Abstracts
A. Dranishnikov
On probabilistic version of topological complexity
Abstract. The topological complexity TC(X) was introduced by Farber as a numerical invariant of robot's configuration space X. Since TC is a homotopy invariant, it can be extended to discrete groups. In the talk we will compare TC with its new cousin dTC, a probabilistic version of TC, and we will discuss the problem of computation of both invariants for discrete groups and some classic manifolds.
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J. Oprea
Robotics, Right Angled Artin Groups and Rationality
Abstract: This talk will focus on the topological complexity of Right-angled Artin groups and questions and conjectures elicited by results about RAAGs. In particular, the Rationality Conjecture, which says that the power series formed by the sequential topological complexities is rational of a certain type, will be discussed.
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Mark Powell
Smoothing codimension two submanifolds
Given a locally flat submanifold of codimension two, is it homotopic, or even isotopic, to a smooth embedding? I will explain what is known about this problem in all dimensions, before focusing on the cases of submanifolds of dimension 3 and 4. This is joint work with Michelle Daher.
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Vitaliy Kurlin
Title. Can we geometrically sense the shape of a molecule?
Abstract. Can we hear the shape of a drum? This question was negatively answered decades ago [1]. The more general question: can we sense the shape of a rigid object such as a cloud of atomic centers representing a molecule? The SSS theorem from school geometry says that any triangles (clouds of 3 unordered points) are congruent (isometric) if and only if they have the same three sides (ordered by length). An extension of this theorem to more points in higher dimensions was practical only for clouds of m ordered points, which are uniquely determined up to isometry by a matrix of m x m distances. If points are unordered, comparing m! matrices under all permutations of m points is impractical. We will define a complete (under rigid motion) and Lipschitz continuous invariant for all clouds of m unordered points, which is computable in polynomial time of m in any fixed Euclidean space [2]. For the QM9 database of 130K+ molecules with 3D coordinates, the more recent invariants distinguished all clouds of atomic centers without chemical elements, which confirmed that the shape of a molecule including its chemistry is determined from sufficiently precise atomic geometry.
[1] C Gordon, DL Webb, S Wolpert. One cannot hear the shape of a drum Bull. AMS, v.27 (1992), p.134-138.
[2] D Widdowson, V Kurlin. Recognizing rigid patterns of unlabeled point clouds by complete and continuous isometry invariants with no false negatives and no false positives. Proceedings of CVPR 2023, 1275-1284.