Fall 2024 Redbud Topology Conference

The talks are in Price Hall room 2040 on the University of Oklahoma campus. Abstracts are below.

Time	Speaker	Talk
8:30 − 9:00		coffee + bagels
9:00 − 9:50	Ben Stucky	Pin the loop taut: a one-player topological game
10:00 − 10:50	Jorge Robinson Arrieta	An explicit section of the Laudenbach exact sequence of the mapping class group of connect sums of S²xS¹
11:00 − 11:50	Dami Lee	Cylinder decompositions on geometric armadillo tails
		lunch
2:00 − 2:50	Alex He	Practical algorithms in 3-manifold topology
3:00 − 3:50	Tomoya Tatsuno	Abstract Group Automorphisms of Two-Step Nilpotent Lie Groups
4:00 − 4:50	Yanwen Luo	Drawing and Morphing Graphs on Surfaces

Abstracts:

Alex He, Oklahoma State University
Title: Practical algorithms in 3-manifold topology
Abstract: For any fixed compact 3-manifold M, there are infinitely many ways to triangulate M. So given two 3-manifold triangulations, how can we algorithmically decide whether or not they triangulate the same 3-manifold? This is the 3-manifold homeomorphism problem, which is (to say the least) very hard. Much attention is therefore given to simpler problems known as recognition problems: after fixing a 3-manifold M, how can we algorithmically decide whether a given triangulation actually triangulates M? In other words, how can we algorithmically recognize triangulations of M? This talk will survey various 3-manifold recognition problems, and discuss techniques for designing "practical" algorithms to solve these problems; the challenge is that the algorithms need to be both simple enough to implement in software and efficient enough to actually execute.

Dami Lee, Oklahoma State University
Title: Cylinder decompositions on geometric armadillo tails
Abstract: In this talk, we discuss a particular type of finite area, infinite-type translation surfaces, and find explicit examples of types of cylinder decompositions on these surfaces which do not manifest on finite-type translation surfaces. This is based on joint work with Josh Southerland.

Yanwen Luo, Oklahoma State University
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 a joint work with Tianqi Wu and Xiaoping Zhu.

Jorge Robinson Arrieta, University of Arkansas
Title: An explicit section of the Laudenbach exact sequence of the mapping class group of connect sums of S²xS¹
Abstract: Laudenbach proved that the mapping class group of the connect sum of n copies of S²xS¹ is an extension of Out(F_n) by a finite group. Brendle-Broaddus-Putman proved that this exact sequence splits. We provide an explicit section s of this split exact sequence.

Ben Stucky, Beloit College
Title: Pin the loop taut: a one-player topological game
Abstract: Consider a closed surface S with a set of punctures P, and fix a homotopy class of loops C in S-P. Among the isotopy classes of its representatives, some are simpler than others. A result of Neumann-Coto says that an isotopy class c contains a geodesic for a complete Riemannian metric on S-P if and only if it is taut (realizes the minimal self-intersection in its homotopy class C). We investigate the inverse problem: given a generic isotopy class c in the closed surface S, where should we place the "pins" P to ensure that it is taut, and how many do we need? We show that it is NP-complete to bound this "pinning number," even when S is the sphere. On the one hand, we provide two separate polynomial algorithms to check if a given set of points is pinning. The first is geometric-group-theoretic, involving an adaptation of a method of Birman--Series for computing intersection numbers of curves in surfaces. The second is purely topological, using an improvement of a theorem of Hass--Scott characterizing taut loops in surfaces to convert a pinning problem instance to a boolean formula whose solutions correspond to the pinning sets. On the other hand, we reduce the planar vertex cover problem for graphs to the pinning problem for loops in the sphere. This is based on joint work with Christopher-Lloyd Simon.

Tomoya Tatsuno, University of Oklahoma
Title: Abstract Group Automorphisms of Two-Step Nilpotent Lie Groups
Abstract: Cartan (1930) and van der Waerden (1933) independently proved that any abstract (not necessarily continuous) group automorphism of a compact, simple Lie group must be continuous. This classical rigidity result is called automatic continuity, and it has been generalized by many authors.
In this talk, we consider abstract group automorphisms of two-step nilpotent Lie groups N. For a large class of N, we prove the following rigidity: any abstract group automorphism of N is a product of a (possibly discontinuous) central automorphism and a Lie group automorphism. In other words, discontinuity only occurs at the center. Our examples of N include Heisenberg groups arising from symmetric spaces of rank 1, low dimensional examples, and a generic group whose commutator subgroup is large. In particular, we cover uncountably many groups that do not admit a rational structure.