Mathematics and Statistics

His current research interests are mixed linear models and variance components. This includes developing confidence interval and point estimation procedures for a function of variance components in mixed linear models.

Confidence intervals may be evaluated in terms of bias and expected length. Expected length computations depend on approximations of the cumulative distribution function of a linear combination of independent chi-square random variables.

This work has been undertaken with the assistance of Hari K. Lyer from Colorado State University. Point estimators of the intraclass correlation (a ratio of variance components) in the balanced one-way random effects model have evolved from confidence interval techniques.

Currently, joint work is being done with Ian R. Harris with the prospect of extending the results to more complex mixed linear models.

His research centers around the interplay between topological, algebraic, geometric, and combinatorial properties of arrangements of hyperplanes in complex vector spaces.

This focus leads to an interest in a wide variety of topics, including, in no particular order, algebraic topology, Lie algebras and root systems, combinatorial group theory, homotopy theory, geometry, algebraic geometry, singularities, representation theory, matroid theory, and polytopes.

In the early 1990's, he used methods from geometric group theory to develop a weight test for $K(\pi,1)$ line arrangements and to construct nontrivial homotopy equivalences of complements of line arrangements from combinatorial moves.

In joint research with Hiroaki Terao, he applied methods from the homology of posets to produce basic differential forms for multivariate hypergeometric integrals.

He has published joint papers and lectured on research conducted with undergraduates in Northern Arizona Univeristy's REU program, Carrie Eschenbrenner, Nick Proudfoot, and Cahmlo Olive and Eric Samansky.

Current interests involve resonance varieties and characteristic varieties, and nonlinear fiberings of arrangements, braid groups and artin groups, and applications of arrangements in quantum field theory.

His research area is now measure and integration theory, dealing especially with finitely additive measures, integral relationships such as Radon-Nikodym theorems, and Lp s paces.

His attentions have been divided between this area of research and changes in calculus instruction. In the latter, he has led workshops on calculus reform, written Mathematica laboratory exercises, and contributed to the redesign of the calculus program at Northern Arizona University.

His research in mathematics education centers on preservice teacher education and inservice teacher professional development.

Much of his work involves developing learning communities that facilitate praxis regarding the art and science of effective mathematics teaching.

His dissertation focused on the interactions between teachers' understandings and practices regarding content, pedagogy, and technology and their instructional practices.

His doctoral research concerned Sign-Changing Solutions of Superlinear Elliptic Boundary Value Problems.

He uses Variational Methods, Degree theory, and tools from Nonlinear Functional Analysis to investigate Uniqueness and Multiplicity of solutions. In general, he seeks to understand the Nodal Structure of solutions.

Bifurcation Theory is both a tool to be used and an area to be contributed to in his research.

Owing to his industrial experience at E-Systems, he uses Numerical Analysis to further investigate the above facets of Elliptic PDEs.

In particular, his method of projecting sign-changing functions onto an infinite dimensional submanifold of the Hilbert Space H_0 ^{1,2} has proven successful in numerically calculating solutions.

He is currently adapting the algorithm to investigate solutions on the disk, annulus, and arbitrary regions, as well as trying to prove corresponding analytic results. He is presently collaborating with several other mathematicians who have independently developed distinct and complementary numerical algorithms.

His other research interests include Neural Networks, Bifurcation of Water Waves, Nonradial Solutions on Radially Symmetric Regions, Parabolic and Hyperbolic PDEs, and extensions of previous results to include a wider class of nonlinearities such as Semipositone Problems.

His research interests are in linear and nonlinear regression models, in particular the development of regression diagnostics for assessing the influence individual (or groups of) observations exert on model fit in a nonlinear regression model.

Diagnostic methods in common use in linear regression analysis are often (mis)applied to nonlinear regression problems.

Some of the topics his research addresses are: characterizing when diagnostic methods designed for linear models may be appropriately applied to nonlinear models; and generalizing common diagnostic methods to nonlinear regression models.

Also, he has an interest in the development of procedures for evaluating agreement (or concordance) between two methods of measurement of a physical quantity, particularly when one such method is a gold standard. 

The topic of his doctoral research was Regular Maps. These are embeddings of graphs on surface that have large groups of symmetry.

His contributions to this theory include papers on operators in maps, branched coverings, smooth coverings, chiral maps, constructions from matrix groups and permutation groups related to Grünbaum’s Conjecture, and more recently, regular maps and hypermaps on non-orientable surfaces.

Work on the question of which graphs occur in regular maps, as well as experiences in the REU program, have led to investigations into symmetry in graphs. These include papers in semi-symmetric and 1/2-tansitive graphs, that is, graphs whose groups are transitive on edges, but not on darts.