Descriptions of Projects

During your REU time at Cal poly, you will have the
opportunity to work with a number of our faculty members. Each of the
participating faculty
members has described a possible project. It is important that
you
choose a project you find at least a little bit interesting, so that we
can
match you with a faculty member. The project you actually work on
may
differ a bit from those described below - it will be decided by you and
your
faculty advisor together once the program begins.

Properties of Toeplitz,
Hankel, and
Composition Operators

Jonathan Shapiro

The Hardy spaces are collections of analytic
functions on the unit disk. The Hardy space H^{2}, for example, is the space of all functions, analytic on
the unit disk, which have
square-summable Taylor coefficients. Functions in H^{2} also satisfy a certain special condition limiting the
growth
of their magnitudes near the boundary of the disk.

H^{2} is of
particular
interest because, with the appropriate inner product, it can be made
into
a Hilbert space.

Our project will be to understand better the actions
of the Toeplitz operators T_{f },
Hankel operators H* _{f }*, and
composition operators C

In recent summers, REU students have worked with me
to
make conjectures and prove results concerning the continuity of the
function
which assigns to an analytic self-map of the disk the norm of the
associated
compositon operator. Others have used matrix methods to analyze
the
norms of composition operators and prove interesting results concerning
compactness
of certain composition operators.

Joseph Borzellino

In this project, undergraduates would study the question of existence of a smooth closed geodesics on compact Riemannian 2-orbifolds using theoretical and computational techniques along with computer visualization. Roughly speaking a Riemannian orbifold is a metric space locally modeled on quotients of Riemannian manifolds by finite groups of isometries. The 2-orbifolds we consider are orbifolds whose underlying space is a compact 2-manifold without boundary. One can think of such Riemannian orbifolds as surfaces with some distinguished singular cone points whose neighborhoods are isometric to a quotient of the unit disc with some metric by a finite cyclic group of isometries fixing the center of the disc. The 2-orbifolds we consider fall into two categories. The first case is when the underlying space of the orbifold is simply connected (in the usual topological sense), that is, the underlying space of the orbifold is the 2-sphere. These 2-orbifolds are examples of what are commonly referred to as teardrops and footballs. The second class of 2-orbifolds are those whose underlying space is not simply connected in the usual sense.

To better understand the context of the proposed project, one should recall the classical theorem of Fet and Lyusternik: On any compact Riemannian manifold there exists at least one closed geodesic. The essential tool in proving this classical result is to develop a process of curve shortening. Theoretical existence of such closed geodesics has been investigated using techniques from topology, geometry and dynamical systems, but a concrete analysis on specific Riemannian 2-orbifolds is still lacking.

Using a symbolic computation package such as Maple, undergraduates would formulate and numerically solve the equations governing geodesic propagation on specific Riemannian 2-orbifolds, such as those that arise as surfaces of revolution or those that admit flat structures. Additional use of the visualization capabilities of Maple would lead them to conjectures on qualitative aspects of geodesics on Riemannian 2-orbifolds and corresponding proofs.

The Geometry of Coxeter
Groups and their Automorphisms

Anton Kaul

Coxeter groups, which are a natural generalization of the symmetry groups of polyhedra, are especially amenable to study via geometric methods. For example, it is known that every Coxeter group acts "geometrically" on a CAT(0) space (and therefore has solvable congugacy problem). In contrast, their automorphism groups are not, in general, well-understood. In this project we will consider the following question: To what extent are the combinatorial and geometric properties of Coxeter groups inherited by their automorphism groups?

Let W be a Coxeter group and let Aut(W) denote the group of automorphisms of W. Examples of topics to be explored include:

(1) Does Aut(W) have a "nice" set A of generators?

(2) Is Aut(W) almost convex (with respect to A)?

(3) Is Inn(W) quasiconvex in Aut(W) (with respect to A)?

For the Summer 2006 program, we expect to have the three projects above available, an possibly more.

Return to the Cal Poly Mathematics REU homepage.