MIT Geometry & Topology Seminar

This is the website for the weekly Geometry & Topology seminar at MIT.
The seminar meets 3:00-4:00PM, Mondays at 2-132 .

It is organized by the MIT Geometry and Topology group.
The seminar focuses on the areas of geometric topology, contact and symplectic topology,
geometric analysis and gauge theory among others.

Seminars Spring 2017

Date Time Speaker Title
Feb 13 3:00 Eli Grigsby (Boston College) Cancelled due to Snowstorm
Feb 20 3:00 Presidents Day George Washington's Birthday (1732)
Feb 27 3:00 Cagatay Kutluhan (Buffalo U) Filtering the Heegaard Floer contact invariant
Mar 6 3:00 Denis Auroux (IAS-UC Berkeley) Monotone Lagrangian tori in vector spaces and projective spaces
Mar 13 3:00 Matthew Hedden (MSU) Floer homology and fractional Dehn twists
Mar 20 3:00 Maylis Limouzineau (IMJ-PRG) Operations on Legendrian submanifolds and Generating Functions
Apr 3 3:00 Nathan Dunfield (UIUC) Floer homology, group orders, and taut foliations of hyperbolic 3-manifolds
Apr 10 3:00 Tim Perutz (UT Austin) Floer theory in moduli spaces of stable pairs over Riemann surfaces
Apr 17 3:00 Patriots' Day Battles of Lexington and Concord (1775)
May 1 3:00 Honghao Gao (NU-Paris 6) Augmentations and sheaves for knot conormals
May 8 3:00 Oleg Lazarev (Stanford) Contact manifolds with flexible fillings

Abstracts Spring 2017
  1. Feb 13: Annular Khovanov-Lee homology, Braids, and Cobordisms,
    by Eli Grigsby.

    Khovanov homology associates to a knot K in the three-sphere a bigraded vector space arising as the homology groups of an abstract chain complex. Using a deformation of Khovanov's complex, due to Lee, Rasmussen defined an integer-valued knot invariant he called s(K) that gives a lower bound on the 4-ball genus of knots, sharp for knots that can be realized as quasipositive braid closures.


  2. Feb 27: Filtering the Heegaard Floer contact invariant,
    Cagatay Kutluhan.

    I will talk about a refinement of the contact invariant in Heegaard Floer homology. Using the Heegaard Floer chain complex, we define a contact invariant, called (spectral) order, which behaves well under Legendrian surgery and is computable from an arbitrary supporting open book decomposition. After giving some background and explaining the origin of our invariant, I will talk about some of its key properties and present some examples.This is joint work with Gordana Matic, Jeremy Van Horn-Morris, and Andy Wand.


  3. Mar 6: Monotone Lagrangian tori in vector spaces and projective spaces,
    Denis Auroux.

    A basic open problem in symplectic topology is to classify Lagrangian submanifolds (say, up to Hamiltonian isotopy) in a given symplectic manifold. In recent years, ideas from mirror symmetry have led to the realization that even the simplest symplectic manifolds (eg. vector spaces or complex projective spaces) contain many more Lagrangian tori than previously thought. We will present some of the recent developments on this problem, and conjectures.


  4. Mar 13: Floer homology and fractional Dehn twists,
    Matthew Hedden.

    The fractional Dehn twist coefficient is a quasimorphism on the mapping class group of a surface with boundary which, in some sense, measures the distance to the Nielsen-Thurston representative. It is closely related to contact geometry in dimension 3. I'll discuss this invariant and show that it offers a geometric interpretation of Heegaard Floer homology invariants. This is joint work with Tom Mark, and more recent work in progress with John Baldwin.


  5. Mar 20: Elementary operations on Legendrian submanifolds and Generating Functions,
    Maylis Limouzineau.

    We see Legendrian submanifolds as generalized functions. This point of view inspires naturally the operations of sum and convolution of Legendrian submanifolds. These two operations are linked by a contact transformation, a kind of Legendre transform. We compare those geometric notions with three classical operations of convex analysis: the sum, the inf-convolution and the Legendre-Fenchel transformed of functions. More than Legendrian geometry, this comparison requires the use of generating function theory, and the introduction of the notion of min-max selector.


  6. Apr 3: Floer homology, group orders, and taut foliations of hyperbolic 3-manifolds,
    Nathan Dunfield.

    A bold conjecture of Boyer-Gorden-Watson and others posit that for any irreducible rational homology 3-sphere M the following three conditions are equivalent: (1) the fundamental group of M is left-orderable, (2) M has non-minimal Heegaard Floer homology, and (3) M admits a co-orientable taut foliation. Very recently, this conjecture was established for all graph manifolds by the combined work of Boyer-Clay and Hanselman-Rasmussen-Rasmussen-Watson. I will discuss a computational survey of these properties involving half a million hyperbolic 3-manifolds, including new or at least improved techniques for computing each of these properties.


  7. Apr 10: Floer theory in moduli spaces of stable pairs over Riemann surfaces,
    Tim Perutz.

    The SU(2) representation variety of a closed Riemann surface is a singular symplectic space; the singularities give rise to notorious difficulties in formulating the Floer cohomology for the Lagrangians arising from a pair of bounding handlebodies. Can one get circumvent such difficulties using spaces of rank 2 stable pairs over the Riemann surface, as studied by Bradlow and by Thaddeus? Is there an analogue of Heegaard Floer theory for 3-manifolds based on these spaces? I present some preliminary explorations of these questions. ln joint work with Andrew Lee, we isolate particular degree and stability parameters for the stable-pair space that seem most promising - the analogue of the g-fold symmetric product in Heegaard Floer theory. We do not yet have a Heegaard Floer-type theory, but we study instead the fixed-point Floer homology for the action of the mapping class group on the space of stable pairs. Our main result is a calculation showing that, in genus 1, the fixed-point Floer homology is (essentially) a summand in the monopole Floer homology of the mapping torus.


  8. May 1: Augmentations and sheaves for knot conormals,
    Honghao Gao.

    Knot invariants can be defined using Legendrian isotopy invariants of the knot conormal. There are two types of invariants raised in this way: one is the knot contact differential graded algebra together with augmentations associated to this dga, and the other one is the category of simple sheaves microsupported along the knot conormal. Nadler-Zaslow correspondence suggests a connection between the two types of invariants. Moreover, augmentations specialized to Q=1 have been understood through KCH representations. I will present a classification result of simple sheaves, and relate it to KCH representations and two-variable augmentation polynomials. I will also present a Radon transform for sheaf categories, and explain how it corresponds to the specialization of Q on the sheaf side.


  9. May 8: Contact manifolds with flexible fillings,
    Oleg Lazarev (Stanford).

    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.


Seminars Fall 2016

Date Time Speaker Title
Sep 12 3:00 Adam Jacob (Harvard) A generalization of special Lagrangian graphs
Sep 19 3:00 Boyu Zhang (Harvard) A monopole invariant for foliations without transverse invariant measure
Sep 26 3:00 Paul Feehan (Rutgers) The Lojasiewicz-Simon gradient inequality and applications to energy discreteness and gradient flows in gauge theory
Oct 3 3:00 John Baldwin (BC) Stein fillings and SU(2) representations
Oct 17 3:00 Ziva Myer
(Bryn Mawr)
An A-Algebra for Legendrian Submanifolds with Generating Families
Oct 24 3:00 Daniel Ruberman (Brandeis) Codimension-one Heegaard Floer invariants
Oct 31 3:00 Lucas Culler (Princeton)
Nov 7 3:00 Yi Xie
(Simons Center)
Unitary Representations of the Knot Group and Floer Homology
Nov 14 3:00 Dan Rutherford
(Ball State)
Cellular Legendrian contact homology for surfaces
Nov 21 3:00 Luis Diogo (Columbia) Monotone Lagrangians in cotangent bundles
Nov 28 3:00 Jo Nelson (Columbia) Symplectic embeddings of four-dimensional polydisks into balls
Dec 5 3:00 Eva Miranda (UPC) Desingularizing bm-symplectic structures

Abstracts Fall 2016
  1. Sep 12: A generalization of special Lagrangian graphs,
    by Adam Jacob.

    In this talk I will define a complex version of the special Lagrangian graph equation with potential introduced by Harvey-Lawson. I will discuss existence of solutions on a compact Kahler manifold under certain analytic assumptions. I will also introduce a conjecture linking existence in general to an algebro-geometric notion of stability, and describe how the setup relates to SYZ mirror symmetry. This is joint work with Tristan C. Collins and S.-T. Yau.


  2. Sep 19: A monopole invariant for foliations without transverse invariant measure,
    by Boyu Zhang.

    Let M be a three manifold. Given a taut foliation F on M, there is a natural way to define a symplectic structure on MxR. If we further assume that the foliation does not admit any transverse invariant measure, then the symplectic structure can be taken to be exact. In the talk I will define an invariant for such a foliation F by counting solutions of a version of the Seiberg-Witten equations on MxR perturbed by this symplectic structure. The invariant takes value in the monopole Floer homology groups of M. It is invariant under deformations of F and and its image in the reduced Floer homology group is always nonzero. I will then give some topological applications of the invariant.


  3. Sep 26: The Lojasiewicz-Simon gradient inequality and applications to energy discreteness and gradient flows in gauge theory,
    by Paul Feehan.

    The Lojasiewicz-Simon gradient inequality is a generalization, due to Leon Simon (1983), to analytic or Morse-Bott functionals on Banach manifolds of the finite-dimensional gradient inequality, due to Stanislaw Lojasiewicz (1963), for analytic functions on Euclidean space. We shall discuss several recent generalizations of the Lojasiewicz-Simon gradient inequality and a selection of their applications, such as global existence and convergence of Yang-Mills gradient flow over four-dimensional manifolds and discreteness of the energy spectrum for harmonic maps from Riemann surfaces into analytic Riemannian manifolds.


  4. Oct 3: Stein fillings and SU(2) representations,
    by John Baldwin.

    In recent work, Sivek and I defined invariants of contact 3-manifolds with boundary in sutured instanton Floer homology. I will sketch the proof of a theorem about these invariants which is analogous to a result of Plamenevskaya in Heegaard Floer homology: if a 4-manifold admits several Stein structures with distinct Chern classes, then the invariants of the induced contact structures on its boundary are linearly independent. As a corollary, we conclude that if a homology sphere Y admits a Stein filling which is not a homology ball, then its fundamental group admits a nontrivial representation to SU(2). This is joint work with Steven Sivek.


  5. Oct 17: An A-Algebra for Legendrian Submanifolds with Generating Families,
    by Ziva Myer.

    In contact geometry, invariants of Legendrian submanifolds in 1-jet spaces have been obtained through a variety of techniques. I will discuss how I am enriching one invariant, Generating Family Cohomology, to an A- algebra. 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. I will focus my talk on the construction of a 2-to-1 product and discuss how it lays the foundation for the A-algebra.


  6. Oct 24: Codimension-one Heegaard Floer invariants,
    by Daniel Ruberman.

    Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented 4-manifold X with the homology of S1xS3. Froyshov showed that if there is a rational homology sphere Y embedded in X representing a generator of H3(X), then his h-invariant (equivalently the Ozsvath-Szabo d-invariant) depends only on X. We show how to define such an invariant without the assumption that Y be a homology sphere, using a suitable version of the Heegaard Floer d-invariant of Y, defined using twisted coefficients. We show how this invariant can be used to obstruct embeddings of certain types of 3-manifolds, including those obtained as a connected sum of a rational homology 3-sphere and any number of copies of S1xS2. We also give similar obstructions to embeddings in certain open 4-manifolds, including exotic R4. This is joint work with Adam Levine.


  7. Oct 31: Link Surgery and the Tate Curve,
    by Lucas Culler.

    In this talk I will describe the derived category of coherent sheaves on a family of degenerating Abelian varieties, the n-fold self product of the Tate curve, and explain how this category is relevant to the link surgery formula in Heegaard Floer homology.


  8. Nov 7: Unitary Representations of the Knot Group and Floer Homology,
    by Yi Xie.

    A sutured manifold is a 3-manifold with a certain decoration on its boundary. As an example, the complement of any knot determines a sutured manifold. In this talk, I will discuss a Floer homology for sutured manifolds. The definition of this invariant is based on studying a version of instanton Floer homology for the product of a Riemann surface and a circle. Our main tool to study this Floer homology group is the generalization of Donaldson invariants to higher rank bundles. Our Sutured Floer homology gives rise to a knot invariant and I will also explain how this invariant is related to SU(3) representations of the knot group. This talk is based on a joint work with Aliakbar Daemi.


  9. Nov 14: Cellular Legendrian contact homology for surfaces,
    by Dan Rutherford.

    This is joint work with Mike Sullivan. We consider a Legendrian surface L in R5 or more generally in the 1-jet space of a surface. Such a Legendrian can be conveniently presented via its front projection which is a surface in R3 that is immersed except for certain standard singularities. We associate a differential graded algebra (DGA) to L by starting with a cellular decomposition of the base projection to R2 of L that contains the projection of the singular set of L in its 1-skeleton. A collection of generators is associated to each cell, and the differential is determined in a formulaic manner by the nature of the singular set above the boundary of a cell. Our cellular DGA is equivalent to the Legendrian contact homology DGA of L whose construction was carried out in this setting by Etnyre-Ekholm-Sullivan with the differential defined by counting holomorphic disks in C2 with boundary on the Lagrangian projection of L. Equivalence of our DGA with LCH is established using work of Ekholm on gradient flow trees. Time permitting, we will discuss constructions of augmentations of the cellular DGA from two parameter families of functions.


  10. Nov 21: Monotone Lagrangians in cotangent bundles,
    by Luis Diogo.

    We show that there is a 1-parameter family of monotone Lagrangians in cotangent bundles of spheres with the following property: every orientable spin closed monotone Lagrangian with non-trivial Floer cohomology is not Hamiltonian-displaceable from either the zero-section or one of the Lagrangians in the family. The proof involves studying a version of the wrapped Fukaya category of the cotangent bundle which includes monotone Lagrangians. This is joint work with Mohammed Abouzaid.


  11. Nov 28: Symplectic embeddings of four-dimensional polydisks into balls,
    by Jo Nelson.

    I will talk about recent joint work with Katherine Christianson, which yields new sharp obstructions to symplectic embeddings of the four-dimensional polydisk into the ball, extending results of Hind-Lisi, Hutchings, and Schlenk. Our proof relies on connections between combinatorial toric geometry and embedded contact homology by way of a necessary criterion for one "convex toric domain" to symplectically embed into another, which was introduced by Hutchings in 2015.


  12. Dec 5: Desingularizing bm-symplectic structures,
    by Eva Miranda.

    Several problems in celestial mechanics (like the elliptic restricted 3-body problem) and their singularities (collisions) can be described using symplectic forms away from a critical set (known in the literature of celestial mechanics as the line at infinity or the collision manifold). In these examples the symplectic form either vanishes or goes to infinity along the critical set. It is possible to give a global description of these objects using bm-symplectic forms and folded symplectic forms. In this talk we will explain a desingularization procedure called deblogging which associates a family of symplectic forms or folded symplectic forms to a given bm-symplectic form depending on the parity of m. Time permitting, several applications of this procedure will be discussed. This talk is based on joint work with Victor Guillemin and Jonathan Weitsman.