The SGGTC seminar meets on Fridays in Math 407 at 1pm, unless noted otherwise (in red).
Previous semesters: 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.
Our e-mail list.
January 20, 2017: Wai-kit Yeung " Perverse sheaves, knot contact homology and relative Calabi-Yau completions"
Abstract: In this talk, I will present joint work [arXiv:1610.02438] with Yu. Berest and A. Eshmatov, where we give a universal construction, called homotopy braid closure, that produces invariants of links in R^3 starting with a braid group action on objects of a (model) category. Applying this construction to the natural action of the braid group B_n on the category of perverse sheaves on the two-dimensional disk with singularities at n marked points, we obtain a differential graded (DG) category that gives knot contact homology in the sense of L. Ng. As an application, we show that the category of finite-dimensional modules over the 0-th homology of this DG category is equivalent to the category of perverse sheaves on R^3 with singularities at most along the link. If time allows, I will also discuss my recent work [arXiv:1612.06352] that generalizes this picture using non-commutative algebraic geometry.
Abstract: This talk is an introduction as well as a survey about 2-representation theory of Coxeter groups. The motivation to study such 'higher' representations is as follows: In groundbreaking work Chuang-Rouquier and, independently, Khovanov-Lauda introduced 2-representation theory of Lie algebras and their quantum analogs. As a 'higher' version of classical representation theory of Lie groups, their ideas have already led to many successive works as well as applications throughout mathematics (and physics). Historically speaking, what `should have come first' is 2-representation theory of finite groups. Sadly the story of `higher' representations of finite groups is not understood at all at the moment. But, quite recently, Mazorchuk-Miemietz made very good progress towards a 'higher' analog of representations of finite-dimensional algebras (as e.g. groups rings of finite groups), and applied it quite successful to one of the most well-behaved family of finite groups: finite Coxeter groups. Already in this case a lot of interesting new phenomena show up, most of which are neither present in classical representation theory nor in the story of Chuang-Rouquier, Khovanov-Lauda, and which might lead to interesting connections and applications in the years to come. However, in this talk we will focus on one (completely explicit) example, i.e. the first non-trivial family of examples given by the dihedral groups, where already several new phenomena are visible. Based on joint work with Marco Mackaay, Volodymyr Mazorchuk and Vanessa Miemiet.
February 10, 2017: Pavel Mnev " Abelian and non-abelian BF theory on cobordisms endowed with cellular decomposition"
Abstract: We will present an example of a topological field theory living on cobordisms endowed with CW decomposition (this example corresponds to the so-called BF theory in its abelian and non-abelian variants), which satisfies the Batalin-Vilkovisky master equation, satisfies (a version of) Segal's gluing axiom w.r.t. concatenation of cobordisms and is compatible with cellular aggregations. In non-abelian case, the action functional of the theory is constructed out of local unimodular L-infinity algebras on cells; the partition function carries the information about the Reidemeister torsion, together with certain information pertaining to formal geometry of the moduli space of local systems. This theory provides an example of the BV-BFV programme for quantization of field theories on manifolds with boundary in cohomological formalism. This is a joint work with Alberto S. Cattaneo and Nicolai Reshetikhin.
Abstract: The Reshetikhin-Turaev construction for the standard representation of the quantum group gl(1|1) sends tangles to C(q)-linear maps in such a way that a knot is sent to its Alexander polynomial. After a brief review of this construction, I will give an introduction to tangle Floer homology — a combinatorial generalization of knot Floer homology which sends tangles to (homotopy equivalence classes of) bigraded dg bimodules. Finally, I will discuss how to see tangle Floer homology as a categorification of the Reshetikhin-Turaev invariant. This is joint work with Alexander Ellis and Vera Vertesi.
Abstract: The mirror of a complete toric variety is a Landau-Ginzburg model. The oscillatory integral of the superpotential function on a Lagrangian cycle mirror to a coherent sheaf is a B-model genus 0 invariant. I will describe a way to compute it, and it is mirror to a genus 0 Gromov-Witten descendant potential with certain Gamma class of the coherent sheaf inserted. This result is related to Iritani's result identifying the integral structures on both sides. When the toric variety is P^1, this result can be extended to compute all genus descendants from Eynard-Orantin's topological recursion theory on the mirror of P^1.
Abstract: Although not every knot in the three-sphere can bound a smooth embedded disk in the three-sphere, it must bound a PL disk in the four-ball. This is not true for knots in the boundaries of arbitrary smooth contractible manifolds. We give new examples of knots in homology spheres that cannot bound PL disks in any bounding homology ball and thus not concordant to knots in the three-sphere. This is joint work with Jen Hom and Adam Levine.
Abstract: Symplectic homology for a Liouville cobordism (possibly filled at the negative end) generalizes simultaneously the symplectic homology of Liouville domains and the Rabinowitz-Floer homology of their boundaries. I will explain its definition, some of its properties, and give a sample application which shows how it can be used in order to obstruct cobordisms between contact manifolds. Based on joint work with Kai Cieliebak and Peter Albers.
Abstract: Seidel's famous theorem describes a mapping cone relation induced by Lagrangian Dehn twists along spheres in Fukaya categories. Recently, this was generalized to Lagrangian submanifolds diffeomorphic to projective spaces in my earlier joint work with Cheuk-Yu Mak using Lagrangian cobordisms. In this talk, we further generalize the cone relation to Lagrangians diffeomorphic to finite quotients of spheres and projective spaces by techniques of SFT. Algebraically, these Dehn twists are all twists along certain spherical functors. This is an ongoing joint work with Cheuk-Yu Mak.
Abstract: We define graded spin surfaces which are marked Riemann surfaces with boundary and some extra structure, and consider their moduli spaces. We prove that these spaces are canonically oriented and are associated with a canonical intersection theory, extending the one defined by Pandharipande, Solomon and the speaker for moduli of disks. If time permits we will discuss the orientability problem for stable maps from graded spin domains to a target pair (M,L) where M is symplectic and L is Lagrangian of even real dimension. Based on a joint work with J. Solomon.
Abstract: I'll talk about recent progress in re-formulating Morse theory as a deformation problem. A central player is a stack classifying broken Morse trajectories, over which all Morse theory seems to live. This is joint work with Jacob Lurie.
Abstract: Let N and d be two coprime positive integer numbers. Given a Riemann surface, the moduli space of stable bundles with a fixed determinant and of rank N and degree d forms a smooth projective variety. One can also use Gromov-Witten invariants to associate a quantum cohomology ring to this variety. In the case that N is equal to two, the ring structure of quantum cohomology was completely characterized by a group of physicists (Bershadsky, Johansen, Sadov and Vafa) and then was verified rigorously by Munoz. In this talk, I will explain how Yang-Mills gauge theory can be used to obtain information about this ring in the case of higher rank bundles. I will also discuss some implications of these results in 3-manifold topology. This talk is mainly based on a joint work with Yi Xie. I'll also borrow some results from another work with Fukaya and Lipyanskiy.
April 14, 2017: Andrew Manion " Douglas-Manolescu's pairing formula and tensor products of 2-representations "
Abstract: I will discuss joint work in progress with Raphael Rouquier on a relationship between higher representation theory and some algebraic constructions appearing in bordered and cornered Heegaard Floer homology. On the representation theory side, one can define a tensor product of 2-representations of the positive half of gl(1|1). On the topological side, strands algebras from bordered Floer homology can be upgraded to 2-representations using the algebra of cornered Floer homology introduced by Douglas and Manolescu. Douglas-Manolescu's pairing formula for the strands algebra can be upgraded to an isomorphism between the strands algebra of a pairing, as a 2-representation, and the tensor product of the 2-representations associated to the pieces. I will outline as much of this project I can in the available time.
Abstract: We explain how to construct an explicit topological model for every two-block Springer fiber of type D. These so-called topological Springer fibers are homeomorphic to their corresponding algebro-geometric Springer fiber. They are defined combinatorially using cup diagrams which appear in the context of finding closed formulas for parabolic Kazhdan-Lusztig polynomials of type D with respect to a maximal parabolic of type A. As an application it is discussed how the topological Springer fibers can be used to reconstruct the famous Springer representation in an elementary and combinatorial way.
Abstract: Rasmussen’s knot concordance invariant is defined in terms of a Z-filtration on the Khovanov-Lee complex. In my talk, I will show that for a link diagram drawn on an annulus, the Khovanov-Lee complex carries a second Z-filtration. Using ideas of Ozsváth-Stipsicz-Szabó as reinterpreted by Livingston, I will use this structure to define a family of annular Rasmussen invariants that yield information about annular and non-annular link cobordisms. Focusing on the special case of annular links obtained as braid closures, I will use the behavior of the annular Rasmussen invariants to obtain a necessary condition for braid quasipositivity and a sufficient condition for right-veeringness. This is joint work with Elisenda Grigsby and Tony Licata.
Abstract: In this talk we describe the quantum cohomology of a sphere with three orbifold points. This is done by defining an isomorphism between the quantum cohomology ring and the Jacobian ring of a certain power series built from the Lagrangian Floer theory of an immersed circle. This is joint work with Cho, Hong and Lau.