Events
Geometry and Topology Seminar
- Events
- Geometry and Topology Seminar
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…
Irina Kogan, NC State, A Generalization of an Integrability Theorem of Darboux
SAS 4201In his monograph “Systèmes Orthogonaux” (Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, Gauthier-Villars, Paris, 1910), Darboux stated three theorems providing local existence and uniqueness of solutions to first order systems of PDEs, where for each unknown function a certain subset of partial derivatives is prescribed and the values of the unknown functions are prescribed along the corresponding transversal coordinate…
Tye Lidman, NC State, Homology three-spheres and SU(2) representations
One way to effectively show a group is non-trivial is to find a non-trivial representation. A major open question in low-dimensional topology is whether the fundamental group of a closed three-manifold other than S^3 has a non-trivial SU(2) representation, and this is a strategy for an alternate proof of the three-dimensional Poincare conjecture. We will…
Jonathan Campbell, Duke University, The Scissors Congruence Problem and the Algebraic K-theory of the Complex Numbers
In this talk I'll explain a surprising relationship between the objects in the title. Two n-dimensional polytopes, $P$, $Q$ are said to be scissors congruent if one can cut $P$ along a finite number of hyperplanes, and re-assemble it into $Q$. The scissors congruence problem asks: when can we do this? what obstructs this? In…
Donald Sheehy, NC State, On the Cohomology of Impossible Figures, Revisited
The Penrose triangle, also known as the impossible tribar is an icon for cohomology. It is literally the icon for Cech cohomology on Wikipedia. The idea goes back to a paper by Roger Penrose in 1992, but was first reported by Penrose several years earlier. There, he shows how the impossibility of the figure depends…
Ákos Nagy, Duke University, Complex Monopoles
Self-duality equations in gauge theory can be complexified in many inequivalent ways, but there are two obvious options: One can extend Hodge duality in either a complex linear fashion, or in a conjugate linear one. In general, the two cases result in two very different equations. The first case was first studied by Haydys, while…