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Research Experience for Undergraduates 
Project Descriptions - Summer 2008

(1)  Jim Swift:  The Gradient Newton Galerkin Algorithm

Here at NAU we have a small group of professors, Swift, Neuberger and Sieben, who have developed the Gradient Newton Galerkin Algorithm (GNGA) to find numerical approximations to solutions of nonlinear systems of equations and nonlinear Partial Differential Equations.
The REU students working with Prof. Swift this summer will do computational projects to advance this research program.

The focus of this summer will be to modify the GNGA to work efficiently on systems with the symmetry of the circle. The first example will be to numerically solve the nonlinear ordinary differential equation u'' + s u + u^3 = 0 where u satisfies periodic boundary conditions u(x) = u(x + 2 pi). The goal is to find solutions u(x) as a function of the parameter s, using continuation methods and bifurcation theory. After this, the problem will switch to the partial differential equation Laplacian u + s u + u^3 = 0 where the domain of u is the disk in the plane. A few student projects on this problem have already been done. The goal this summer is to write a more efficient algorithm using techniques that have not been applied to this problem. The students will write programs in C++, MATLAB, or some other language.

Background: The successful candidate will have taken Differential Equations and Linear Algebra from the math department, or will have had courses in another department that covers these topics. A strong background in both math and physics is especially desirable. Some familiarity with partial differentialequations is helpful. The student should have programming experience.

Reference: The original paper where the GNGA was introduced is "Newton's method and Morse index for semilinear elliptic PDEs." by John M. Neuberger and James W. Swift, International Journal of Bifurcation and Chaos, Vol. 11, No. 3 (2001), pp. 801-820.
A pdf of this paper can be downloaded from http://jan.ucc.nau.edu/~jmn3/paper.pdf

(2) Michael Falk - Arrangements

In this project students will work with mentor Michael Falk on aspects of  his research on arrangements of  complex hyperplanes. Algebraic, geometric and combinatorial techniques all come into play in understanding the topological structure of the complement of the union of a finite set of codimension-one linear subspaces of a complex vector space. Motivating examples and results come from the theory of the braid group, studies of complex plane curves and singularity theory, and applications to generalized hypergeometric functions and the Knizhnik-Zamolodchikov equation of mathematical physics.

Through the combinatorial notion of matroid, arrangements can be thought of as generalizations of graphs. Many open problems in the theory of arrangements can be approached using only the combinatorics of graphs or matroids and certain vector spaces, rings, or groups associated with them. Some major conjectures in the field can be reduced to elementary problems involving presentations of these groups and rings. Oftentimes methods developed by studying structures associated with graphs can be extended to the context of matroids and arrangements. Also, interesting examples can often be described in terms of graph- or matroid-theoretic constructions. In this way, research problems in this active area of topology are made accessible to undergraduates with background in linear and abstract algebra.

Previous REU projects have resulted in publications that have had a direct impact on research in the field:

M. Falk. Resonance varieties over fields of characteristic p, Int. Math. Research Notices, Vol. 2007, article ID rnm009, 25 pages, doi:10.1093/imrn/rnm009.

M. Falk and S. Yuzvinsky, Multinets, resonance varieties, and pencils of plane curves, Compositio mathematica, 143 No. 4 (Steenbrink issue) (2007), 1069-1088.

M. Falk and N. Proudfoot. Parallel connections and bundles of arrangements, Topology and Its Applications, 118 (2002), 65-83.

Tutte polynomials and Orlik-Solomon algebras, with C. Eschenbrenner, J. Algebraic Combinatorics, 10 (1999), 189-199.

The latter two papers were co-authored with REU students. The first two were helped along greatly by examples discovered by REU students.

(3) Steve Wilson – Semi-transitive and Bi-transitive Graphs

A symmetry (or automorphism) of a graph is a permutation of its vertices which preserves edges. The automorphisms of a graph G from a group Aut(G) under composition. Of particular interest are those graphs G for which Aut(G)  is large. For example, we say G is vertex-transitive (V-T) provided Aut(G) is large enough to have symmetries that take any one vertex onto all other vertices. Similarly we can talk about graphs which are edge-transitive (E-T) or those which are dart-transitive (a dart is a directed edge). Now certainly, if G is dart-transitive it must be V-T and E-T as well. Is the converse true? The answer is no, and the graphs which are V-T and E-T but not dart-transitive from a subclass called semi-transitive graphs. The project will offer investigations into many aspects of semi-transitive graphs and their close cousins, the bi-transitive graphs (E-T but not V-T_. students will investigate families of semi-transitive graphs, aspects of group actions on them, coverings and projections, families of balanced bi-transitive graphs and the eligibility of a given bi-transitive graph to be a building block for a semi-transitive graph.

Prerequisites: Graph theory; group theory.

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