# Topology Seminar

Date Time |
Location | Speaker | Title – click for abstract | |
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08/264:00pm |
Zoom | Hongbin Sun Rutgers University - New Brunswick |
Subgroup separability and subgroup distortion of 3-manifold groups For a (finitely generated) subgroup of a (finitely generated) group, we will consider two properties of this subgroup: the separability of this group and its subgroup distortion. The separability of a subgroup measures whether the property of an element not lying in this subgroup is visible by taking some finite quotient. We will give a characterization on whether a subgroup of a 3-manifold group is separable. The subgroup distortion compares the intrinsic and extrinsic geometries of a subgroup. For an arbitrary subgroup of a 3-manifold group, we prove that the subgroup distortion can only be linear, quadratic, exponential and double exponential. It turns out that these two properties (subgroup separability and subgroup distortion) are closed related for subgroups of 3-manifold groups. The subgroup distortion part is joint work with Hoang Nguyen. | |

09/164:00pm |
Zoom | Guangbo Xu Texas A&M University |
Introduction to the Atiyah-Floer conjectureThe Atiyah-Floer conjecture states that the instanton Floer homology of certain three-dimensional manifold M (constructed using gauge theory) is isomorphic to the Lagrangian Floer homology associated to M and a separating surface Σ (constructed using symplectic geometry). In this talk I will review the construction of these two Floer theories and explain Atiyah's original idea about the correspondence between them. The talk will be expository and accessible to graduate students. Video recording available at: https://tamu.zoom.us/rec/share/n2pv7TxIWfOU2jq-BSpI0Z3sUPey1RW5YTQkiHzfYnBhzFpgRv8LehuPiHjqW5on.kCygmvYBTG_2fHW3 (Access Password: =S6$Y7GA) | |

09/234:00pm |
Zoom | Tianqi Wu Harvard CMSA |
Koebe circle domain conjecture and the Weyl problem in hyperbolic 3-spaceIn 1908, Paul Koebe conjectured that every open connected set in the plane is conformally diffeomorphic to an open connected set whose boundary components are either round circles or points. The Weyl problem, in the hyperbolic setting, asks for isometric embedding of surfaces of curvature at least -1 into the hyperbolic 3-space. We show that there are close relationships among the Koebe conjecture, the Weyl problem and the work of Alexandrov and Thurston on convex surfaces. This is a joint work with Feng Luo. Video recording available at: https://tamu.zoom.us/rec/share/t8t6lUEjibukQk5UPeAh66Rx_jGOec7_UYSi_WfrnesukxcH-bI86x1a30Sz73pC.6BM95cH7-fL_aLiu (Access Password: GqBQ^y85)
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10/074:00pm |
ZooM | Prasit Bhattacharya University of Notre Dame |
The stable Adams conjectureThe Adams conjecture, perhaps one of the most celebrated results in the subject of stable homotopy theory, was resolved by Quillen and Sullivan independently in the 1970s. Essentially, the Adams conjecture says that the q-th Adams operation composed with the J-homomorphism can be deformed continuously to the J-homomorphism itself if localized away from q. The stable enhancement of the Adams conjecture (which is only possible in the complex case) claims that this deformation can be achieved within the space of infinite loop maps from BU to the classifying space of spherical bundles. We recently found that the only accepted proof of the stable Adams conjecture, which is due to Friedlander (1980), has a mistake. In this talk, I will explain the mistake, reformulate the statement of the stable Adams conjecture, sketch our new proof of the stable Adams conjecture and discuss some of the ramifications. This is joint work with N. Kitchloo. | |

10/144:00pm |
Zoom | Shaoyun Bai Princeton University |
Equivariant Cerf theory and SU(n) Casson invariantsI will present joint work with Boyu Zhang on defining perturbative SU(n)-Casson invariant for integer homology 3-spheres. Ideas from the studies of J-holomorphic curves in symplectic manifolds play important roles in the construction, in particular issues related to equivariant transversalities. Time permitting, I will discuss about possible generalizations and topological applications. Video recording available at https://tamu.zoom.us/rec/share/GoNJG5eazmSjWS8r3p_eXLSjQ48Zr90BerbhmecnJal8jCO1Dd84e1-wIjTYE-gm.5eOBWKDp_eiopFPR (Access Password: u5d2W?mS) | |

10/214:00pm |
Zoom | Sara Maloni University of Virginia |
Convex hulls of quasicircles in hyperbolic and anti-de Sitter space.Thurston conjectured that quasi-Fuchsian manifolds are determined by the induced hyperbolic metrics on the boundary of their convex core and Mess generalized those conjectures to the context of globally hyperbolic AdS spacetimes. In this talk I will discuss a universal version of these conjectures (and prove the existence part) by considering convex sets spanning quasicircles in the boundary at infinity of hyperbolic and anti-de Sitter space. This work generalizes Alexandrov and Pogorelov's results about the characterization metrics induced on the boundary of a compact convex subset of hyperbolic space. Time permitting, we will discuss why in hyperbolic space quasicircles can't be characterized by the width of their convex hulls, except when the convex hulls have small width. This is different than the anti-de Sitter setting, as Bonsante and Schlenker showed. (This is joint work with Bonsante, Danciger and Schlenker.) Video recording is available at https://tamu.zoom.us/rec/share/_A6ZUaZTjFW0EvL90F_z22eLAHewUPSV4dQZ9kLHTYNLsG0VX61lv1kWtidJ1j3R.fia_NZuVr1sEYoMO (Access Password: eEniSG*4)
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10/284:00pm |
zoom | Paul VanKoughnett Purdue University |
Topological modular forms and their co-operationsTopological modular forms (TMF) is a cohomology theory built out of elliptic curves. I'll describe what TMF is, how it's constructed, and how it's been useful in topology, as well as some of the more speculative recent attempts to extend its construction. I'll then talk about the problem of calculating TMF co-operations, which are an essential input to serious computations using TMF. Part of this object -- the part that is obtained from ordinary elliptic curves -- admits a simple algebraic description, as well as an interesting relationship with number theory. This is joint work with Dominic Culver. | |

11/044:00pm |
Zoom | Sanjay Kumar Michigan State University |
Fundamental shadow links realized as links in the 3-sphere In this talk, I will discuss two conjectures which relate quantum topology and hyperbolic geometry. Chen and Yang conjectured that the asymptotics of the Turaev-Viro invariants determine the hyperbolic volume of the 3-manifold, and Andersen, Masbaum, and Ueno (AMU) conjectured for a surface that the asymptotics of the quantum representations reflect certain geometric properties of the mapping class group. For a manifold M(f) constructed as the mapping tori of an element f in the mapping class group, Detcherry and Kalfagianni showed that M(f) satisfying the Turaev-Viro invariant volume conjecture implies that f satisfies the AMU conjecture. Using techniques from Turaev's shadow theory, I construct infinite families of links in the 3-sphere with complement homeomorphic to the complement of fundamental shadow links which are a class of links in connected sums of S^2 times S^1 that satisfy the Turaev-Viro invariant volume conjecture. Through homeomorphisms, these link complements in S^3 can be realized as the mapping tori for explicit elements in the mapping class group providing families that satisfy the AMU conjecture. Video recording is available at https://tamu.zoom.us/rec/share/NVDt4tmOzD6DyGdOfO8XTjBCMBG0d_BK4grFJPN35hrSa7b-N_0rD1MS_OfLEOWK.OV72bu9Mc4g3cwyL (Access Password: 13TuT+XF) | |

11/114:00pm |
Zoom | Fenglong You University of Alberta |
Relative Gromov-Witten theory and mirror symmetryGromov-Witten invariants are rational numbers that count curves in algebraic varieties or symplectic manifolds. Given a smooth projective variety X and a codimension one subvariety (i.e. a divisor) D, relative Gromov-Witten invariants count curves in X with tangency conditions along D. While absolute Gromov-Witten theory is known to have many nice structural properties, such as quantum cohomology, WDVV equation, Givental's formalism, Cohomological field theory (CohFT) etc., parallel structural properties were unknown for relative Gromov-Witten theory until recently. In this talk, I will give an overview of some recent progress in relative Gromov-Witten theory including several structural properties and applications to mirror symmetry. | |

11/184:00pm |
Zoom | Honghao Gao Michigan State University |
Legendrian invariants, Lagrangian fillings and cluster algebrasClassifications of Legendrian knots and their exact Lagrangian fillings are central questions in low-dimensional contact and symplectic topology. Recent development suggests that one can use cluster seeds to distinguish exact Lagrangian fillings. It requires a filling-to-cluster functoriality over a moduli space of Legendrian invariants. This invariant can be sheaf theoretic (Shende-Treumann-Williams-Zaslow) or Floer theoretic (joint work with Linhui Shen and Daping Weng). As an application, I will explain how to use Legendrian loops and cluster algebras to construct infinitely many exact Lagrangian fillings for most torus links (joint work with Roger Casals, using sheaf-theoretic invariants), and for most positive braid links (joint work with Linhui Shen and Daping Weng, using Floer-theoretic invariants). |