The SGGTC seminar meets on Fridays in Math 520, at 10:45 am unless noted otherwise (in red).
Christopher Cornwell, "Knot contact homology and representations of knot groups"
Abstract: We study certain linear representations of the knot group that induce augmentations of knot contact homology. This perspective on augmentations enhances our understanding of the relationship between the augmentation polynomial and the $A$-polynomial of the knot. For example, we show that for 2-bridge knots the two polynomials agree. We also show this is never the case for (non-2-bridge) torus knots, nor for a family of 3-bridge pretzel knots. Moreover, we show that these representations provide a lower bound on the meridional rank of the knot. As a consequence, our results give another proof that torus knots have meridional rank and bridge number that are equal.
Kristen Hendricks, "Localization and the link Floer homology of doubly-periodic knots"
Abstract: We construct localization spectral sequences for the link Floer homology of doubly-periodic knots, and discuss how they give a simultaneous generalization of two classical results of Kunio Murasugi and Allan Edmonds concerning, respectively, the Alexander polynomial and genus of the knot.
Matt Hedden, "Topologically slice knots"
Abstract: I'll discuss the world of topologically slice knots: those knots which bound topologically flat disks in the 4-ball. These knots generate a particularly interesting subgroup of the smooth concordance group which can be used to produce examples of smooth 4-manifolds homeomorphic, but not diffeomorphic, to R^4. I'll survey the history of this subgroup and the flurry of recent progress on its understanding. I'll then discuss the first examples of (two-) torsion elements in this subgroup. This is joint work with Se-Goo Kim and Charles Livingston.
Ben Webster, "Category O for quiver varieties and the categorification of Lie algebras"
Abstract: I'll expand on my talk from Wednesday to describe the relationship between quantizations of quiver varieties and the categorification of Lie algebras; in particular, I'll describe how this philosophy leads us to new generalizations of KLR algebras and their cyclotomic quotients, the weighted KLR algebras, as well as a new understanding of classic categories of representation theory, such as symplectic reflection algebras and BGG categories O.
Marco Mackaay, "sl3 web algebras and categorified Howe duality"
Abstract: In a joint paper (arXiv:1206.2118) with Weiwei Pan and Daniel
Tubbenhauer, I introduced an sl3 analogue of Khovanov's arc algebras,
which we called the sl3 web algebras. The definition uses Kuperberg's sl3
webs and Khovanov's sl3 foams, which are important ingredients for the
sl3 knot polynomials and knot homologies respectively.
We showed that the sl3 web algebras are graded Frobenius algebras and that they are Morita equivalent to direct summands of certain cyclotomic Khovanov-Lauda-Rouquier algebras. The latter result uses a categorification of quantum skew Howe duality and allows us to prove interesting results about the sl3 web algbras, e.g. the center of an sl3 web algebra is isomorphic to the cohomology ring of a certain Spaltenstein variety and its Grothendieck group is isomorphic to a certain space of quantum sl3 tensors. Under the latter isomorphism the Grothendieck classes of the indecomposable projective modules correspond to the dual canonical basis elements.
In my talk, I will try to explain these results and comment on further developments, such as Lauda, Queffelec and Rose's recent preprint (arXiv:1212.6076), in which they use the same categorified quantum skew Howe duality in order to relate Webster's and Khovanov's sl3 link homologies, and generalizations to sln.
Jen Hom, "The Seifert form, Alexander module and bordered Floer homology"
Abstract: We study the bordered Floer homology of the 3-manifold with boundary obtained by cutting S^3 along a Seifert surface Sigma for a knot K. In particular, we show that this bordered invariant determines both the Seifert form and Alexander module. This is joint work in progress with Sam Lewallen, Tye Lidman and Liam Watson.
Doris Hein, "Reeb dynamics and symplectically degenerate extrema"
Abstract: In this talk, I will explain how the notion of a symplectically degenerate maximum can be transferred to the contact setting and Reeb flows. As in the proof of the Hamiltonian Conley conjecture, the existence of such an orbit implies the existence of infinitely many closed Reeb orbits. I will also discuss how this result can be used to reprove a recent theorem of Cristofaro-Gardiner and Hutchings asserting that every Reeb flow on the standard contact three-sphere has at least two periodic orbits.
Michael Usher, "Lagrangian Floer theory and Hofer geometry"
Abstract: I will discuss some ways in which Lagrangian submanifolds and their Floer complexes can be used to detect properties of the Hofer metric on the Hamiltonian diffeomorphism group. This story begins in the late 1990's with work of Chekanov and Oh on the displacement energy of compact Lagrangian submanifolds; I'll explain a simple modification of their argument which gives results about certain noncompact Lagrangians such as cotangent fibers as well. I'll also discuss an application of the Lagrangian-Floer-theoretic boundary depth which shows that the Hamiltonian diffeomorphism groups of some cotangent bundles admit infinite-rank subgroups that are quasi-isometrically embedded with respect to the Hofer metric.
Yin Tian, "A categorification of super Hopf algebra U_t sl (1|1) via contact topology"
Abstract: Representation theory of quantum groups has profound applications to low-dimensional topology in the framework of Reshetikhin-Turaev invariants. The Alexander polynomial of knots can be recovered from the representation theory of super quantum group U_q sl (1|1); moreover, the knot Floer homology gives rise to a categorification of the Alexander polynomial. In this talk, we will construct DG categories motivated by contact topology to categorify U_t sl (1|1) as a variant of U_q sl(1|1) and its tensor product representations.
Eleny Ionel, "A natural Gromov-Witten virtual fundamental class"
Abstract: The moduli space of pseudo-holomorphic maps into closed symplectic manifolds carries several natural structures which induce relations between the Gromov-Witten invariants. The moduli space comes in several flavors, including the orbifold version (for a finite group actions) and a relative version (relative a symplectic normal crossing divisor). In this talk, based on joint work with Tom Parker, we discuss how these different structures on the moduli space interact to induce a unique virtual fundamental class that satisfies certain naturality conditions. This allows one to reduce the construction of the virtual fundamental class to one involving only Gromov-type perturbations (after introducing stabilizing divisors and twisted G-structures). This approach can also be used to define a Real version of Gromov-Witten invariants for closed symplectic manifolds with anti-symplectic involutions.
Chris Wendl, "Spinal open books and algebraic torsion in contact 3-manifolds"
Abstract: By the Giroux correspondence, contact structures on a closed manifold can be understood in terms of open book decompositions that support them. A "spinal" open book is a more general notion that also supports contact structures, and arises naturally e.g. on the boundary of a Lefschetz fibration whose fibers and base are both oriented surfaces with boundary. One can learn much about symplectic fillings by studying spinal open books: for instance, using holomorphic curve methods, we can classify the symplectic fillings of S^1-invariant contact structures on any circle bundle over a surface (joint work with Sam Lisi and Jeremy Van Horn-Morris). One can also use them to compute an invariant that lives in Symplectic Field Theory and measures the "degree of tightness" of a contact manifold (joint work with Janko Latschev).
Brendan Owens, "Immersed disks and knot invariants"
Abstract: The four-dimensional clasp number of a knot K in the three-sphere is the minimum number of double points of an immersed disk in the four-ball bounded by K. We obtain bounds on this and on two related invariants: the slicing number and the concordance unknotting number. The main tool we use is Heegaard Floer homology, and in particular a slightly strengthened version of Ozsváth-Szabó's unknotting number one obstruction. This is joint work with Sašo Strle.
Hyun Kyu Kim, "Central extensions of the Ptolemy-Thompson group from quantization of the universal Teichmuller space"
Abstract: Quantization of the Teichmuller space of a surface yields projective representations of the mapping class group and hence central extensions of it. Using the Chekhov-Fock-Goncharov quantization of the universal Teichmuller space, Funar and Sergiescu computed the central extension of the Ptolemy-Thompson group T, which is a universal version of the mapping class groups. We compute the central extension of T using the Kashaev quantization, and compare the two extensions. Also, I'll briefly show a graphical proof of the isomorphism between this central extension and the relative abelianization of the braided Ptolemy-Thompson group of Funar and Kapoudjian. Possibility of construction of a representation of the full braided Ptolemy-Thompson group, in relation to the author's joint work with Igor Frenkel on the representation-theoretic construction of quantum Teichmuller space, may be discussed too. Based on arXiv:1211.4300.
Somnath Basu, "Transverse strings, knots and configuration spaces"
Abstract: The study of free loop space of manifolds is classical. A loop in M can be thought of as a closed arc in M x M with end points in the diagonal. We define the space of transversal open strings, a subspace of free loops thought of as closed arcs. We then analyze algebraic structures on such strings. On one hand, we recover interesting invariants in the case of knots. On the other hand, we relate this new structure to the Pontrjagin product on the based loop space of configuration spaces, which we show is not a homotopy invariant.
Corrin Clarkson, "3-Manifold Mutations Detected by Heegaard Floer Homology"
Abstract: Given a self-diffeomorphism γ of a closed surface S and an embedding f of S into a three manifold M, we construct a mutant manifold Mγ by cutting M along f(S) and regluing by γ. We will consider whether there are any gluings γ such that for any embedding, the manifold M and its mutant Mγ have isomorphic Heegaard Floer homology. In particular, we will demonstrate that if γ is not isotopic to the identity and it induces the identity map on homology, then there exists an embedding of S into a three manifold M such that HF(M)≇ HF(Mγ). Similarly, if the map induced by γ on homology is -id and the genus of S is greater than 3, then there exists an embedding of S into a three manifold M such that HF(M)≇ HF(Mγ).
Sheila Sandon, "The discriminant metric on the contactomorphism group"
Abstract: I will explain the construction of a bi-invariant metric on the universal cover of the contactomorphism group of any contact manifold, and discuss its relations to other rigidity phenomena for contactomorphisms such as the contact non-squeezing theorem, orderability of contact manifolds, and a contact analogue of the Arnold conjecure on fixed points of Hamiltonian symplectomorphisms. This is joint work with Vincent Colin.
Other relevant information
FRG Workshop on Gromov-Witten Theory
April 12 – 14, 2013
- Columbia Geometric Topology Seminar
- Columbia Algebraic Geometry Seminar
- Eilenberg lecture series
- Princeton Topology Seminar