MIT Geometry and Topology Seminar

This is the website for the weekly Geometry & Topology seminar at MIT.

Although in-person seminar are canceled for the remainder of the semester, we will keep organizing virtual seminars.

If you would like to join the zoom seminar at 3pm, please email the organizers to get the password to the zoom link.

To find more seminars in the Boston area, please find in the shared google calendar.

The previous talks can be found here.

  • Spring 2020

     Date Time Speaker Title
     Feb. 3 3pm Sucharit Sarkar (UCLA)  Khovanov homology detects split links
     Feb. 10 3pm Andrew Manion (USC) Heegaard Floer algebras, hypertoric varieties, and the amplituhedron
     Feb. 24 3pm Mariano Echeverria (Rutgers)A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for Tori.
     Mar. 2 3pm Ina Petkova  (Dartmouth)A contact invariant from bordered Heegaard Floer homology
     Mar. 9 3pm Piotr Suwara (MIT) Constructing Floer homologies: Morse theory vs. infinite-dimensional cycles
     Mar. 16
     No speaker 
     Mar. 30 3pm Sherry Gong (UCLA) (Virtual) On the Kronheimer-Mrowka concordance invariant
     Apr. 6 3pm Jack Smith (Cambridge)(Virtual) Exterior algebras and local mirror symmetry
     Apr. 13 3pm Kouichi Yasui (Osaka U)* (Canceled) 
     Apr. 27  No speaker 
     May 4 3pm

    Jeff Meier (Western Washington University)(Virtual)

     Generalized square knots and homotopy 4-spheres
     May 113pm Honghao Gao (MSU)(Virtual)Infinitely many Lagrangian fillings

* to be confirmed.
  • May 11 Infinitely many Lagrangian fillings
    By Honghao Gao 

    Abstract: A filling is an oriented surface bounding a link. Lagrangian fillings can be constructed via local moves in finite steps, but it was unknown whether a Legendrian link could admit infinitely many Lagrangian fillings. In this talk, I will show that Legendrian torus links other than (2,m), (3,3), (3,4), (3,5) indeed have infinitely many fillings. These fillings are constructed using Legendrian loops, and proven to be distinct using the microlocal theory of sheaves and the theory of cluster algebras. This is a joint work with Roger Casals.

  • Apr. 6 Exterior algebras and local mirror symmetry
    By Jack Smith

    Abstract: The exterior algebra plays an important role in mirror symmetry, as the Floer algebra of a Lagrangian torus bounding no holomorphic discs and as the Ext-algebra of the corresponding smooth point on the mirror.  In the presence of holomorphic discs one obtains an A-infinity deformation of this picture, and I'll explain how to classify such deformations under a monotonicity hypothesis.  This leads to a simple proof that the Floer algebra of a monotone torus is the endomorphism algebra of the expected matrix factorization of its superpotential, as well as a purely algebraic result generalizing the classification of Clifford algebras by quadratic forms.

  • Mar 30  On the Kronheimer-Mrowka concordance invariant
    By Sherry Gong

    Abstract: We will talk about Kronheimer and Mrowka’s knot concordance invariant, $s^\sharp$. We compute the invariant for various knots. Our computations reveal some unexpected phenomena, including that $s^\sharp$ differs from Rasmussen's invariant $s$, and that it is not additive under connected sums. We also generalize the definition of $s^\sharp$ to links by giving a new characterization of the invariant 
    in terms of immersed cobordisms.

  • Mar.9 Constructing Floer homologies: Morse theory vs. infinite-dimensional cycles
    By Piotr Suwara

    Abstract: Floer's idea of obtaining homological invariants in symplectic and low-dimensional topology via a construction of a Morse-Smale-Witten chain complex, computing the "middle-dimensional homology" of some specific infinite-dimensional spaces, led to the construction of groundbreaking invariants known collectively under the name of "Floer homology". Soon after the emergence of these ideas Atiyah suggested thinking about Floer theories as describing the homology of "semi-infinite dimensional cycles". Following the (unpublished) work of Mrowka and Ozsváth, Lipyanskiy defined Floer homology and Floer bordism groups using semi-infinite dimensional cycles for a class of so-called Floer spaces. The construction is appealing since it does not require considering the compactification of moduli spaces of trajectories, which is a subtle analytical procedure, or perturbing the Floer functional, which is usually a serious obstacle for defining equivariant Floer homologies. I will describe the construction of this semi-infinite Floer homology and argue why it recovers the tilde flavor (i.e. the non-equivariant version) of Monopole Homology (assuming no reducible solutions).

  • Mar. 2   A contact invariant from bordered Heegaard Floer homology
    By Ina Petkova

    Abstract: Given a contact structure on a bordered 3-manifold, we describe an invariant which takes values in the bordered sutured Floer homology of the manifold. This invariant satisfies a nice gluing formula, and recovers the Oszvath-Szabo contact class in Heegaard Floer homology. This is joint work with Alishahi, Foldvari, Hendricks, Licata, and Vertesi.

  • Feb. 24 A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for Tori.
    By Mariano Echeverria

    Abstract: Given a knot K inside an integer homology sphere Y, the Casson-Lin-Herald invariant can be interpreted as a signed count of conjugacy classes of irreducible representations of the knot complement into SU(2) which map the meridian of the knot to a fixed conjugacy class. It has the interesting feature that it determines the Tristram-Levine signature of the knot associated to the conjugacy class chosen. Turning things around, given a 4-manifold X with the integral homology of S1 × S3, and an embedded torus which is homologically nontrivial, we define a signed count of conjugacy classes of irreducible representations of the torus complement into SU(2) which satisfy an analogous fixed conjugacy class condition to the one mentioned above for the knot case. Our count recovers the Casson-Lin-Herald invariant of the knot in the product case, thus it can be regarded as implicitly defining a Tristram-Levine signature for tori. This count can also be considered as a singular Furuta-Ohta invariant, and it is a special case of a larger family of Donaldson invariants which we also define. 

  • Feb. 10  Heegaard Floer algebras, hypertoric varieties, and the amplituhedron (joint with A. Lauda and A. Licata)
    By Andrew Manion

    Abstract: Recently, Ozsvath-Szabo showed that their 2016 theory of "bordered knot Floer homology" does indeed compute knot Floer homology. This theory has a rich algebraic structure with many relationships to other areas of mathematics. I will sketch the physical framework into which Heegaard Floer homology is supposed to fit, along with the expected role of Ozsvath-Szabo's bordered knot Floer homology in the overall framework. Then I will discuss a new observation that apparently goes beyond the existing physical framework, namely that Ozsvath-Szabo's algebras from bordered knot Floer homology can be viewed as convolution algebras for certain hypertoric varieties whose associated polytopes are relatives of the "amplituhedron" as introduced by Arkani-Hamed and Trnka to compute scattering amplitudes in maximally supersymmetric gauge theory.

  • Feb. 3 Khovanov homology detects split links
    By Sucharit Sarkar

    Extending ideas of Hedden-Ni, we show that the module structure on Khovanov homology detects split links. The proof is via an analogous result for untwisted Heegaard Floer homology of the branched double cover. Along the way, we will describe an interpretation of the module structure on untwisted Heegaard Floer homology in terms of twisted Heegaard Floer homology. This is joint with Robert Lipshitz.