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. A possible project would be to relate the Viral Tiling Theory to the properties of the non-crystallographic Coxeter group H3 (which is the point group of the icosahedron), and explore the importance of its affine extension in the description of the content of the viral capsid (RNA genome). PrerequisitesA good grasp of linear algebra and a reasonable geometrical intuition.ResourcesYou can start by looking at the online overviews on virus structure here. A brief review with plenty more detailed references can be found here . - The Group Theory relevant for projects in this area is presented in several books: `Reflection Groups and Coxeter Groups', J. E. Humphreys, CUP (1990), ISBN 0-521-43613-3 `Group Theory in Physics', J. Cornwell, Academic Press (1986), ISBN 0-978-0121898045 `Group Theory and Physics', S. Sternberg, CUP (1994), ISBN 0-521-55885-9.
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email: Anne Taormina