## Departmental PhD Thesis Exam – Mario Morfin Ramirez

Friday, April 30, 2010, 2-3 p.m., in BA 6183, 40 St. George St.

PhD Candidate:  Mario Morfin Ramirez

PhD Advisors:    Michael Shub and Charles Pugh

Thesis title: Grassmann dynamics

Thesis Abstract:

The present work is divided in two parts. The first is concerned with the
dynamics on the Grassmann manifold of k-dimensional subvector spaces
of an n dimensional real or complex vector space induced by a linear
invertible transformation A of the vector space into itself.

In 1968 Steve Batterson attacked a similar problem in his Northwestern
University doctoral dissertation. He considered the action induced by
diagonal transformations whose eigenvalues have distinct moduli. He gives
a topological classification of these maps and proves that they are
Morse-Smale.

In the current work, I considere the non diagonal case and give a complete
topological classification of the conjugacy classes of the linearly
induced maps on the Grassmann manifold. Also, I define an extension of the
Morse-Smale diagrams to describe these conjugacy classes, and I discuss
the structural stability and bifurcation properties among and within the
conjugacy classes.

In the second part, I consider dynamics induced by a linear cocycle
covering a diffeomorphism f:N --> N of a compact
manifold, acting on the Grassmann bundle of k-dimensional linear
subspaces of T N.

Two problems are attacked: A Kupka-Smale theorem for the space of cocycles
covering diffeomorphisms of a compact manifold. The proof of this theorem
implies the same type of results for derived cocycles parametrized in the
space of diffeomorphisms. The results of the second part can be
generalized without effort to cocycles covering endomorphisms of N.