Configuration spaces and related topics

Invited Speakers

* The schedule may vary.

Speaker: Ben Knudsen (Northeastern University)
Title: Farber's conjecture and beyond
Abstract: Topological complexity is a numerical invariant quantifying the difficulty of motion planning; applied to configuration spaces, it measures the difficulty of collision-free motion planning. In many situations of practical interest, the environment is reasonably modeled as a graph, and the topological complexity of configuration spaces of graphs has received significant attention for this reason. This talk will discuss a proof of a conjecture of Farber, which asserts that this invariant is as large as possible in the stable range. Time allowing, we will touch on recent work establishing an analogue of this result in the setting of unordered configuration spaces.

Speaker: Sangrok Oh (Kyungpook National University)
Title: Large Scale geometry of graph 2-braid groups
Abstract: Graph $n$-braid groups or $n$-braid groups over graphs are the fundamental groups of configuration spaces on graphs. Unlike configuration spaces on higher dimensional spaces, there is a discrete version of configuration space on a graph, which is a locally CAT(0) cube complex, and in particular, graph braid groups act geometrically on CAT(0) cube complexes. In this talk, using quasi-isometry invariants of CAT(0) square complexes, called intersection complexes, we will talk about the quasi-isometric classification of 2-braid groups over special kind of graphs.

Speaker: Jihoon Park (Korea University)
Title: Combinatorial HHS and its modification
Abstract: Hierarchically hyperbolicity is a powerful tool for studying the large-scale geometry of wide range of groups (including mapping class groups and cubical grops), but finding explicit HH structure is usually challanging due to complicated definition of HHS. Recently Behrstock-Hagen-Martin-Sisto introduced a combinatorial HHS as a simpler combinatorial criterion for verifying whether a given curve graph induces a HHS structure. Such criterion is, however, not the optimal one as it cannot cover the case of the curve graph of CAT(0) cube complex, which has more complicated intersection patterns of product regions. In this talk, we will briefly review the (combinatorial) HHS theory and explain the modification of combinatorial HHS to cover cubical curve graphs and using this, extend the Behrstock-Hagen-Sisto style factor system machinary to the CAT(0) prism complex.

Speaker: Arthur Soulie (IBS-CGP)
Title: On homological representations for braid groups and mapping class groups
Abstract: I will describe a general construction of homological representations for families of motion groups or mapping class groups, including the families of braid groups, surface braid groups and loop braid groups. This recovers the well-known constructions of Lawrence-Bigelow, and in this sense it unifies these constructions. I will also discuss indecomposability and irreducibility of these representations.
The construction is moreover “global” in the sense that, for each dimension d, it is a functor on a category whose automorphism groups are all d-dimensional motion groups and mapping class groups, and which also carries a richer structure. Using this richer structure, I will discuss polynomiality of these families of representations, and use this to prove twisted homological stability for the braid groups with coefficients in any one of the Lawrence-Bigelow representations.
All this represents a joint work with Martin Palmer.

Speaker: Anderson Vera (IBS-CGP)
Title: A double Johnson filtration for the Goeritz group of the sphere and for the mapping class group of a surface
Abstract: For a triple (K,X,Y) consisting of a group K and two normal subgroups X and Y of K, we introduce a double-indexed family of normal subgroups of K which we call the double lower central series. In particular, if K=XY we show that this family allows us to recover the lower central series of K. If G is a group acting on K preserving X and Y, we show that the double lower central series induces a double-indexed filtration of G. We apply this theory to the group of isotopy classes of self-homeomorphisms of the 3-sphere S^3 which preserves the standard decomposition of S^3 as the gluing of two handlebodies (Goeritz group) we then extend this double-indexed filtration to the whole mapping class group of a surface common to the two handlebodies. (Joint work with Kazuo Habiro.)