Events
Geometry and Topology Seminar
- Events
- Geometry and Topology Seminar
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…
Yu-Min Chung, UNC Greensboro, Summaries of persistence diagrams and their applications to data science
Topological Data Analysis (TDA) is a relatively young field in both algebraic topology and machine learning. Tools from TDA, in particular persistent homology, have proven successful in many scientific disciplines. Persistence diagrams, a typical way to study persistent homology, contain fruitful information about the underlying objects. However, performing statistical methods directly on the space of persistence diagrams is…
Shan Gao, Beijing Institute of Technology, Discrete Geometrically-Exact Beams
A geometrically-exact beam is a nonlinear field-theoretic model for elongated elastic objects. It utilizes moving frames to reduce the number of system’s independent spatial variables, which is a further development of Euler’s approach to the rotational dynamics of rigid bodies. The talk will discuss the dynamics and geometrically-inspired discretization for structure-preserving numerical simulations of free,…
Christine Ruey Shan Lee, Plamenevskaya’s invariant from the stable Khovanov homology of twisted torus knots
A transverse link is a link in the 3-sphere that is everywhere transverse to the standard contact structure. Transverse links are considered up to transverse isotopy, with classical invariants such as the self-linking number and regular isotopy class. One of the first connections between transverse links and quantum invariants was made by Plamenevskaya in 2006,…
Peter McGrath, NC State, Quantitative Isoperimetric Inequalities on Riemannian Surfaces
ZoomIn this talk, we introduce a scattering asymmetry which measures the asymmetry of a domain on a surface by quantifying its incompatibility with an isometric circle action. We prove a quantitative isoperimetric inequality involving the scattering asymmetry and characterize the domains with vanishing scattering asymmetry by their rotational symmetry. We also give a new proof…
Adam Lowrance, Vassar College, Extremal Khovanov homology of Turaev genus one links
ZoomThe Turaev surface of a link diagram is a surface built from a cobordism between the all-A and all-B Kauffman states of the diagram. The Turaev surface can be seen as a Jones polynomial analogue of the Seifert surface. The Turaev genus of a link is the minimum genus of the Turaev surface for any…
Geometry/Topology Social Hour
ZoomCome chat with other geometers/topologists. This is a good chance for graduate students to meet the geometry/topology faculty, especially our newest members, Peter McGrath and Teemu Saksala. Host: Tye Lidman (tlid@math.ncsu.edu) Instructions to join: Zoom invitation is sent to the geometry and topology seminar list. If you are not on the list, please, contact the…
Teemu Saksala NC State, Probing an unknown elastic body with waves that scatter once. An inverse problem in anisotropic elasticity.
ZoomWe consider a geometric inverse problem of recovering some material parameters of an unknown elastic body by probing with elastic waves that scatter once inside the body. That is we send elastic waves from the boundary of an open bounded domain. The waves propagate inside the domain and scatter from an unknown point scatterer. We measure the entering…
Miruna-Stefana Sorea, Max-Planck-Institut für Mathematik in den Naturwissenschaften, The shapes of level curves of real polynomials near strict local minima
ZoomWe consider a real bivariate polynomial function vanishing at the origin and exhibiting a strict local minimum at this point. We work in a neighbourhood of the origin in which the non-zero level curves of this function are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries…
Darrick Lee Affiliation, University of Pennsylvania, Path Signatures on Lie Groups
ZoomPath signatures are powerful nonparametric tools for time series analysis, shown to form a universal and characteristic feature map for Euclidean valued time series data. The theory of path signatures can be lifted to the setting of Lie group valued time series while retaining their universal and characteristic properties. This talk will introduce these generalized path signatures on Lie groups and…
Roman Aranda, University of Iowa, Diagrams of $\star$-trisections
ZoomA trisection of a smooth, connected 4-manifold is a decomposition into three standard pieces. Like the case of Heegaard splittings in dimension three, a trisection is described by a trisection diagram: three sets of curves in a surface satisfying some properties. In general, it is not evident whether two trisection diagrams represent the same decomposition…
Alex Chandler, University of Vienna, Torsion in Thin Regions of Khovanov Homology
ZoomIn the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is links whose Khovanov homology is supported on two adjacent diagonals, are known to contain only 2-torsion. In this paper, we prove a local version of this result. If the Khovanov homology of a link is supported…
Beibei Liu, Max Planck Institute / Georgia Tech, The four-ball genus of links via the Heegaard Floer homology
ZoomThe Heegaard Floer homology introduced by Ozsvath and Szabo provides a lot of link invariants to study links in the three-sphere and its surgery manifolds. In this talk, we exact some invariants from the package to give bounds for the four-ball genus of links in the three-sphere. We also give a family of links where the…