LIT Workshop: Abstracts

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In unpublished notes from 1960, Richard Thompson defined the three groups F < T < V, which act respectively on the interval, circle and Cantor set. Their definition might appear very specific, but these groups have an incredibly rich theory, bizarre combinations of properties, intricate dynamics, and many mysteries surrounding them. I will focus on F, and the long-standing open question of whether it is amenable or not. My hope is that by the end of this minicourse, you will have a new group in your personal library of favourite groups, and a better appreciation of just how strange some groups can be. 

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A classical problem related to Nielsen theory deals with the question of determining the minimum number of fixed points among all maps homotopic to a given continuous map f from a compact space to itself. This problem motivated the definition of an equivalence relation on the set of fixed points of f separating the fixed points into Nielsen equivalence classes. In this talk we will focus on the strong Nielsen equivalence, which is a "stronger" equivalence relation concerning periodic points of a surface homeomorphism. We will focus on orientation-preserving homeomorphisms of the 2-disc D2that fix the boundary pointwise and leave invariant a finite subset in the interior of D2. We will study the strong Nielsen equivalence of periodic points of such homeomorphisms f and we will give a necessary and sufficient condition for two periodic points to be strong Nielsen equivalent, with respect to the given f-invariant set, in the context of braid group theory. 

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We will describe a new strategy for constructing a pair of closed, smooth 4-manifolds that are homotopy equivalent but not homeomorphic. The key input for the construction is a pair of surfaces in a given 4-manifold. This talk is based on joint work with Daniel Kasprowski and Mark Powell.

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In this talk, I will present joint work with Alessio Di Prisa on the study of equivariant rational concordance of strongly invertible knots. I will discuss the new theory with several explicit constructions (with a refinement of Kawauchi's theorem) and obstructions (with a refinement of the Fox-Milnor condition). I will also introduce the equivariant rational concordance group and discuss its relations with the other concordance groups. 

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