The SGGTC seminar meets on Fridays in Math 407 at 1pm, unless noted otherwise (in red).
Previous semesters: Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007.
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Abstract: We will discuss how to study 2n-dimensional Weinstein (gradient-like exact symplectic) manifolds via a core n-dimensional stratified complex called the skeleton. We show that the Weinstein structure can be homotoped to admit a skeleton with a unique symplectic neighborhood. Then we further analyze and divide the remaining singularities with a goal (partially achieved, generally in progress) of reducing the singularity types to a finite combinatorial list in each dimension, corresponding to (a signed version of) Nadler's arboreal singularities. We will discuss how arboreal singularities are natural in a Lagrangian skeleton, and what information about the symplectic manifold one might hope to extract out of an arboreal complex.
Abstract: To a Legendrian knot, one can associate an A∞ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials.
Abstract: I will tell several stories surrounding the Landau-Ginzburg potential, a basic enumerative invariant of a monotone Lagrangian submanifold. In the first part, I will explain how to mutate Lagrangian tori in toric Fanos and compute the LG potentials using the wall-crossing formula (joint with James Pascaleff). In dimension 4, this recasts a known cluster-algebraic story in the symplectic world. In the second part, time permitting, I will explain a connection of LG potentials with symplectic cohomology and its ring structure.
Abstract: It is conjectured that if Dehn surgery on a hyperbolic knot in S^3 yields a Seifert fibered space, then the surgery slope must be an integer. An equivalent formulation is that the only knots in S^3 with non-integer Seifert fibered surgeries are torus knots and cables of torus knots. I will discuss results relating to this conjecture that can be derived from Heegaard Floer homology. This includes joint work with Ahmad Issa on classifying the Seifert fibered L-spaces can arise by non-integer surgery on S^3.
Abstract: The monodromy of a variation of Hodge structures is generated by integer matrices and it can be finite or infinite index in the appropriate arithmetic group. Brav & Thomas showed that for several examples of interest in mirror symmetry, the monodromy is of infinite index. I will discuss several further geometric properties of these "thin" monodromy groups, from there construct a complex 3-fold, and relate all of this to Lyapunov exponents, which are invariants coming from dynamical systems. The mirror quintic family will serve as a guiding example.
October 13, 2017: Sander Kupers " Cellular E_2-algebras and the unstable homology of mapping class groups"
Abstract: We discuss joint work with Soren Galatius and Oscar Randal-Williams on the application of higher-algebraic techniques to classical questions about the homology of mapping class groups. This uses a new "multiplicative" approach to homological stability -- in contrast to the "additive" one due to Quillen -- which has the advantage of providing information outside of the stable range.
Abstract: The cosmetic surgery conjecture, which has been around since the early 1990s, asks when surgery along two different framing curves for the same knot produce the same 3-manifold. The virtual cosmetic surgery conjecture is a generalization of this question to coverings between surgeries. I will explain the conjecture, discuss how far you can get with elementary techniques, and present an application of hyperbolic geometry to the conjecture. Time permitting, I will also discuss the relevance of equivariant Heegaard Floer homology and monopole Floer homology, and ongoing work.
Abstract: I will discuss a proof that the Legendrian conormal torus is a complete invariant of knots in R^3. This proof uses holomorphic curves, knot contact homology, and string topology, and is joint work with Tobias Ekholm and Vivek Shende. If time permits, I'll discuss recent progress in the circle of ideas connecting knot contact homology, colored HOMFLY-PT polynomials, and topological strings.
Abstract: In this talk I will introduce a class of Legendrian wavefronts associated to surface triangulations. First, I will explain the interplay between the Legendrian isotopy type and the combinatorics of the triangulation. In particular, we will be connecting symplectic geometry and graph theory. Then I will discuss the Floer theory of these wavefronts and provide a description of their dg-algebras. The talk will conclude with two geometric applications.
Abstract: Pillowcase homology is a geometric construction, which was developed in order to better understand and compute a knot invariant called singular instanton knot homology. We will introduce an algebraic extension of pillowcase homology. First we will associate an algebra A to the pillowcase. Then to an immersed curve L inside the pillowcase we will associate an A∞ module M(L) over A. Then we will show how, using these modules, one can recover and compute Lagrangian Floer homology for immersed curves. This can also be viewed as a way to compute the partially wrapped Fukaya category of the pillowcase.
Abstract: The spherical manifold realization problem asks which spherical three-manifolds arise from surgeries on knots in the three-sphere. In recent years, the realization problem for C, T, O, and I-type spherical manifolds has been solved, leaving the D-type manifolds (also known as the prism manifolds) as the only remaining case. Every prism manifold can be parametrized as P(p, q), for a pair of relatively prime integers p > 1 and q. We determine a complete list of prism manifolds P(p, q) that can be realized by positive integral surgeries on knots in the three-sphere. The methodology undertaken to obtain the classification relies on tools from Floer homology and lattice theory, and is primarily combinatorial in nature. This is joint work with Ballinger, Ni, and Ochse.
November 17, 2017: Sherry Gong " Marked link invariants: Khovanov, instanton, and binary dihedral invariants for marked links"
Abstract: We introduce a version of Khovanov homology for alternating links with marking data, $\omega$, inspired by instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology (Kronheimer and Mrowka: Khovanov homology is an unknot-detector) collapses on the $E_2$ page for alternating links. We moreover show that the Khovanov homology we introduce for alternating links does not depend on $\omega$; thus, the instanton homology also does not depend on $\omega$ for alternating links. Finally, we study a version of binary dihedral representations for links with markings, and show that for links of non-zero determinant, this also does not depend on $\omega$.
December 8, 2017: Daniel Alvarez-Gavela " The simplification of singularities of Lagrangian and Legendrian fronts"
Abstract: We present a full h-principle (relative, parametric, C^0-close) for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we show that if there is no homotopy theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respect to an ambient foliation by Lagrangian or Legendrian leaves, then the simplification can be achieved by means of an ambient Hamiltonian isotopy. We present complete calculations of the homotopy theoretic obstruction in certain concrete examples. We also discuss several applications to symplectic and contact topology, including to Nadler's program for the arborealization of singularities of Lagrangian skeleta. The relevant papers are arXiv:1605.07259 and arXiv:1605.07258.