov-Rozansky link homology I will outline a graphical calculus of Soergel bimodules, developed by B. Elias and M. Khovanov, and describe how it can be used to construct an integral version of sl(n) and HOMLFYPT link homology, as well as prove functoriality of the latter. Caltech Sloan 257 2010-04-23 16:00-18:00 Yanki Lekili (MSRI) Quilted Floer homology of 3-manifolds We introduce quilted Floer homology (QFH), a new invariant of 3-manifolds equipped with a circle valued Morse function. This is a natural extension of Perutz's 4-manifold invariants, making it a (3+1) theory. We relate Perutz's theory to Heegaard Floer theory by showing an isomorphism between QFH and HF^+ for extremal spin^c structures with respect to the fibre of the Morse function. As applications, we give new calculations of Heegaard Floer theory and a characterization of sutured Floer homology. Tao Li (Boston College) Rank and genus of amalgamated 3-manifolds A fundamental question in 3-manifold topology is whether the rank of the fundamental group a 3-manifold is equal to its Heegaard genus. We use hyperbolic JSJ pieces to construct closed 3-manifolds with rank smaller than genus. UCLA MS 6627 2010-03-01 16:00-17:00 Christopher Douglas (Berkeley) 2-dimensional algebra and 3-dimensional local field theory Witten showed that the Jones polynomial invariants of knots can be computed in terms of partition functions of a (2+1)-dimensional topological field theory, namely the SU(2) Chern-Simons theory. Reshetikhin and Turaev showed that this theory extends to a (1+1+1)-dimensional topological field theory---that is, there is a Chern-Simons-type invariant associated to 3-manifolds, 3-manifolds with boundary, and 3-manifolds with codimension-2 corners. I will explain the notion of a local or (0+1+1+1)-dimensional topological field theory, which has, in addition to the structure of a (1+1+1)-dimensional theory, invariants associated to 3-manifolds with codimension-3 corners. I will describe a notion of 2-dimensional algebra that allows us to construct and investigate such local field theories. Along the way I will discuss the geometric classification of local field theories, and explain the dichotomy between categorification and algebraification. Caltech Sloan 257 2009-12-04 16:00-18:00 Yi Liu (UC Berkeley) Tiny groups and the simplicial volume A group is called tiny if it cannot map onto the fundamental group of any aspherical 3-manifold of negative Euler characteristic. For example, knot groups are tiny. In this talk we show that if a finitely presented tiny group G maps onto the fundamental group of a compact aspherical 3-manifold N, then the simplicial volume of N is bounded above in terms of G. This is joint work with Ian Agol. Shelly Harvey (Rice University) Filtrations of 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. 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. USC KAP 145 2009-11-02 16:00-18:00 Kefeng Liu (UCLA) Recent results on moduli spaces Paul Melvin (Bryn Mawr College) Degree formulas for higher order linking The linking number of a pair of closed curves in 3-space can be expressed as the degree of a map from the 2-torus to the 2-sphere, by means of the linking integral Gauss wrote down in 1833. In the early 1950's, John Milnor introduced a family of higher order linking numbers (the "mu-bar invariants"). In this talk I will describe a formula for Milnor's triple linking number as the "degree" of a map from the 3-torus to the 2-sphere. This is joint work with DeTurck, Gluck, Komendarczyk, Shonkwiler and Vela-Vick. UCLA MS 6229 2009-10-09 16:00-18:00 Scott Morrison (UC Berkeley) Blob homology Blob homology is a new gadget that takes an n-manifold and an n-category with duals, and produces a graded vector space. It's a simultaneous generalization of two important constructions: the 0-th graded piece recovers the usual "skein module" invariant, and in the special case of n=1, where the manifold is the circle, blob homology reduces to Hochschild homology. I'll begin by reviewing these ideas, then give the definition of blob homology. Finally, I'll describe some of its nice formal properties, including an action of chains of diffeomorphisms generalizing the action of diffeomorphisms on a skein module, and a nice gluing formula in terms of A_\infty modules. Yi Ni (Caltech) Some applications of Heegaard Floer homology to Dehn surgery In recent years, Heegaard Floer homology has become a very powerful tool for 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 consists of some results about cosmetic surgeries.