Research Problems for 2008

• Braid Groups of Graphs
• Surgery Obstruction Groups
• Positivity of Generalized Matrix Functions on Totally Positive Matrices
• Soliton solutions for reductions of the KP and discrete KP hierarchies
• Dihedral Geodesics in the Hyperbolic Plane
• Orbits of matrix group actions

1. Braid Groups of Graphs: Students will compute the fundamental groups of configuration spaces of planar graphs and will construct classifying spaces of minimal dimension for some of these. They will try to extend some known results concerning the conjectured nature of these groups.
   
   
2. Surgery Obstruction Groups: Students will compute the odd dimensional surgery obstruction groups of certain imaginary quadratic rings of integers. This will involve learning about certain matrix groups and about modules over principal ideal domains. Particular attention will be given to Z [ζ] where ζ is a primitive third root of unity.
   
   
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6. An action of a group G on a space M is a map from G x M to M that "respects" the multiplication in G. Matrix-vector multiplication and conjugation of a matrix by an invertible matrix are examples of group actions.
   
   Many of natural actions of various matrix group are of great importance in geometry and mathematical physics. The problem of describing orbits of ( i. e. the minimal subspaces left invariant by ) such actions can be very difficult. For example, in the case of a so-called co-adjoint action on lower triangular matrices by invertible upper triangular ones, this problem is known to be "wild". One can, however, try to describe particular kinds of orbits, e.g. the most general ones, or indecomposable orbits of smallest possible dimension. The answer to these questions in the case of triangular matrices is already quite interesting. We suggest a problem of describing minimal orbits in the case of generalization of a triangular co-adjoint action to the case of symplectic and oprthogonal matrices.
   
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   Long Version of Problem #7