Vitaliy Kurlin (University of Liverpool)
Title:聽Can we geometrically sense the shape of a molecule?
Abstract:聽Can we hear the shape of a drum? This question was聽negatively answered decades ago by many authors including Gordon,聽Webb, Wolpert, who constructed non-isometric planar shapes that have聽the identical eigenvalues of the Laplace operator (Bull. AMS, v.27 (1992), p.134-138). 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, published in CVPR 2023. 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. The relevant papers are at聽.
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