Abstracts

Mark Grant

Title: Comparison of equivariant cohomological dimensions

Abstract:

Let G be a Γ-group (meaning G is a group with a fixed action of a group Γ by automorphisms). There are at least 3 definitions of the Γ-equivariant group cohomology of G in the literature, corresponding to 3 different flavours of relative group cohomology of the semi-direct product relative to the subgroup Γ. There arise 3 definitions of equivariant cohomological dimension (ecd). In this talk we will compare the dimensions coming from Takasu cohomology and from Bredon cohomology, showing by examples that both inequalities can occur. This will include a discussion of equivariant Stallings--Swan theorems which characterize when a Γ-group has ecd less than or equal to 1. We will also discuss an example of a group G for which TC(G) is less than cd_\D(GxG), the Bredon dimension of the product with respect to the family of subgroups generated by the diagonal subgroup.

This is joint work with Kevin Li, Ehud Meir and Irakli Patchkoria.

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Stephan Mescher

Title: Geodesic complexity and decompositions of cut loci

Abstract:

Geodesic complexity is an isometry invariant of geodesic spaces that is motivated by Farber’s notion of topological complexity. It is seen as a mathematical abstraction of the notion of efficient robot motion planning, i.e. of motion planning along length-minimizing paths. The difficulties in computing geodesic complexity obviously arise from pairs of points which are connected by more than one length-minimizing path. Such points occur in the cut loci of one another, which yields that cut loci are the crucial objects of study in computing or estimating geodesic complexity. In this talk we shall focus on complete Riemannian manifolds as geodesic spaces. In general, the cut loci of two distinct points in the same manifold might differ wildly from each other. However, under certain homogeneity assumptions on the Riemannian metric, the cut loci of all points in the manifold might be arranged as a finite disjoint union of total spaces of fiber bundles. Studying the structures of these bundles, one can derive interesting upper bounds for geodesic complexity. We will present this method of decomposing cut loci in certain settings as well as examples for which these bounds can be computed explicity. In particular, we will see that the geodesic complexities of complex projective spaces with their standard symmetric metrics coincide with their topological complexities. This is a joint work with Maximilian Stegemeyer.

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Vitaliy Kurlin

Title. Geometric Data Science for unlabeled point clouds modulo Euclidean isometry

Abstract.

Geometric Data Science studies continuous metrics and paramerisations on modulo spaces of data objects considered up to practical equivalence relations. The basic example of real data is afinite cloud on unlabeled points modulo Euclidean isometry or rigid motion. A complete and continuous isometry invariant of three points is a triple of pairwise distances (sides of a triangle) known as the side-side-side theorem in school geometry. The pairwise distances distinguish all non-isometric clouds in general position, though their completeness fails already for singular configurations of four points. The more complicated and slower isometry invariant of point clouds is persistent homology, which recently turned out to be weaker thapreviously anticipated. The talk will introduce new isometry invariants that are complete for all clouds in any Euclidean space and are continuous under perturbations of points, see the key papers at http://kurlin.org/research-papers.php#Geometric-Data-Science.

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Ian Leary

Title: Contractibility of acyclic 2-complexes

Abstract:

The question of whether there can be an algorithm to decide whether afinite 2-complex is contractible is a well-known open problem. I will discuss this problem and solve an infinite analogue.

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Lucile Vandembroucq

Title: On the weak topological complexity and the TC-Ganea conjecture

Abstract:

By analogy with the classical Ganea conjecture, which has been disproved by N. Iwase, the TC-Ganea conjecture asks whether the equality TC(X x S^n)=TC(X)+TC(S^n) holds for all finite CW complexes X and all positive integers n. In a previous work in collaboration with J. González and M. Grant, we have constructed a space satisfying TC(X x S^n)=TC(X)+1 for all n>1, which disproves the TC-conjecture for n even. In this talk, we will use the notion of weak topological complexity to establish some sufficient conditions for a space X to satisfy

TC(X x S^n)=TC(X)+TC(S^n). This is a joint work with J. Calcines.

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Debasis Sen

Title: Simplicial and combinatorial versions of higher symmetric topological complexity

Abstract:

In this talk, we will introduce higher symmetric simplicial complexity SC^{\Sigma}_{n}(K) of a simplicial complex K and higher symmetric combinatorial complexity CC^{\Sigma}_{n}(P) of a finite poset P. These are simplicial and combinatorial approaches to symmetric motion planning of Basabe - Gonz\'{a}lez - Rudyak - Tamaki. We prove that the symmetric simplicial complexity \SC^{\Sigma}_{n}(K) is equal to symmetric topological complexity TC^{\Sigma}_{n}(|K|) of the geometric realization of K and the symmetric combinatorial complexity CC^{\Sigma}_{n}(P) is equal to symmetric topological complexity TC^{\Sigma}_{n}(|\mathcal{K}(P)|) of the geometric realization of the order complex of P.

This is joint work with Dr. Amit Kumar Paul.

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Amit Kumar Paul

Title: SEQUENTIAL PARAMETRIZED MOTION PLANNING AND ITSCOMPLEXITY

Abstract:

The approach of parametrized motion planning was introduced recently by M. Farber and S. Weinberger. In this presentation, we introduce sequential parametrized motion planning. A sequential parametrized motion planning algorithm produces a motion of the system which is required to visit a prescribed sequence of states, in a certain order, at specified moments of time. The sequential parametrized algorithms are universal as the external conditions are not fixedin in advance but constitute part of the algorithm’s input. In this talk, we present an upper and lower bound for the sequential parametrized topological complexity. Further, we obtain the sequential parametrized topological complexity of the Fadell- Neuwirth fibration. In the language of robotics, sections of the Fadell - Neuwirth fibration are algorithms for moving multiple robots avoiding collisions with other robots and obstacles in the Euclidean space. (This is jont work with Prof. Michael Farber)