Time and venueThe LSU Virtual Topology Seminar meets in 233 Lockett Hall, at 3:30 pm CT on Wednesdays.

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Schedule The schedule is updated throughout the academic year, and only confirmed speakers are listed below.Date | Speaker | Institution | Title |
---|---|---|---|

Aug 29, 2018 | Ignat Soroko | Louisiana State University | Dehn functions of subgroups of right-angled Artin groups |

Sep 5, 2018 | Andrew Zimmer | Louisiana State University | Limit sets of discrete subgroups |

Sep 26, 2018 | Yilong Wang | Louisiana State University | Modular tensor categories and Reshetikhin–Turaev TQFTs I |

Oct 10, 2018 | Yilong Wang | Louisiana State University | Modular tensor categories and Reshetikhin–Turaev TQFTs II |

Oct 17, 2018 | Joshua Sabloff | Haverford College | Length and width of Lagrangian cobordisms |

Oct 24, 2018 | Scott Baldridge | Louisiana State University | A new cohomology for planar trivalent graphs with perfect matchings |

Oct 31, 2018 | Matthew Haulmark | Vanderbilt University | Non-hyperbolic groups with Menger curve boundary |

Nov 7, 2018 | Andrew McCullough | Georgia Institute of Technology | Legendrian large cables and non-uniformly thick knots |

Nov 14, 2018 | Marco Marengon | UCLA | Strands algebras and Ozsváth–Szabó’s Kauffman states functor |

Jan 9, 2019 | Genevieve Walsh | Tufts University | Relatively hyperbolic groups with planar boundary |

Jan 30, 2019 | Francesco Lin | Princeton University | Monopole Floer homology and spectral geometry |

Mar 13, 2019 | Caitlin Leverson | Georgia Institute of Technology | DGA representations, ruling polynomials, and the colored HOMFLY–PT polynomial |

Mar 27, 2019 | Tye Lidman | North Carolina State University | Splices, Heegaard Floer homology, and Seifert manifolds |

Apr 3, 2019 | Linh Truong | Columbia University | An infinite rank summand of the homology cobordism group |

Apr 10, 2019 | Peter Lambert-Cole | Georgia Institute of Technology | Bridge trisections and the Thom conjecture |

## Abstracts

Date Aug 29, 2018

Speaker Ignat Soroko

Title Dehn functions of subgroups of right-angled Artin groups

Abstract The question of what is a possible range for the Dehn functions (a.k.a. isoperimetric profile) for certain classes of groups is a natural and interesting one. Due to works of many authors starting with Gromov, we know a lot about the isoperimetric profile for the class of all finitely presented groups. Much less is known for many natural subclasses of groups, such as subgroups of right-angled Artin groups. We prove that polynomials of arbitrary degree are realizable as Dehn functions of subgroups of right-angled Artin groups. The key step is to construct for each natural k a free-by-cyclic group with the monodromy automorphism growing as n^k, which is virtually special in the sense of Haglund and Wise. Then its double will have Dehn function growing as n^{k+2}. This is a joint work with Noel Brady.

Date Sep 5, 2018

Speaker Andrew Zimmer

Title Limit sets of discrete subgroups

Abstract Given a discrete group of matrices one can define an associated limit set in projective space. In this talk I'll describe some results concerning the regularity of this limit set when the discrete group satisfies certain geometric properties.

Date Sep 26, 2018

Speaker Yilong Wang

Title Modular tensor categories and Reshetikhin–Turaev TQFTs I

Abstract In this talk, we give a detailed introduction to modular tensor categories and the Reshetikhin–Turaev TQFT associated to them. Time permitted, I will talk about algebraic properties of the RT-TQFTs.

Date Oct 10, 2018

Speaker Yilong Wang

Title Modular tensor categories and Reshetikhin–Turaev TQFTs II

Abstract In this talk, I will define ribbon and modular categories, and show how modular categories give rise to representations of the modular group SL(2,Z) using the graphical calculus introduced last time. Time permitted, I will explain how to generalize the construction to obtain a TQFT for closed surfaces.

Date Oct 17, 2018

Speaker Joshua Sabloff

Title Length and width of Lagrangian cobordisms

Abstract In this talk, I will discuss two measurements of Lagrangian cobordisms between Legendrian submanifolds in symplectizations: their length and their relative Gromov width. The Gromov width, in particular, is a fundamental global invariant of symplectic manifolds, and a relative version of that width helps understand the geometry of Lagrangian submanifolds of a symplectic manifold. Lower bounds on both the length and the width may be produced by explicit constructions; this talk will concentrate on upper bounds that arise from a filtered version of Legendrian contact homology, a Floer-type invariant. This is joint work with Lisa Traynor.

Date Oct 24, 2018

Speaker Scott Baldridge

Title A new cohomology for planar trivalent graphs with perfect matchings

Abstract In this lecture, I will describe a simple-to-compute polynomial invariant of a planar trivalent graph with a perfect matching (think: Jones polynomial for graphs). This polynomial is interesting because of what it detects: If the polynomial is non-zero when evaluated at one, then the perfect matching is even. Such a perfect matching implies that the graph can be 4-colored. I will then show how to categorify this polynomial to get a Khovanov-like cohomology theory for planar trivalent graphs and compute a couple of simple examples. If time permits, I will talk about some consequences of the cohomology theory.

Date Oct 31, 2018

Speaker Matthew Haulmark

Title Non-hyperbolic groups with Menger curve boundary

Abstract In the setting of hyperbolic groups, groups with Menger curve boundary are known to be abundant. Given the prevalence of negatively curved groups, it was a surprising observation of Ruane that there were no known examples of non-hyperbolic groups with Menger curve boundary found in the literature. Thus Ruane posed the problem (early 2000s) of finding examples (alt. interesting classes) of non-hyperbolic groups with Menger curve boundary. In this talk I will discuss the first class of such examples. This is joint work with Chris Hruska and Bakul Sathaye.

Date Nov 7, 2018

Speaker Andrew McCullough

Title Legendrian large cables and non-uniformly thick knots

Abstract We will define the notion of a knot type having Legendrian large cables, and discuss the fact that having this property implies that the knot type is not uniformly thick. In this case, there are solid tori in this knot type that do not thicken to a solid torus with integer slope boundary torus, and that exhibit new phenomena; specifically, they have virtually overtwisted contact structures. We will give an example of an infinite family of ribbon knots that have Legendrian large cables which fail to be uniformly thick in several ways not previously seen.

Date Nov 14, 2018

Speaker Marco Marengon

Title Strands algebras and Ozsváth–Szabó’s Kauffman states functor

Abstract Ozsváth and Szabó introduced in 2016 a knot invariant, which they announced to be isomorphic to the usual knot Floer homology. Their construction is reminiscent of bordered Floer homology: for example, their invariant is defined by tensoring bimodules over certain algebras. During the talk I will introduce a more geometric construction, closer in spirit to bordered sutured Floer homology, based on strands on a particular class of generalised arc diagrams. The resulting strands algebras are quasi-isomorphic to the Ozsváth–Szabó algebras, suggesting that Ozsváth and Szabó’s theory may be part of a hypothetical generalisation of bordered sutured Floer homology. This is a joint work with Mike Willis and Andy Manion.

Date Jan 9, 2019

Speaker Genevieve Walsh

Title Relatively hyperbolic groups with planar boundary

Abstract I will describe what a relatively hyperbolic group is, and give a lot of examples where the boundary is planar. Furthermore, I will explore some of the interesting phenomena that can occur and explain the significance of cut points in the boundary. Lastly I will discuss restrictions on the peripheral groups when the boundary is planar and without cut points.

Date Jan 30, 2019

Speaker Francesco Lin

Title Monopole Floer homology and spectral geometry

Abstract By studying the Seiberg–Witten equations, Kronheimer and Mrowka defined a package of invariants of three-manifolds called monopole Floer homology. In this talk, we discuss some interactions between this topological invariant and the spectral geometry of the Laplacian on the underlying Riemannian manifold, with the goal of understanding concrete examples of hyperbolic and Solv manifolds.

Date Mar 13, 2019

Speaker Caitlin Leverson

Title DGA representations, ruling polynomials, and the colored HOMFLY–PT polynomial

Abstract Given a pattern braid beta in J

^{1}(S

^{1}), to any Legendrian knot Lambda in R

^{3}with the standard contact structure, we can associate the Legendrian satellite knot S(Lambda, beta). We will discuss the relationship between counts of augmentations of the Chekanov–Eliashberg differential graded algebra of S(Lambda, beta) and counts of certain representations of the algebra of Lambda. We will then define an m-graded n-colored ruling polynomial from the m-graded ruling polynomial, analogously to how the n-colored HOMFLY–PT polynomial is defined from the HOMFLY–PT polynomial, and extend results of the second author, to show that the 2-graded n-colored ruling polynomial appears as a specialization of the n-colored HOMFLY–PT polynomial. (Joint work with Dan Rutherford.)

Date Mar 27, 2019

Speaker Tye Lidman

Title Splices, Heegaard Floer homology, and Seifert manifolds

Abstract A natural way to construct three-manifolds is to glue two knot exteriors together. We will study properties of the Heegaard Floer homology of such manifolds. We then use this to characterize homeomorphisms between a special class of three-manifolds. This is joint work with Cagri Karakurt and Eamonn Tweedy.

Date Apr 3, 2019

Speaker Linh Truong

Title An infinite rank summand of the homology cobordism group

Abstract We show that the homology cobordism group of integer homology three-spheres contains an infinite rank summand. The proof uses an algebraic modification of the involutive Heegaard Floer package of Hendricks–Manolescu and Hendricks–Manolescu–Zemke. This is inspired by Hom's techniques in the setting of knot concordance. This is joint work with Irving Dai, Jen Hom and Matt Stoffregen.

Date Apr 10, 2019

Speaker Peter Lambert-Cole

Title Bridge trisections and the Thom conjecture

Abstract The classical degree-genus formula computes the genus of a nonsingular algebraic curve in the complex projective plane. The well-known Thom conjecture posits that this is a lower bound on the genus of smoothly embedded, oriented and connected surface in CP

^{2}. The conjecture was first proved twenty-five years ago by Kronheimer and Mrowka, using Seiberg–Witten invariants. In this talk, we will describe a new proof of the conjecture that combines contact geometry with the novel theory of bridge trisections of knotted surfaces. Notably, the proof completely avoids any gauge theory or pseudoholomorphic curve techniques.