Collaborative Research: Homological Invariants in Crystallography

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Proposal: DMS-0204823, DMS-0204845Principal Investigators: Benji Fisher, David Rabson A B S T R A C TFisher and Rabson are classifying the symmetry types of quasicrystalsin two and three dimensions and studying the physical consequences ofthese symmetries. The classification generalizes the work begun inthe nineteenth century on space groups of crystals. The investigatorsfollow the Fourier-space approach to crystallography, which has theadvantage of avoiding space groups in higher dimensions. Thus theclassification starts from the known list of finite subgroups of theorthogonal groups O(2) and O(3). They introduce techniques of groupcohomology, some of which have long been used in "direct space," tothe Fourier-space formulation. These techniques lead to boththeoretical simplifications and efficient computational methods. Oneimportant application of these ideas is the description of certaingauge invariants in terms of group homology. The two simplest typesof homology class are connected with known physical phenomena:systematic extinctions in diffraction patterns and crossing ofelectronic bands ("band sticking"). The next simplest type firstoccurs in a rank-five tetragonal modulated crystal and should beconnected with some similar phenomenon. Tiling models are produced,and the ideas are also extended to magnetic and color groups.Crystallography underlies and informs much of physics, chemistry, andgeology. The present research has applications to recent experimentsin liquid crystals (related to the popular LCD displays onwristwatches and other electronic equipment), plasmas, and modulatedcrystals, highly symmetric systems that cannot be described by theclassical theory of crystals. Ever since 1784, when the French abbotRene Just Hauy deduced the microscopic structure of crystals, it hadbeen believed impossible for a crystal to have the symmetries of anicosahedron (or a soccer ball). Precisely 200 years later, suchmaterials, called "quasicrystals," were discovered, and much researchhas ensued into their properties. Quasicrystals possess strangeelectronic and physical properties and have already found applicationas high-quality, non-stick coatings on electrosurgical blades. Thepresent research aims to classify the symmetry types of crystals andto study physical properties associated with these symmetries.This project is funded jointly by the Division of Mathematical Sciencesand the Division of Materials Research.