## Research areas

### Algebra, combinatorics, geometry and topology

Research in this area focuses on commutative algebra, combinatorial geometry, numerical semi-groups, combinatorics and geometry.

People active in this area include Michael Falk, Guenther Huck, Jeff Rushall, and Nandor Sieben.

### Applied mathematics

Research in this area focuses on ordinary and partial differential equations, numerical analysis, dynamical systems and operations research.

People active in this area include Terence Blows, Shafiu Jibrin, John Neuberger, Nandor Sieben and Jim Swift.

### Mathematics education

People active in this area include Brian Beaudrie, Barbara Boschmans, Terry Crites, Shannon Guerrero, Jeff Hovermill and April Judd.

### Statistics

Research in this area focuses on mixed linear models, variance components, regression and measures of agreement.

People active in this area include Brent Burch, Roy St. Laurent, Derek Sonderegger and Jin Wang.

### Measure theory and functional analysis

Research in this area focuses on operator algebras and measure theory.

People active in this area include John Hagood and Nandor Sieben.

### More on faculty research

Need more? Click on the faculty names below to get an in-depth view of their research backgrounds.

Terence R. Blows—nonlinear differential equations

Received PhD from University College of Wales, Aberystwyth, 1982.

His research area is Nonlinear Differential Equations. Specifically, limit cycles of planar autonomous differential equations (Hilberts Sixteenth Problem). Such differential equations include those of Lienard type, and many that are used in applications, for example in population ecology.

Brent D. Burch—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.

Michael J. Falk—topology and combinatorics

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.

John W. Hagood—analysis

Received PhD from the University of Utah in 1977 in Probability 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.

Jeffrey Hovermill—mathematics education

Received PhD from University of Colorado-Boulder in 2003 in Instruction and Curriculum in Mathematics Education.

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.

Shafiu Jibrin

His main research interests are on linear matrix inequality (LMI) constraints in the field of positive semidefinite programming. This includes finding ways to eliminate redundant constraints from large LMI constraints sets and developing methods for the analysis of LMI constraints sets. It has been established that the removal of redundancies can improve the performance of interior point algorithms, thereby increasing the size of the problems that can be solved.

A related important question is how to quickly find a feasible point or a near-feasible point of the LMI constraints set.  Did joint work with Dr. Richard Caron and Dr. Tim Traynor, both of the University of Windsor in Canada. He is currently working with a colleague, Dr. Jim Swift, on constructing constraint consensus methods for finding strictly feasible points for LMI constraints sets.

Dr. Jibrin is also interested in developing new algorithms for solving semidefinite programming problems that uses weighted analytic centers for LMIs.

Janet M. McShane—commutative algebra

Received PhD from The University of Arizona in 1992.

Her doctoral dissertation involved the computation of polynomial invariants of finite groups. Currently her focus is on the computation of polynomial semi-invariants of finite groups.

John M. Neuberger—partial differential equations and numerical analysis

Received PhD from University of North Texas in 1995.

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.

Roy T. St. Laurent—statistics

Received PhD from University of Minnesota 1987 in Statistics.

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.

Nándor Sieben—operator algebras and combinatorics

Received PhD from Arizona State University 1997 in Operator Algebras.

His research interest is in C*-algebras; C*-dynamical systems; groupoid, inverse semigroup and graph algebras; combinatorial game theory; graph pebbling; partial differential equations.

James W. Swift—dynamical systems

Received PhD in physics from UC Berkeley in 1984.