This is the website for the weekly Geometry & Topology seminar at MIT.
Although inperson 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.
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 TristramLevine Knot Signatures as a Singular FurutaOhta 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. infinitedimensional cycles  Mar. 16 
 No speaker   Mar. 30  3pm  Sherry Gong (UCLA) (Virtual)  On the KronheimerMrowka 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 4spheres  May 11  3pm  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 Extalgebra of the corresponding smooth point on the mirror. In the presence of holomorphic discs one obtains an Ainfinity 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 KronheimerMrowka 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. infinitedimensional cycles
By Piotr Suwara
Abstract: Floer's idea of obtaining homological invariants in symplectic and lowdimensional topology via a construction of a MorseSmaleWitten chain complex, computing the "middledimensional homology" of some specific infinitedimensional 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 "semiinfinite dimensional cycles". Following the (unpublished) work of Mrowka and Ozsváth, Lipyanskiy defined Floer homology and Floer bordism groups using semiinfinite dimensional cycles for a class of socalled 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 semiinfinite Floer homology and argue why it recovers the tilde flavor (i.e. the nonequivariant 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 3manifold, 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 OszvathSzabo contact class in Heegaard Floer homology. This is joint work with Alishahi, Foldvari, Hendricks, Licata, and Vertesi.
 Feb. 24 A Generalization of the TristramLevine Knot Signatures as a Singular FurutaOhta Invariant for Tori.
By Mariano Echeverria
Abstract: Given a knot K inside an integer homology sphere Y, the CassonLinHerald 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 TristramLevine signature of the knot associated to the conjugacy class chosen. Turning things around, given a 4manifold 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 CassonLinHerald invariant of the knot in the product case, thus it can be regarded as implicitly defining a TristramLevine signature for tori. This count can also be considered as a singular FurutaOhta 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, OzsvathSzabo 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 OzsvathSzabo'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 OzsvathSzabo'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 ArkaniHamed and Trnka to compute scattering amplitudes in maximally supersymmetric gauge theory.
 Feb. 3 Khovanov homology detects split links
By Sucharit Sarkar
Abstract: Extending ideas of HeddenNi, 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.

