Seminars and Colloquia by Series

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.

Joint UGA-GT Topology Seminar at GT: Upper bounds on the topological slice genus via twisting operations

Series
Geometry Topology Seminar
Time
Monday, October 7, 2019 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Duncan McCoyUQAM
I will explain how null-homologous twisting operations can be used to obtain bounds on the topological slice genus. In particular, I will discuss how one can obtain upper bounds on the topological slice genera of torus knots and satellite knots using these operations.

Joint UGA-GT Topology Seminar at GT: Smooth 4-Manifolds and Higher Order Corks

Series
Geometry Topology Seminar
Time
Monday, October 7, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Paul MelvinBryn Mawr College

It is a remarkable fact that some compact topological 4-manifolds X admit infinitely many exotic smooth structures, a phenomenon unique to dimension four.  Indeed a fundamental open problem in the subject is to give a meaningful description of the set of all such structures on any given X.  This talk will describe one approach to this problem when X is simply-connected, via cork twisting.  First we'll sketch an argument to show that any finite list of smooth manifolds homeomorphic to X can be obtained by removing a single compact contractible submanifold (or cork) from X, and then regluing it by powers of a boundary diffeomorphism.  In fact, allowing the cork to be noncompact, the collection of all smooth manifolds homeomorphic to X can be obtained in this way.  If time permits, we will also indicate how to construct a single universal noncompact cork whose twists yield all smooth closed simply-connected 4-manifolds.  This is joint work with Hannah Schwartz.

Geometry Topology Seminar : Surface bundles and complex projective varieties by Corey Bregman

Series
Geometry Topology Seminar
Time
Monday, September 30, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Corey BregmanBrandeis University

Kodaira, and independently Atiyah, gave the first examples of surface bundles over surfaces whose signature does not vanish, demonstrating that signature need not be multiplicative.  These examples, called Kodaira fibrations, are in fact complex projective surfaces admitting a holomorphic submersion onto a complex curve, whose fibers have nonconstant moduli. After reviewing the Atiyah-Kodaira construction, we consider Kodaira fibrations with nontrivial holomorphic invariants in degree one. When the dimension of the invariants is at most two, we show that the total space admits a branched covering over a product of curves.

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