The SGGTC seminar meets on Fridays in Math 407 at 1pm, unless noted otherwise (in red).
Previous semesters: Fall 2017, 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.
Our e-mail list.
January 19, 2018: Mark McLean " Birational Calabi-Yau Manifolds have Isomorphic Hamiltonian Floer Algebras "
Abstract: We show that any two birational projective Calabi-Yau manifolds admit Hamiltonians with isomorphic Hamiltonian Floer cohomology algebras after a certain change of Novikov rings. As a result, we show that such Calabi-Yau manifolds have isomorphic small quantum cohomology algebras over Novikov rings of any characteristic. The proof is inspired by ongoing work of Borman and Sheridan and uses a version of symplectic cohomology defined by Groman, Venkatesh and Varolgunes.
Abstract: The knot concordance group C consists of knots in S^3 modulo knots that bound smooth disks in B^4. We consider C_Z, the group of knots in homology spheres that bound homology balls modulo knots that bound smooth disks in a homology ball. Matsumoto asked if the natural map from C to C_Z is an isomorphism. Adam Levine answered this question in the negative by showing the map is not surjective. We show that the quotient of C_Z by the image of C is infinitely generated and contains elements of infinite order. The proof relies on Heegaard Floer homology. This is joint work with Adam Levine and Tye Lidman.
Abstract: Work of Li-Li, McDuff, and others makes it possible to determine algorithmically whether a given rational four-dimensional ellipsoid E(1,a) symplectically embeds into a given rational polydisk P(b,c). Even so, the dependence of the answer on a, b, and c is sufficiently complicated that a number of features of the embedding capacity functions governing embeddings of ellipsoids into polydisks are not well-understood. I will describe a method for constructing families of exceptional spheres in blowups of CP^2 that give rise to sharp obstructions to embeddings of E(1,a) into dilates of P(1,b) for certain a and b, which collectively give rise to infinite staircases in the embedding capacity functions of certain irrational polydisks that are analogous to the infinite staircases found by McDuff and Schlenk for balls and Frenkel and Muller for cubes.
Abstract: Thom spectra of virtual bundles give important examples of ring spectra. I'll discuss a project describing Hochschild-type invariants of such ring spectra, and talk about what this has to do with Lagrangian immersions. This talk will include an introduction to factorization homology via labeled configuration spaces.
Abstract: In this thesis talk, we will describe progress towards proving homological mirror symmetry (HMS) for the genus 2 curve in an abelian variety. HMS is known for the genus 2 surface as a symplectic manifold by work of Seidel. Here we consider it on the complex manifold side. We describe a fully faithful embedding of the bounded derived category of coherent sheaves on the genus 2 curve to the Fukaya-Seidel category of a generalized SYZ mirror constructed via methods described in Abouzaid-Auroux-Katzarkov’s paper on SYZ for hypersurfaces of toric varieties. HMS would be that these two categories are equivalent.
February 23, 2018: Chris Scaduto " Yang-Mills theory and definite intersection forms bounding homology 3-spheres "
Abstract: Using Yang-Mills instanton Floer theory, we find new constraints on the possible definite intersection forms of smooth 4-manifolds that bound integer homology 3-spheres. We will give examples of 3-manifolds such that the set of all bounding negative definite lattices consists of essentially two distinct non-standard lattices. The methods used follow the work of Froyshov.
Abstract: Recently constructed by Hendricks and Manolescu, involutive Heegaard Floer homology provides several new tools for studying the three-dimensional homology cobordism group. In this talk, we will describe the structure of the involutive Floer homology for certain connected sums of plumbed three-manifolds, and give some applications of these computations. In particular, we show that if Y is a connected sum of Seifert fibered spaces with Rokhlin invariant \mu(Y) = 1, then Y is not torsion in the homology cobordism group. This is joint work with Matthew Stoffregen.
Abstract: We study applications of Heegaard Floer homology to homology cobordism. In particular, to a homology sphere Y, we define a module HF_conn(Y), called the connected Heegaard Floer homology of Y, and show that this module is invariant under homology cobordism and isomorphic to a summand of HF_red(Y). The definition of this invariant relies on involutive Heegaard Floer homology. We use this to define a new filtration on the homology cobordism group, and to give a reproof of Furuta's theorem. This is joint work with Jen Hom and Tye Lidman.
March 9, 2018: Ziva Myer " Product Structures for Legendrian Submanifolds with Generating Families "
Abstract: Algebraic invariants of Legendrian submanifolds in standard contact 1-jet spaces have been defined through a variety of techniques. I will discuss how I am using the Morse-theoretical technique of generating families to enrich one such invariant. In particular, I have constructed a ``two-to-one" product on generating family cohomology and I will discuss current work on ways to extend this construction to A-infinity algebras and categories of generating families. The construction uses moduli spaces of Morse flow trees: spaces of intersecting gradient trajectories of functions whose critical points encode Reeb chords of the Legendrian submanifold. If time permits, I will discuss conjectured relations of these constructions to others in the field.
Abstract: Khovanov homology is a link invariant introduced by Mikhail Khovanov in 1999 as a categorification of Jones polynomial, and nicely reinterpreted by Viro in terms of Kauffman states. While conceptually simple, this definition becomes impractical when increasing the number of crossings of a link diagram. In this talk we present two alternative approaches to extreme and almost-extreme Khovanov homology. Moreover, we compare them with the Khovanov homotopy type constructed by Lipshitz and Sarkar and show some explicit examples.
Abstract: It is well known that all contact manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. What is not so well understood is what properties of a contact structure are preserved by positive contact surgeries (the case for negative contact surgeries is fairly well understood now). In this talk we will discuss some new results about positive contact surgeries and in particular completely characterize when contact r surgery is symplectically fillable when r is in (0,1].
Abstract: The braid index of a knot is the least number of strands necessary to represent it as the closure of a braid. If we view a braid as an element of the mapping class group of the punctured disk, its fractional Dehn twist coefficient measures, informally, the amount of twisting it exerts about the boundary. In this talk I will discuss joint work with Peter Feller showing that if an n-braid has fractional Dehn twist coefficient greater than n-1, then its closure is of minimal braid index, which draws a connection between braids as topological and geometric objects.
Abstract: I will discuss joint work in progress with Raphaël Rouquier on a relationship between tensor products for 2-representations of U_q(gl(1|1)) and some constructions appearing in Douglas-Manolescu's cornered Heegaard Floer homology, a type of twice-extended TQFT framework for Heegaard Floer homology.
Abstract: An important difference between high dimensional smooth manifolds and smooth four-manifolds is the ability to represent any middle dimensional homology class with a smoothly embedded sphere. For four-manifolds this is not always possible even among the simplest cases: four-manifolds $X_0(K)$, called $0$-traces, obtained by attaching an $0$-framed 2-handle to the 4-ball along a knot $K\in S^3$. The $0$ shake genus of $K$ records the minimal genus of any smooth embedded generator of the second homology of $X_0(K)$ and is clearly bounded above by the slice genus of $K$. It is conjectured that the $0$-shake genus can be strictly less than the slice genus. We prove that slice genus is not a $0$-trace invariant, and thereby provide infinitely many examples of knots with $0$-shake genus strictly less than slice genus. This resolves problem 1.41 from the Kirby list. As a corollary we give examples, via the satellite operation, of bijective maps on the smooth concordance group which fix the identity but don't preserve slice genus.
Abstract: Planar contact manifolds have been intensively studied to understand several aspects of 3-dimensional contact geometry. In this talk, we define "iterated planar contact manifolds", a higher-dimensional analog of planar contact manifolds, by using topological tools such as "open book decompositions" and "Lefschetz fibrations”. We provide some history on existing low-dimensional results regarding Reeb dynamics, symplectic fillings/caps of contact manifolds and explain some generalization of those results to higher dimensions via iterated planar structure. This is partly based on joint work in progress with J. Etnyre and B. Ozbagci.
Abstract: We shall give an overview of the information about smooth knot concordance so far extracted from quantum knot cohomologies. Joint work with Lukas Lewark. No prior knowledge assumed.
Abstract: Arboreal singularities are a class of singular Lagrangians, originally due to Nadler, which are generic in a certain sense. On the other hand, loose Legendrians are a class of Legendrians which can approximate certain "wrinkle" singularities, and from this they satisfy an h-principle by which their isotopy classes can be completely understood. This talk will focus on the relationship between them, looking at the Legendrian which is the link of an arboreal singularity. After reviewing the basics of both of these objects, we will present the main theorem, which shows that the category of constructable sheaves detects looseness for this class of Legendrians coming from arboreal singularities.
Abstract: 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 Floer’s 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.