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
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.
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.
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.