Cascade Topology Seminar - Abstracts

Geoffroy Horel (Paris 13): Operad of genus zero curves and the Grothendieck-Teichmüller group.

The Grothendieck-Teichmüller group is a profinite group that contains the absolute Galois group of the rational numbers and is conjecturally isomorphic to it. In this talk I will explain how one can understand this group using the homotopy theory of operads. This is joint work with Pedro Boavida de Brito and Marcy Robertson.

Keiichi Sakai (Shinshu U.): The space of short ropes and the classifying space of the space of long knots

We prove affirmatively the conjecture raised by J. Mostovoy; the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in R3. We make use of techniques developed by S. Galatius and O. Randal-Williams to construct a manifold space model of the classifying space of the space of long knots, and we give an explicit map from the space of short ropes to the model in a geometric way. This is joint work with Syunji Moriya (Osaka Prefecture University).

Liam Watson (Sherbrooke U.): Modules from Heegaard Floer theory as curves in a punctured torus

Heegaard Floer theory is a suite of invariants for studying low-dimensional manifolds. In the case of punctured torus, for instance, this theory constructs a particular algebra. And, the invariants associated with three-manifolds having (marked) torus boundary are differential modules over this algebra. This is structurally very satisfying, as it translates topological objects into concrete algebraic ones. I will discuss a geometric interpretation of this class of modules in terms of immersed curves in the punctured torus. This point of view has some surprising consequences for closed three-manifolds that follow from simple combinatorics of curves. For example, one can show that, if the dimension of an appropriate version of the Heegaard Floer homology (of a closed manifold) is less than 5, then the manifold does not contain an essential torus. Said another way, this gives a certificate that the manifold admits a geometric structure à la Thurston. This is joint work with Jonathan Hanselman and Jake Rasmussen.

Jeff Meier (U. Georgia): Bridge trisections of knotted surfaces in four-manifolds

The theory of knotted surfaces in four-manifolds (the natural analogue of knot theory to dimension four) is one of the richest and least-explored domains of low-dimensional topology.  In this talk, I'll outline some of the most intriguing open problems in this area, and I'll discuss a new approach to four-dimensional knot theory that is inspired by the theory of trisections.  Particular focus will be placed on the study of complex curves in the simplest complex four-manifolds: CP2 and S2xS2  In this setting, the theory of bridge trisections has produced surprisingly beautiful pictures, which intriguing implications to the study of exotic smooth structures on (complex) four-manifolds.

Tomasz Kacynski (Sherbrooke U.): Persistent Homology in Topological Data Mining

In the past decade, persistent homology became an important tool in Data and Network Analysis. Connections between distinct elements of data can be expressed in terms of finite combinatorial structures such as simplicial complexes, also carrying a geometric flavor. Homology provides information on properties of complexes such as connectivity, cycles, tunnels or voids. Persistent homology detects those properties which persist over changes of a chosen parameter, say, a resolution scale. Following an introduction to the topic and a guiding example, I will pass to my joint work with Claudia Landi and Madjid Allili on multidimensional persistent homology.  It has been introduced with the purpose of analyzing and comparing data according to several parameters simultaneously. Its effective computation remains a challenge due to the huge size of complexes built from data. A reduction of the size of a complex can be achieved by suitable pairing of cells, called partial matching. I will present an algorithm that constructs an acyclic partial matching on cells of a given simplicial complex. This is used to obtain a reduced cell complex with the same multidimensional persistent homology as the original one.

 

Gabriel Islambouli (U. Virginia): Trisections and the Pants Complex

Given two smooth, oriented, closed 4-Manifolds M and N with the same Euler characteristic, we will construct an invariant D(M,N) coming from a distance in the pants complex. To do this, we adapt the work of Johnson on Heegaard splittings of 3-manifolds to the trisections of 4-manifolds introduced by Gay and Kirby. This naturally leads us to various graphs of 4-manifolds and we will briefly discuss their properties.

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