DescriptionThis project concerns some simple and not-so-simple aspects of geometry and group theory with applications to problems in biology. The common theme is tiling, the idea of covering a surface (or a higher-dimensional space) with many copies of a small number of basic shapes, or tiles. This might seem an old subject, but it is still developing in interesting directions.An area where ideas of tiling have an unexpected application is the subject of virus structure. Viruses consist of an outer protein shell, or capsid, which protects the genetic material inside. The shells are made by assembling basic subunits, which "tile" the surface of the virus. The geometric principles of the assembly of these subunits were laid down by Caspar and Klug in 1962, an interesting story in its own right. However, some viruses fall outside Caspar and Klug's scheme, and very recently it has been suggested that ideas inspired by Penrose tilings can help to resolve these mysteries. The project focusses on group theory techniques relevant to the study of viruses with icosahedral symmetry. One would first review the basic theoretical aspects of Viral Tiling Theory, for which there are some informative articles listed below. Possible directions for a project within this topic are, for instance, a study of the normal modes of vibration of a virus with icosahedral symmetry, a study of how the viral genome is packaged within an icosahedral capsid, an understanding of how to relate the Viral Tiling Theory to the properties of the non-crystallographic Coxeter group H3 (which is the point group of the icosahedron). The emphasis of the project will be tailored to each student, and other directions can be followed within the set context. PrerequisitesA good grasp of linear algebra and a reasonable geometrical intuition.ResourcesYou may want to watch this short video as a quick introduction. A brief review with plenty more detailed references can be found here . More resources will be shared with students taking the project. Another interesting resource is `Quasicrystals and geometry', Marjorie Senechal CUP (1996);(book available in the Bill Bryson Library) - The Group Theory relevant for projects in this area is presented in several books: `Groups and Symmetry', M.A. Armstrong, Springer (2008);(book available in the Bill Bryson Library) `Reflection Groups and Coxeter Groups', J. E. Humphreys, CUP (1990); (available on line through Durham University Library) `Group Theory in Physics: an introduction', J. Cornwell, Academic Press (1997); (book available in the Bill Bryson Library) `Group Theory and Physics', S. Sternberg, CUP (1994) (book available in the Bill Bryson Library)
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email: Anne Taormina