Kyushu University Institute of Mathematics for Industry

The IMI Colloquium Report in January 13, 2021

The IMI Colloquium in January was held on January 13, 2021.

The IMI Colloquium in January
Title : Discover of Mobius Crystals and Topological Crystallography and Topological Science
Speaker : Prof. Satoshi Tanda, Dept.of Applied Physics, Hokkaido Univ.
Place : Only by Zoom

Prof. Satoshi Tanda, Hokkaido University, gave a talk entitled “Discover of Mobius Crystals and Topological Crystallography and Topological Science”.

This talk is an overview of his challenging research activities connecting and expanding physics and mathematics, inspired by a discovery of materials with a new structure through experiments. A typical microscopic structure in materials is crystal. Nowadays geometrically various structures, observed in, say fullerene and carbon nanotubes, are found as a collection of crystal-like structures. Tanda’s first topic is an introduction and the construction of crystals with ring structures, and a discovery of crystals with torsional structure observed in NbSe3, which are called “Mobius crystals”. The new structure, Mobius crystal, is constructed by taking chemical symmetry into account, but there is a non-trivial problem if this eccentric structure geometrically different from typical crystal structures should be called “a crystal”. He then goes to the next topic, the Bragg reflection property which “crystal structures” should possess. During researches in his group, a texture pattern which is not observed in usual (quasi-)crystals has been observed in the new structure. and his group has proposed a new category of crystals called “topological crystals”. He introduced, related to this category, a classification among topological crystals by means of topological invariants such as linking numbers, as well as embedding manifolds, which characterizes different types of topological crystals including Mobius crystals and “Hopf-link crystals” consisting of two rings. Linking numbers relate to an invariant “helicity”. His talk goes to the topic related to helicity observed in many scientific scenes, including crystal structures. Helicity itself is defined for fluid fields, gauge fields, spacetime and so on, which characterize various macroscopic and microscopic phenomena. Tanda’s group has proved the existence of helicity in Charged-Density Wave fields (CDW for short) which are known to be observed in topological crystals. This result induces a non-trivial relationship of helicity to materials structures in topological crystals. Tanda’s last topic consists of his idea about the origin of helicity. He thinks of “fluctuations” in some sense as the origin, which can determine physical quantities such as helicity and vorticity through mathematical structures, and characterize the nature of materials, fluids and the universe. He has ended his talk by drawing a big picture of a research field named “Topological Science” characterizing various scientific natures through topology.

Attendance: Staff 14, Students 15