Cal Poly Mathematics REU
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 H2, for example, is the space of all functions, analytic on the unit disk, which have square-summable Taylor coefficients. Functions in H2 also satisfy a certain special condition limiting the growth of their magnitudes near the boundary of the disk.

H2 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 Tf , Hankel operators Hf , and composition operators Cf on H2. These are each classes of operators, and their properties as operators depend on the particular function f chosen. Much is known about the relationship between the function-theoretic properties of f and the operator-theoretic properties of Tf , Hf , and Cf . There are still many unknown questions, however. For example, one of the most fundamental questions one asks about an operator, "What is its norm?'' is not, in general, known for the composition operator Cf , or particularly easy to answer for Tf  and Hf . Investigations into questions involving norms, essential norms, spectra, and other operator-theoretic properties can make use of computer programs such as Maple, since questions about many of these properties can be reduced easily to questions about certain (real) integrals on the boundary of the unit disk. Information about the operators which is obtained experimentally in this way can lead to insight into interesting theorems.

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.  

Visualization and Analysis of Closed Geodesics on 2-Orbifolds
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

The incorporation of geometric methods has lead to significant advances in the theory of groups.  This approach has opened many avenues for new research and provided insight into long-standing problems.

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.

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