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MIT Geometry and Topology Seminar
The seminar will run on Mondays at 3:05pm-3:55pm.
Upcoming talks:
Past talks:
February 14 (Zoom) Hokuto Konno (University of Tokyo) (UNUSUAL TIME: 6pm-7pm)
Title: Floer K-theory for knots.
Abstract: This talk is based on joint work with Jin Miyazawa and Masaki Taniguchi where we established a version of Seiberg-Witten Floer K-theory for knots. This framework is used to prove a version of “10/8-inequality for knots”, which effectively extracts difference between topological and smooth categories in knot theory. I will explain concrete applications including stabilizing numbers and relative genus bounds, as well as how we constructed this framework.
February 28 (Room 2-449): Marc Kegel (HU-Berlin).
Title: Stein traces
Abstract: Every Legendrian knot leaves a traces in the 4-dimensional symplectic world. In this talk we will investigate whetheer a 4-dimensional tracker (with the necessary mathematical education) can determine the 3-dimensional creature that left the trace. This is based on joint work with Roger Casals and John Etnyre.
March 7 (Room 2-449): Joan Licata (Australian National University).
Title: Bordered Contact Invariants via Foliated Open Books
Abstract: Open books are topological decompositions of three-manifolds that encode isotopy classes of contact structures. In this talk I'll introduce foliated open books, a recent version adapted to cutting and gluing contact manifolds with boundary. Foliated open books allow us to reinterpret familiar operations (e.g., stabilization) as instances of cutting and gluing open books, and we use this framework to define a contact invariant in bordered sutured Floer homology. This includes joint work with Alishahi, Földvári, Hendricks, Petkova, and Vértesi.
March 14 (Room 2-449): Daniel Alvarez-Gavela (MIT).
Title: K-theoretic obstructions for the smooth structure of nearby Lagrangians
Abstract: The nearby Lagrangian conjecture predicts that every exact Lagrangian embedding of a closed manifold L in the cotangent bundle of a closed manifold M is Hamiltonian isotopic to the zero section. In particular, it predicts that L and M should be diffeomorphic. It is known that L and M must have the same simple homotopy type, but so far the only known restrictions on the smooth structure are in the case of odd dimensional homotopy spheres. I will present new restrictions on the smooth structure of nearby Lagrangians, including the first known restrictions beyond the case of homotopy spheres. The method of proof uses Waldhausen's algebraic K-theory of spaces. Joint work with M. Abouzaid, S. Courte and T. Kragh.
March 28 (Room 2-449): Ailsa Keating (University of Cambridge).
Title: Symplectomorphisms of some Weinstein 4-manifolds.
Abstract: (Joint work with Paul Hacking.) We introduce two new families of four-dimensional symplectomorphisms, inspired by mirror symmetry. The basic symplectic manifolds which support these are smoothings of cusp singularities. Their topology can be described explicitly, as can their mirrors, which are associated to the dual cusps. The new symplectomorphisms are a) `Lagrangian translations', which we prove are mirror to tensors with line bundles; and b) `nodal slide recombinations', which we prove are mirror to biholomorphisms. Together with spherical twists, these are expected to generate the symplectic mapping class groups of our manifolds.
April 4 (Room 2-449): Rachael Boyd (University of Cambridge).
Title: Spaces of configurations of links .
Abstract: This talk reports on work in progress with Corey Bregman. We study the homotopy type of embedding spaces of unparameterised links, inspired by work of Brendle and Hatcher. Our main innovation is a semi-simplicial space of separating spheres, which is a combinatorial object that provides a gateway to studying the homotopy type of configurations of totally split links via the homotopy type of their individual pieces. We apply this tool to find a simple description of the fundamental group.
Friday April 22 (Room 2-449): Orsola Capovilla-Searle (UC Davis).
!!Unusual Day!! Still 3pm.
Title: Infinitely many planar Lagrangian fillings of Legendrian links .
Abstract: We provide a new family of Legendrian links with infinitely many distinct exact orientable Lagrangian fillings up to Hamiltonian isotopy. This family of links includes the first examples of Legendrian links with infinitely many distinct planar exact Lagrangian fillings, which can be viewed as the smallest Legendrian links currently known to have infinitely many distinct exact Lagrangian fillings. As an application we find examples of infinitely many exact Lagrangian spheres and tori 4-dimensional Milnor fibers of isolated hypersurface singularities with positive modality.
May 2 (Room 2-449): Tye Lidman (North Carolina State University ).
Title: The nullhomotopic knot complement problem .
Abstract: Gordon and Luecke proved that knots in the three-sphere are determined by their complements. Boileau asked if the same is true for nullhomotopic knots in three-manifolds. We extend this result to nullhomotopic knots in some other three-manifolds using standard ideas in gauge theory and low-dimensional topology. This is joint work with Ali Daemi.
Previous Semesters.
September 13 (Room 2-449): Hyunki Min (MIT)
Title: Non-Loose torus knots.
Abstract: One of the important problems in 3-dimensional contact geometry is to classify Legendrian knots or study various properties of Legendrian knots since they are a rich source of producing new contact manifolds. Ever since Eliashberg classified Legendrian unknots in the standard contact 3-sphere, there has been several attempts to extend this result. For example, Eliashberg and Fraser classified Legendrian unknots in overtwisted contact 3-spheres and Entyre and Honda classified Legendrian torus knots in the standard 3-sphere. In this talk, we will classify Legendrian torus knots in overtwisted 3-spheres. This is a joint work with John Entyre and Anubhav Mukherjee.
September 20 (Zoom): Angela Wu (Louisiana State University)
Title: Obstructing Lagrangian concordance for closures of 3-braids.
Abstract: Two knots are said to be concordant if they jointly form the boundary of a cylinder in four-dimensional Euclidean space. In the symplectic setting, we say they are Lagrangian concordant if the knots are Legendrian and the cylinder is Lagrangian. In this talk I'll show that no Legendrian knot which is both concordant to and from the unstabilized Legendrian unknot can be the closure of an index 3 braid except the unknot itself.
September 27 (Room 2-449): Kyle Hayden (Columbia University)
Title: From corks to complex curves.
Abstract: This is a story about one embedded disk in the 4-ball, and how it ties together exotic 4-dimensional phenomena, complex curves, and Khovanov homology. After explaining this disk's origins, I will use it to build properly embedded surfaces in 4-space (indeed, complex curves in C²) that are exotically knotted, i.e., isotopic through ambient homeomorphisms but not ambient diffeomorphisms. Time permitting, I'll preview how the complex curves perspective inspires the proof that Khovanov homology can detect exotic surfaces in the 4-ball (the latter of which is joint work that Isaac Sundberg will explain on Oct 4).
October 4 (Zoom): Isaac Sundberg (Bryn Mawr)
Title: The Khovanov homology of slice disks.
Abstract: A smooth, oriented surface that is properly embedded in the 4-ball can be regarded as a cobordism between the links it bounds, namely, the empty link and its boundary in the 3-sphere. To such link cobordisms, there is an associated linear map between the Khovanov homology groups of the boundary links, and moreover, these maps are invariant, up to sign, under boundary-preserving isotopy of the surface. In this talk, we review these maps and use their invariance to understand the existence and uniqueness of slice disks and other surfaces in the 4-ball. This reflects joint work with Jonah Swann and joint work with Kyle Hayden.
October 18 (Room 2-449): Robert Burklund (MIT)
Title: A classification of metastable manifolds.
Abstract: A basic goal in differential topology is to classify all smooth, closed, oriented n-manifolds up to diffeomorphism. As stated, this goal is too ambitious. However, for large n and with strong restrictions on the topology of the manifold progress becomes possible. In this talk, I will begin by discussing the classification of simply-connected manifolds whose only nonzero homology groups are H_0, H_{n/2}, and H_n, from the seminal work of Wall to its final resolution by the speaker and his collaborators. Then, I will discuss the extension of this classification into the metastable range, where the condition on the homology groups is relaxed so that H_0, H_{n/3 + e}, ..., H_{2n/3 -e} and H_n may be nonzero ( e is an error term growing sublinearly in n ). These manifolds are distinguished by the feature that the ring structure on their cohomology groups is entirely recorded by Poincare duality. Time permitting, I'll indicate how modern homotopy-theoretic methods can exploit metastability. This talk represents joint works with Andy Senger and Jeremy Hahn.
October 25 (Zoom): Christopher Davis (University of Wisconsin Eau Claire).
Title: An obstruction to concordance to boundary links
Abstract: In a pair of celebrated papers from 1990 Cochran-Orr and Livingston produced link with vanishing Milnor invariants which are not concordant to boundary links. The examples they produce sit in a larger family called homology boundary links. It is still unknown if every link with vanishing Milnor invariants is concordant to a homology boundary link. During this talk I will produce an invariant which might be able to detect such a difference. Along the way we will study the interaction of boundary links, homology boundary links, and Milnor's invariants with the solvable filtration of knot concordance introduced by Cochran-Orr-Teichner in 2003, and prove that any homology boundary link is related by concordance and a strong notion of S-equivalence. This work is joint with Shelly Harvey and JungHwan park.
November 1 (Zoom): Jonathan Zung (Princeton).
Title: Reeb flows transverse to foliations
Abstract: Eliashberg and Thurston showed that (almost all) C^2 taut foliations on 3-manifolds can be approximated by tight contact structures. I will explain a new approach to this theorem which allows one to control the resulting Reeb flow and hence produce many hypertight contact structures. Along the way, I will explain how harmonic transverse measures may be used to understand the holonomy of foliations.
November 8 (Zoom): Akram Alishahi (University of Georgia).
Title: Upsilon-like homomorphisms from homology cobordism group of homology cylinders
Abstract: For a surface of genus g with n boundary components, the homology cobordism group of homology cylinders H_{g,n} is defined as an enlargement of the mapping class group by Graoufaldis and Levine. In this talk, we define a family of homomorphisms from H_{0,n} to the real numbers so that they generalize both the d-invariant from Heegaard Floer homology (n=1) and the upsilon invariant from knot Floer homology (n=2).
November 15 (Zoom): Fan Ye (Univesity of Cambridge)
Title: A large surgery formula for instanton Floer homology.
Abstract: In Heegaard Floer homology, Oszváth-Szabó and Rasmussen introduced a large surgery formula computing HF^\hat(S^3_m(K)) for any knot K and large integer m by bent complexes from CFK^-(K). In this talk, I'll introduce a similar formula for instanton Floer homology. More precisely, I construct two differentials on the instanton knot homology KHI(K) and use them to compute the framed instanton homology I^#(S^3_m(K)) for any large integer m. As an application, I show that if the coefficients of the Alexander polynomial of K are not ±1, then there exists an irreducible SU(2) representation of the fundamental group of S^3_r(K)) for all but finitely many rational r. In particular, all hyperbolic alternating knots satisfy this condition. Also by this large surgery formula, I show KHI(K)=HFK^\hat(K) for any Berge knot and I^#(S^3_r(K))=HF^\hat(S^3_r(K)) for any genus-one alternating knot. This is a joint work with Zhenkun Li.
November 22 (Zoom): Sucharit Sarkar (UCLA) (UNUSUAL TIME: 4pm-5pm)
Title: Mixed invariants in Khovanov homology for unorientable cobordisms.
Abstract: Using Bar-Natan's and Lee's deformations of Khovanov homology of links, we define minus, plus, and infinity versions of Khovanov homology. Given an unorientable cobordism in [0,1]\times S^3 from a link L_0 to a link L_1, we define a mixed invariant as a map from the minus version of the Khovanov homology of L_0 to the plus version of the Khovanov homology of L_1. The construction is similar to the mixed invariant in Heegaard Floer homology. This invariant can be used to distinguish exotic cobordisms, that is, two cobordisms which are topologically isotopic but not smoothly isotopic. This is joint with Robert Lipshitz.
November 29 (Room 2-449): Shira Tanny (Institute for Advanced Study),
Title: Floer homology of Hamiltonians supported on subsets.
Abstract: Floer homology is a fundamental construction relating dynamical properties of Hamiltonian flows on symplectic manifolds to the topology of the manifold. Although this construction is global in nature, when the Hamiltonian flow is supported on a subset one would like to "localize" this construction, or at least some of its consequences. I'll discuss several results in this direction. All symplectic preliminaries will be explained.
December 6 (Room 2-449): Agustin Moreno (Institute for Advanced Study),
Title: On theoretical and practical aspects of the restricted 3-body problem.
Abstract: Despite the fact that the 3-body problem is an ancient conundrum that goes back to Newton, it is remarkably poorly understood, and is still a benchmark for modern developments. In this talk, I will give a (very) biased account of this classical problem, both from a modern theoretical perspective, i.e. outlining possible lines of attack coming from Symplectic Geometry, holomorphic curves and Floer theory; as well as comment on practical and numerical aspects, within the context of finding orbits for space mission design and ocean worlds exploration.
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