Seminars and Colloquia by Series

Localization in Khovanov homology

Series
Geometry Topology Seminar
Time
Monday, January 6, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skile 006
Speaker
Melissa ZhangUGA

When a topological object admits a group action, we expect that our invariants reflect this symmetry in their structure. This talk will explore how link symmetries are reflected in three generations of related invariants: the Jones polynomial; its categorification, Khovanov homology; and the youngest invariant in the family, the Khovanov stable homotopy type, introduced by Lipshitz and Sarkar. In joint work with Matthew Stoffregen, we use Lawson-Lipshitz-Sarkar's construction of the Lipshitz-Sarkar Khovanov homotopy type to produce localization theorems and Smith-type inequalities for the Khovanov homology of periodic links.

Classifying contact structures on hyperbolic 3-manifolds

Series
Geometry Topology Seminar
Time
Monday, December 9, 2019 - 14:30 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
James ConwayUC, Berkeley

Please Note: Note time and place of seminar

Two of the most basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. In dimension 3, these questions have been answered for large classes of manifolds, but with a notable absence of hyperbolic manifolds. In this talk, we will see a new classification of contact structures on an family of hyperbolic 3-manifolds arising from Dehn surgery on the figure-eight knot, and see how it suggests some structural results about tight contact structures. This is joint work with Hyunki Min.

Residual Torsion-Free Nilpotence and Two-Bridge Knot Groups

Series
Geometry Topology Seminar
Time
Monday, December 2, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Jonathan JohnsonThe University of Texas at Austin

I will discuss how a graph theoretic construction used by Hirasawa and Murasugi can be used to show that the commutator subgroup of the knot group of a two-bridge knot is a union of an ascending chain of parafree groups. Using a theorem of Baumslag, this implies that the commutator subgroup of a two-bridge knot group is residually torsion-free nilpotent which has applications to the anti-symmetry of ribbon concordance and the bi-orderability of two-bridge knots. In 1973, E. J. Mayland gave a conference talk in which he announced this result. Notes on this talk can be found online. However, this result has never been published, and there is evidence, in later papers, that a proper proof might have eluded Mayland.

Classifying incompressible surfaces in hyperbolic 4-punctured sphere mapping tori

Series
Geometry Topology Seminar
Time
Monday, November 25, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sunny Yang XiaoBrown University

One often gains insight into the topology of a manifold by studying its sub-manifolds. Some of the most interesting sub-manifolds of a 3-manifold are the "incompressible surfaces", which, intuitively, are the properly embedded surfaces that can not be further simplified while remaining non-trivial. In this talk, I will present some results on classifying orientable incompressible surfaces in a hyperbolic mapping torus whose fibers are 4-punctured spheres. I will explain how such a surface gives rise to a path satisfying certain combinatorial properties in the arc complex of the 4-punctured sphere, and how we can reconstruct such surfaces from these paths. This extends and generalizes results of Floyd, Hatcher, and Thurston.

Joint UGA/Tech Topology Seminar at UGA: A generalization of Rasmussen’s invariant, with applications to surfaces in some four-manifolds

Series
Geometry Topology Seminar
Time
Monday, November 18, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Boyd 303
Speaker
Marco MarengonUCLA

Building on previous work of Rozansky and Willis, we generalise Rasmussen’s s-invariant to connected sums of $S^1 \times S^2$. Such an invariant can be computed by approximating the Khovanov-Lee complex of a link in $\#^r S^1 \times S^2$ with that of appropriate links in $S^3$. We use the approximation result to compute the s-invariant of a family of links in $S^3$ which seems otherwise inaccessible, and use this computation to deduce an adjunction inequality for null-homologous surfaces in a (punctured) connected sum of $\bar{CP^2}$. This inequality has several consequences: first, the s-invariant of a knot in the three-sphere does not increase under the operation of adding a null-homologous full twist. Second, the s-invariant cannot be used to distinguish $S^4$ from homotopy 4-spheres obtained by Gluck twist on $S^4$. We also prove a connected sum formula for the s-invariant, improving a previous result of Beliakova and Wehrli. We define two s-invariants for links in $\#^r S^1 \times S^2$. One of them gives a lower bound to the slice genus in $\natural^r S^1 \times B^3$ and the other one to the slice genus in $\natural^r D^2 \times S^2$ . Lastly, we give a combinatorial proof of the slice Bennequin inequality in $\#^r S^1 \times S^2$.

Joint UGA/Tech Topology Seminar at UGA: Concordance invariants from branched coverings and Heegaard Floer homology

Series
Geometry Topology Seminar
Time
Monday, November 18, 2019 - 14:30 for 1 hour (actually 50 minutes)
Location
Boyd 221
Speaker
Antonio AlfieriUBC

I will outline the construction of some knot concordance invariants based on the Heegaard Floer homology of double branched coverings. The construction builds on some ideas developed by Hendricks, Manolescu, Hom and Lidman. This is joint work with Andras Stipsicz, and Sungkyung Kang.

Koszul duality and Knot Floer homology

Series
Geometry Topology Seminar
Time
Monday, November 4, 2019 - 14:00 for
Location
Skiles 006
Speaker
Tom HockenhullUniversity of Glasgow

‘Koszul duality’ is a phenomenon which algebraists are fond of, and has previously been studied in the context of '(bordered) Heegaard Floer homology' by Lipshitz, Ozsváth and Thurston. In this talk, I shall discuss an occurrence of Koszul duality which links older constructions in Heegaard Floer homology with the bordered Heegaard Floer homology of three-manifolds with torus boundary. I shan’t assume any existing knowledge of Koszul duality or any form of Heegaard Floer homology.

Connected Floer homology of covering involutions

Series
Geometry Topology Seminar
Time
Monday, October 28, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skile 006
Speaker
Sungkyung KangChinese University of Hong Kong

Using the covering involution on the double branched cover of S3 branched along a knot, and adapting ideas of Hendricks-Manolescu and Hendricks-Hom-Lidman, we define new knot (concordance) invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots. This is a joint work with A. Alfieri and A. Stipsicz.

The geometry of subgroup combination theorems

Series
Geometry Topology Seminar
Time
Monday, October 21, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jacob RussellCUNY Graduate Center

While producing subgroups of a group by specifying generators is easy, understanding  the structure of such a subgroup is notoriously difficult problem.  In the case of hyperbolic groups, Gitik utilized a local-to-global property for geodesics to produce an elegant condition that ensures a subgroup generated by two elements (or more generally generated by two subgroups) will split as an amalgamated free product over the intersection of the generators. We show that the mapping class group of a surface and many other important groups have a similar local-to-global property from which an analogy of Gitik's result can be obtained.   In the case of the mapping class group, this produces a combination theorem for the dynamically and topologically important convex cocompact subgroups.  Joint work with Davide Spriano and Hung C. Tran.

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