Thematic Einstein Semester on

Geometric and Topological Structure of Materials

Summer Semester 2021

Speaker


Teresa Heiss   (IST Austria)


Title


Geometric and Topological Fingerprints for Periodic Crystals


Abstract


The following application has motivated us to develop new Computational Geometry and Topology methods, involving Brillouin zones and periodic order k persistent homology: Crystalline materials (short: crystals) are often represented by a parallelepiped with a finite point set inside, such that tiling R3 with this parallelepiped yields an infinite periodic point set where each point represents an atom of the crystal. However, this finite representation is not unique. We therefore consider representations up to equivalence: Two representations are equivalent, if they represent the same crystalline material, i.e. if there exists an isometry between the two corresponding periodic point sets. When material scientists want to quantify the difference between two crystals they cannot use the two given representations directly, but instead need to first find the two most similar looking representations of the two crystals. However, this is a difficult open problem in crystallography. We therefore suggest a different approach: Finding an injective and continuous invariant, called a fingerprint map, mapping the space of crystals to a metric space. This enables the comparison of crystal representations by comparing their fingerprints and avoids the open problem of finding best matching representations.

In this talk, I will explain the problem and present a continuous invariant that is generically injective: The density fingerprint, computing the probability that a random ball of radius r contains exactly k points of the periodic point set, for all positive integers k and positive reals r. Afterwards, I'll present ongoing research about how to improve this to an even more promising candidate for a fingerprint map, using the sequence of order k persistence diagrams, newly defined for infinite periodic point sets, for all positive integers k.

Joint work with Herbert Edelsbrunner, Alexey Garber, Vitaliy Kurlin, Georg Osang, Janos Pach, Morteza Saghafian, Phil Smith, and Mathijs Wintraecken.



Contact


tes-summer2021@math.tu-berlin.de