MA 499: Project

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Feel free to contact me about topics for final year projects; I think that solving some problem as a project is more fun, but I can suggest reading–oriented projects as well. Some hints on starting points for several different projects follow.

• Linear algebra: real vs. complex. Consider two 2-dimensional planes in R⁴. Under what conditions there exists a complex structure on R⁴, that is a way to identify R⁴ with C², for which these two planes are complex 1-dimensional subspaces? This is a toy model for some slightly more complicated questions that you will address.
• Real hyperplane arrangements: algebras of locally constant functions. This project is both about reading and solving a problem; in an old paper ("On Heaviside Functions of Configuration of Hyperplanes", Functional Analysis and its Applications, 21:4 (1987), 1-18), Gelfand and Varchenko proved an elegant theorem on regions of a complement to a real hyperplane arrangement. The goal of the project is to read their paper and then prove some theorems that are — without any good reason — missing there.
• Ramified coverings of sphere and transitive factorisations. This project is mostly a reading project, the main purpose being to understand the amazing result of Hurwitz relating the classification of `ramified coverings of a 2-sphere' (you will learn what it is in due course) and elementary combinatorial questions about permutations.
• Graphs on surfaces and applications. This project is about studying embedding of various graphs into Riemann surfaces (spheres with handles). A remarkable theorem of Pontryagin and Kuratowski says, in particular, that the graph formed by the regular pentagon together with its diagonals cannot be drawn on the plane (or 2-sphere) without self-intersections. What is the minimal number of handles one needs to attach to the sphere to draw this graph on the resulting surface? This is a typical question you will learn how to answer.

Year 2010/11

Year 2009/10

Year 2008/09

Year 2007/08