Crystal flexibility: methods from Analysis and Commutative Algebra

Professor Stephen Power, Lancaster University. Part of the pure mathematics colloquium

"The stability of crystal lattices was considered by Max Born and coworkers in the 1940s by essentially linear (small vibration) methods for which "it is not necessary to consider the complete elastic spectrum with its innumerable proper frequencies". The analysis of such mechanical modes (a.k.a. zero modes, rigid unit modes) is an ongoing research topic in both mathematics (infinitesimal and combinatorial rigidity) and condensed-matter science (surface modes and topological modes). I shall survey some of the mathematical rigidity theory and indicate new methods and results from analysis and commutative algebra. In particular we have obtained spectral synthesis for weakly closed shift-invariant subspaces of Z^d \times C^r and this has lead to a (first) characterisation of first-order rigidity for crystal frameworks (Kastis and Power, 2018)."
The talk will be followed by a reception in the School of Maths common room.