Vague Project Areas
Symmetry & Groups
Spring 2010
1. Describe the representation theory behind spectrometers, MRI’s, or another piece of
scientific or medical equipment.
2. Give the complete character theory for a finite group not covered in Chapter 5.
3. Give the decomposition of the regular representation for a compact topological group
such as S 1 and S 2 (or in general).
4. Explore finite groups as they relate to Galois theory.
5. Symmetries of the hydrogen atom, or representation theory behind electron shells.
6. Discrete groups acting on the hyperbolic plane (e.g. Fuchsian, Schottky, Kleinian
groups) What about the geometry is preserved under these group actions?
7. Study groups acting on topological spaces, such as the homotopy groups πn (X). How
can these groups be used to classify the space?
8. Fractal geometry. For example, understand the relationship between the symmetries of
a holomorphic function on the complex plane, and symmetries of the associated Julia
set.
9. In introductory differential equations courses, we learn how to solve first-order linear
systems of equations by finding eigenvalues of matrices. Explain the underlying representation theory. Find other systems of differential equations in which representation
theory can be exploited to find solutions. Explain, using representation theory, why
Fourier transforms can be used to solve differential equations.
10. Potentially solve problems from Parts II, III of Serre, Herstein’s Topics in Algebra, or
graduate-level number theory, algebraic topology, or representation theory texts.
11. Braid groups and gauge theory.
12. Any other application of representation theory to chemistry or physics you can find.
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