Topology Seminar

This fall (2023) we will have an active seminar.
September 14th
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
Date October 12th
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
Date October 19th
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
October 26th
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
November 3rd
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
November 30th
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
March 28th
Speaker: Ren Guo (Oregon State)
Title: Discrete conformal geometry on surfaces with boundary
Abstract: Conformal geometry, a branch of differential geometry concerned with preserving angles locally, finds a discrete counterpart that plays a pivotal role in various computational and theoretical domains. Examples of discrete conformal geometry structures are circle packings and combinatorial Yamabe flow. Discrete conformal geometry has been studied on both closed surfaces and surfaces with boundary. In this talk, we will focus on discrete conformal geometry on surfaces with boundary. Variational principle and relation to the volume of hyperbolic polyhedra will be investigated.
Location: DTB A203. 12:00pm--1:00pm
April 8th
Speaker: Joel Hass (U.C. Davis)
Title: Triangulating surfaces in Mathematics and in Computer Graphics
Abstract: A technique for efficiently describing surfaces was developed to solve the knot recognition problem. This method, using `normal surfaces,' was introduced by Kneser and applied to topological algorithms by Haken. In this talk we will show how normal surfaces can be used to solve a key problem in computer graphics: How to triangulate a surface so that no triangle has an angle that is close to zero. This is joint work with M. Trnkova.
Location: Maclaurin A144 (Department Colloquium) Food and Coffee 3:30pm, talk 4pm--5pm
May 20th
Speaker: Connor Malin (Notre Dame)
Title: A covariant manifold calculus in the style of Goodwillie
Abstract: Embedding calculus is a type of manifold calculus which studies presheaves on the category of manifolds in terms of their values on discs. Embedding calculus is formally dual to the theory of Weiss cosheaves, which serve as a type of manifold calculus for covariant functors. In codimension greater than two, embedding calculus accurately describes the homotopy type of the presheaf Emb(.,N). However, the dual theory of Weiss cosheafification fails spectacularly to study the covariant functor Emb(N,.). We introduce a new version of covariant manifold calculus which rectifies this issue and shares many similarities with the classical Goodwillie calculus of spaces. In this setting, we study the interaction of Weiss cosheafification and embedding calculus in order to unite and extend results of Arone--Ching and Ayala--Francis. Integral to this work is the Koszul self duality of the E_n operad.
Location: DTB A203 11am--12pm.
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Seminar from 2022-2023.