| Speaker: Yanwen Luo (U.Vic) |
| Title: Drawing and Morphing Graphs on Surfaces |
| Abstract: In his famous paper ``How to draw a graph" in 1962, Tutte proposed a simple method to produce a straight-line embedding of a planar graph in the plane, known as Tutte's spring theorem. This construction provides not only one embedding of a planar graph, but infinite many distinct embeddings of the given graph. This observation leads to a surprisingly simple proof of a classical theorem proved by Bloch, Connelly, and Henderson in 1984 stating that the space of geodesic triangulations of a convex polygon is contractible. In this talk, we will introduce spaces of geodesic triangulations of surfaces, review Tutte's spring theorem, and present this short proof. We will briefly report the recent progress in identifying the homotopy types of spaces of geodesic triangulations of more complicated surfaces. This is joint work with Tianqi Wu and Xiaoping Zhu. |
| Location: DSB C130, 4pm--4:50pm |
| Speaker: Yanwen Luo (U.Vic) |
| Title: Combinatorial curvature flows on surfaces |
| Abstract: I will discuss two combinatorial analogues of classical curvature flows in geometric analysis. The first is the combinatorial Ricci flows introduced by Chow and Luo in 2002 for circle packings. The second is the combinatorial Yamabe flows introduced by Luo in 2003, which is further improved by introducing combinatorial surgery. Both flows are gradient flows of some convex energies related to volumes of hyperbolic polyhedra and provide discrete versions of uniformization theorems for discrete surfaces. |
| Location: DSB C130, 3:30pm--4:20pm |
| Speaker: Nicholas Rouse (UBC) |
| Title: Freely Periodic Knots and Symmetric L-space Knots |
| Abstract: Among the various types of knots symmetries are the freely periodic ones which are those all powers of which act freely on the 3-sphere and leave the knot invariant. Examples of freely periodic knots are torus knots and their iterated cables. These knots also happen to be L-space knots and a natural question is whether there are any freely periodic L-space knots. The main tool for obstructing free periodicities is a condition on the Alexander polynomial due to Hartley. This condition allows one to obstruct free periodicities for any particular integer, but it is not well suited to obstructing free periodicities of all orders. The work in this talk will discuss ways to improve Hartley's condition to handle free periodicities of all orders. The main techniques involve refining Hartley's condition for irreducible polynomials and then understanding how it behaves in factorizations. As an application, we show that no freely periodic knot through 16 crossings is an L-space knot. This work is joint with Keegan Boyle. |
| Location: DSB C130 3:30pm--4:20pm |
| Speaker: Tomasz Kaczynski (U.Sherbrooke) |
| Title: Morse theory for multi-filtrations: smooth and discrete. |
| Abstract: In this talk, I will present joint efforts to develop an analogy of Morse Theory for functions with values in Rk, k>1, in the context of multi-filtered persistent
homology. In [1], a Forman-like multidimensional discrete Morse function is defined with the purpose of the matching algorithm for reduction of the underlying complex. The extension
of the theory was partial and missing some geometric insight. It was pointed out in [1] that an appropriate application-driven extension of the Morse theory to multi-filtrations for
smooth functions was not much investigated yet, and it would help in understanding the discrete analogy. My joint work [2] is a step in that direction and the main part of my talk. We
describe the evolution of bi-filtrations in terms of cellular attachments. A concept of persistence path is introduced, analogies of Morse-Conley equation and Morse inequalities along
persistence paths are derived. A scheme for computing path-wise barcodes is proposed. At the end, I will summarise main results of the joint work [3], which is a completion of the
work done in [1], inspired by smooth analogies from [2].
1. M. Allili, T. Kaczynski, C. Landi, and F. Masoni, Acyclic partial matchings for multidimensional persistence: algorithm and combinatorial interpretation, J Math Imaging Vis (61) (2019) 174-192, DOI 10.1007/s10851-018-0843-8. 2. R. Budney and T. Kaczynski, Bi-filtrations and persistence paths for 2-Morse functions, arXiv:2110.08227 [math.AT] Oct 2021, Algebraic & Geometric Topology, 23:6 (2023), 2895–2924, DOI: 10.2140/agt.2023.23.2895. 3. M. Allili, G. Brouillette, and T. Kaczynski, Multidimensional discrete Morse theory, arXiv:2212.02424 [math.GT] Dec 2022. |
| Location: DSB C130 3:30pm--4:20pm |
| Speaker: Henry Segerman (Oklahoma State U.) |
| Title: Avoiding inessential edges |
| Abstract: Results of Matveev, Piergallini, and Amendola show that any two triangulations of a three-manifold with the same number of vertices are related to each other by a sequence of local combinatorial moves (namely, 2-3 and 3-2 moves). For some applications however, we need our triangulations to have certain properties, for example that all edges are essential. (An edge is inessential if both ends are incident to a single vertex, into which the edge can be homotoped.) We show that any two triangulations with all edges essential can be related to each other by a sequence of 2-3 and 3-2 moves, keeping all edges essential as we go. This is joint work with Tejas Kalelkar and Saul Schleimer. |
| Location: DSB C128, 3:30pm--4:20pm |
| Speaker: Yanwen Luo |
| Title: Combinatorial curvature flows on three manifolds |
| Abstract: I will discuss two types of combinatorial curvature flows on three manifolds, combinatorial Yamabe flows introduced by David Glickenstein in 2006 and combinatorial Ricci flows by Feng Luo in 2005. These two flows are essentially the same on surfaces since the curvatures are concentrated on vertices. However, combinatorial Yamabe flows deform three manifolds based on discrete scalar curvatures defined on vertices of the triangulation within a "discrete conformal class", while combinatorial Ricci flows are defined based on discrete Ricci curvatures on edges. I will summarize the known results about the long-time behavior of these two flows and their relations with the geometrization conjecture and the Yamabe problem for three manifolds. |
| Location: DTB A-203. 3:30pm--4:30pm |