Seminars and Colloquia by Series

Gluing data in chromatic homotopy theory

Series
Geometry Topology Seminar
Time
Monday, September 14, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Agnes Beaudry – University of Chicago
Understanding the stable homotopy groups of spheres is one of the great challenges of algebraic topology. They form a ring which, despite its simple definition, carries an amazing amount of structure. A famous theorem of Hopkins and Ravenel states that it is filtered by simpler rings called the chromatic layers. This point of view organizes the homotopy groups into periodic families and reveals patterns. There are many structural conjectures about the chromatic filtration. I will talk about one of these conjectures, the \emph{chromatic splitting conjecture}, which concerns the gluing data between the different layers of the chromatic filtration.

Braid groups, Burau representations, and algebraic curves

Series
Geometry Topology Seminar
Time
Monday, August 31, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Weiyan Chen – U Chicago
The theory of étale cohomology provides a bridge between two seemingly unrelated subjects: the homology of braid groups (topology) and the number of points on algebraic varieties over finite fields (arithmetic). Using this bridge, we study two problems, one from topology and one from arithmetic. First, we compute the homology of the braid groups with coefficients in the Burau representation. Then, we apply the topological result to calculate the expected number of points on a random superelliptic curve over finite fields.

Contact structures and their applications in Finsler geometry

Series
Geometry Topology Seminar
Time
Monday, August 24, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hassan Attarchi – visitor
In this work, a novel approach is used to study geometric properties of the indicatrix bundle and the natural foliations on the tangent bundle of a Finsler manifold. By using this approach, one can find the necessary and sufficient conditions on the Finsler manifold (M; F) in order that its indicatrix bundle has the Sasakian structure.

Veering Dehn surgery

Series
Geometry Topology Seminar
Time
Friday, April 17, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Henry Segerman – Oklahoma State University –
This is joint work with Saul Schleimer. Veering structures onideal triangulations of cusped manifolds were introduced by Ian Agol, whoshowed that every pseudo-Anosov mapping torus over a surface, drilled alongall singular points of the measured foliations, has an ideal triangulationwith a veering structure. Any such structure coming from Agol'sconstruction is necessarily layered, although a few non-layered structureshave been found by randomised search. We introduce veering Dehn surgery,which can be applied to certain veering triangulations, to produceveering triangulationsof a surgered manifold. As an application we find an infinite family oftransverse veering triangulations none of which are layered. Untilrecently, it was hoped that veering triangulations might be geometric,however the first counterexamples were found recently by Issa, Hodgson andme. We also apply our surgery construction to find a different infinitefamily of transverse veering triangulations, none of which are geometric.

Knot invariants and their categorifications via Howe duality

Series
Geometry Topology Seminar
Time
Monday, April 13, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Aaron Lauda – USC –
It is a well understood story that one can extract linkinvariants associated to simple Lie algebras. These invariants arecalled Reshetikhin-Turaev invariants and the famous Jones polynomialis the simplest example. Kauffman showed that the Jones polynomialcould be described very simply by replacing crossings in a knotdiagram by various smoothings. In this talk we will explainCautis-Kamnitzer-Licata's simple new approach to understanding theseinvariants using basic representation theory and the quantum Weylgroup action. Their approach is based on a version of Howe duality forexterior algebras called skew-Howe duality. Even the graphical (orskein theory) description of these invariants can be recovered in anelementary way from this data. The advantage of this approach isthat it suggests a `categorification' where knot homology theoriesarise in an elementary way from higher representation theory and thestructure of categorified quantum groups. Joint work with David Rose and Hoel Queffelec

Tightness of positive rational surgeries

Series
Geometry Topology Seminar
Time
Monday, April 6, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Bulent Tosun – University of Virginia
Existence of a tight contact structure on a closed oriented three manifold is still widely open problem. In this talk we will present some work in progress to answer this problem for manifolds that are obtained by Dehn surgery on a knot in three sphere. Our method involves on one side generalizing certain geometric methods due to Baldwin, on the other unfolds certain homological algebra methods due to Ozsvath and Szabo.

Independence of Whitehead Doubles of Torus Knots in the Smooth Concordance Group

Series
Geometry Topology Seminar
Time
Monday, March 30, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Juanita Pinzon-Caicedo – University of Georgia
In the 1980’s Furuta and Fintushel-Stern applied the theory of instantons and Chern-Simons invariants to develop a criterion for a collection of Seifert fibred homology spheres to be independent in the homology cobordism group of oriented homology 3-spheres. In turn, using the fact that the 2-fold cover of S^3 branched over the Whitehead double of a positive torus knot is negatively cobordant to a Seifert fibred homology sphere, Hedden-Kirk establish conditions under which an infinite family of Whitehead doubles of positive torus knots are independent in the smooth concordance group. In the talk, I will review some of the definitions and constructions involved in the proof by Hedden and Kirk and I will introduce some topological constructions that greatly simplify their argument. Time permiting I will mention some ways in which the result could be generalized to include a larger set of knots.

Lagrangian concordance and contact invariants in sutured Floer theories

Series
Geometry Topology Seminar
Time
Monday, March 23, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
John Baldwin – Boston College
In 2007, Honda, Kazez, and Matic defined an invariant of contact 3-manifolds with convex boundaries using sutured Heegaard Floer homology (SHF). Last year, Steven Sivek and I defined an analogous contact invariant using sutured Monopole Floer homology (SMF). In this talk, I will describe work with Sivek to prove that these two contact invariants are identified by an isomorphism relating the two sutured theories. This has several interesting consequences. First, it gives a proof of invariance for the contact invariant in SHF which does not rely on the relative Giroux correspondence between contact structures and open books (something whose proof has not yet been written down in full). Second, it gives a proof that the combinatorially computable invariants of Legendrian knots in Heegaard Floer homology can obstruct Lagrangian concordance.

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