Events
Geometry and Topology Seminar
- Events
- Geometry and Topology Seminar
Alex Chandler, NC State, Thin Posets and Homology Theories
Inspired by Bar-Natan's description of Khovanov homology, we discuss thin posets and their capacity to support homology and cohomology theories which categorify rank-statistic generating functions. Additionally, we present two main applications. The first, a categorification of certain generalized Vandermonde determinants gotten from the Bruhat order on the symmetric group by applying a special TQFT to…
Guangbo Xu, Simons Center for Geometry and Physics, Mirror Symmetry and Gauged Linear Sigma Model
In this overview talk, I will first explain several mathematical approaches of mirror symmetry. Then I will focus on the mathematical theory of Witten's gauged linear sigma model based on the recent joint work with Gang Tian, which settles the mathematical foundation of the approach of Hori and Vafa.
Alex Zupan, University of Nebraska, A special case of the Smooth 4-dimensional Poincare Conjecture
The smooth version of the 4-dimensional Poincare Conjecture (S4PC) states that every homotopy 4-sphere is diffeomorphic to the standard 4-sphere. One way to attack the S4PC is to examine a restricted class of 4-manifolds. For example, Gabai's proof of Property R implies that every homotopy 4-sphere built with one 2-handle and one 3-handle is standard. …
Juanita Pinzon Caicedo, NC State, Four–manifolds and knot concordance
SAS 4201The main goal of geometric topology is the classification of manifolds within a certain framework (topological, piecewise linear, smooth, simply-connected, symplectic, etc.). Dimension four is special, as it is the only dimension in which a manifold can admit infinitely many non-equivalent smooth structures, and the only dimension in which there exist manifolds homeomorphic but not…
Yakov Berchenko-Kogan, Washington University in St. Louis, Variational numerical methods in geometric PDE
Variational methods can be used to create numerical methods that respect conservation laws. I will discuss applications to electromagnetism, the Yang-Mills equations, and mean curvature flow. I will also discuss some new ideas about finite element spaces of differential forms.
Dustin Leninger, An Introduction to Spectral Sequences
SAS 2201I will describe a homological algebra construction which is fundamental in algebraic topology, algebraic geometry, and related areas: the spectral sequence. Originally developed by Jean Leray in the 1940s, a spectral sequence is a simultaneous higher-dimensional generalization of homology and long exact sequences. I will discuss a few examples of spectral sequences and their applications.
Alex Chandler, NC State, Spectral Sequences Working Seminar part 3
Exact couples and the Bockstein spectral sequence.
Grant Barkley, NC State, Spectral Sequences Working Seminar part 5
Computing Tor with spectral sequences.
Jai Aslam and Ezra Nance, NC State, Spectral Sequences Working Seminar
Knot homology theories
Juanita Pinzon-Caicedo, NC State, Instanton and Heegaard Floer homologies of surgeries on torus knots
SAS 2102A Floer homology is an invariant of a closed, oriented 3-manifold Y that arises as the homology of a chain complex whose generators are either the set of solutions to a differential equation or the intersection points between Lagrangian manifold, and its differential arises as the count of solutions of a differential equation on Y…
Alex Chandler, NC State, Spectral Sequences Working Seminar
Spectral sequences in Khovanov homology.
Jonathan Hanselman, Princeton, The cosmetic surgery conjecture and Heegaard Floer homology
Duke University, Physics 119The cosmetic surgery conjecture states that no two surgeries on a given knot produce the same 3-manifold (up to orientation preserving diffeomorphism). Floer homology has proved to be a powerful tool for approaching this problem; I will survey partial results that are known and then show that these results can be improved significantly. If a…
Yuanan Diao, Department of Mathematics and Statistics, UNC Charlotte, Braid Index Bounds Ropelength From Below
For an un-oriented link K, let L(K) be the ropelength of K. It is known that in general L(K) is at least of the order O((Cr(K))3/4), and at most of the order O(Cr(K) ln5 (Cr(K)) where Cr(K) is the minimum crossing number of K. Furthermore, it is known that there exist families of (infinitely many) links with the property…
Krzysztof Putyra, University of Zurich, An equivalence between gl(2)-foams and Bar-Natan cobordisms
The original construction of the Khovanov homology of a link can be seen as a formal complex in the category of flat tangles and surfaces between them. There is a way to associate a chain map with a link cobordism, but only up to a sign. Blanchet has fixed this by introducing the category of gl(2)-foams, certain singular cobordisms…