This is the website for the joint topology seminar between Duke, NCSU and UNC. The seminar will normally meet on selected Tuesdays with the location rotating between Duke, NCSU and UNC. An informal pretalk seminar aimed at graduate students will also be held before some talks.
Please contact one of the organizers to be added to our mailing list.
(Bryn Mawr College)
|Sept 1 3:30pm||Duke
|"One is enough"|
(University of Virginia)
|Oct 2 4:30pm||NCSU
|Flow and Yamada polynomials, planar triangulations, and TQFT|
(University of Edinburgh)
|Oct 10 4:30pm||UNC
|Hall algebras and the Fukaya category|
(University of Glasgow)
|Oct 17 4:30pm||NCSU
|Modules from Heegaard Floer theory as curves in a punctured torus|
|Oct 23 3:15pm||Duke
|Complex curves through a contact lens|
(California Institute of Technology)
|Nov 7 4:30pm||NCSU
|The prism manifold realization problem|
"One is enough"
Abstract: Using Gabai's recent proof of the 4-dimensional light-bulb theorem, we show that any two homologous surfaces of the same genus embedded with simply-connected complements in a smooth 4-manifold become smoothly isotopic after one stabilization (connected summing with a 2-sphere bundle over the 2-sphere). This is joint work with Auckly, Kim, Ruberman and Schwartz.
Flow and Yamada polynomials, planar triangulations, and TQFT
Abstract: In the 1960s Tutte observed that the value of the chromatic polynomial of planar triangulations at (golden ratio +1) obeys a number of remarkable properties. In this talk I will explain how TQFT gives rise to a conceptual framework for studying planar triangulations. I will discuss several extensions of Tutte's results and applications to the structure of the chromatic and flow polynomials of graphs, and the Yamada polynomial of spatial graphs. This talk is based on joint works with Ian Agol and with Paul Fendley.
Hall algebras and the Fukaya category
Abstract: The Hall algebra is an invariant of an abelian (or triangulated) category C whose multiplication comes from "counting extensions in C." Recently, Burban and Schiffmann defined the "elliptic Hall algebra" using coherent sheaves over an elliptic curve, and this algebra has found applications in knot theory, mathematical physics, combinatorics, and more. In this talk we discuss some background and then give a conjectural description of the Hall algebra of the Fukaya category of a topological surface. This is partially motivated by an isomorphism between the elliptic Hall algebra and the skein algebra of the torus, which we also discuss. (Joint works with H. Morton and with B. Cooper.)
Modules from Heegaard Floer theory as curves in a punctured torus
Abstract: Heegaard Floer theory is a suite of invariants for studying low-dimensional manifolds. In the case of punctured torus, for instance, this theory constructs a particular algebra. And, the invariants associated with three-manifolds having (marked) torus boundary are differential modules over this algebra. This is structurally very satisfying, as it translates topological objects into concrete algebraic ones. I will discuss a geometric interpretation of this class of modules in terms of immersed curves in the punctured torus. This point of view has some surprising consequences for closed three-manifolds that follow from simple combinatorics of curves. For example, one can show that, if the dimension of an appropriate version of the Heegaard Floer homology (of a rational homology sphere) is less than 5, then the manifold does not contain an essential torus. Said another way, this gives a certificate that the manifold admits a geometric structure à la Thurston. This is joint work with Jonathan Hanselman and Jake Rasmussen.
Complex curves through a contact lens
Abstract: Every four-dimensional Stein domain has a height function whose regular level sets are contact three-manifolds. This allows us to study complex curves in the Stein domain via their intersection with these contact level sets, where we can comfortably apply three-dimensional tools. We use this perspective to characterize the links in Stein-fillable contact manifolds that bound complex curves in their Stein fillings. (Some of this is joint work with Baykur, Etnyre, Hedden, Kawamuro, and Van Horn-Morris.)
The prism manifold realization problem
Abstract: The spherical manifold realization problem asks which spherical three-manifolds arise from surgeries on knots in the three-sphere. In recent years, the realization problem for C, T, O, and I-type spherical manifolds has been solved, leaving the D-type manifolds (also known as the prism manifolds) as the only remaining case. Every prism manifold can be parametrized by a pair of relatively prime integers p > 1 and q. We determine a complete list of prism manifolds P(p, q) that can be realized by positive integral surgeries on knots in the three-sphere. The methodology undertaken to obtain the classification relies on tools from Floer homology and lattice theory, and is primarily combinatorial in nature. This is joint work with Ballinger, Ni, and Ochse.