# Research areas

## NAU’s Department of Mathematics & Statistics current projects

Learn about the various research projects taking place with Northern Arizona University’s mathematics and statistics faculty.

### Areas of focus and researchers active in these areas

#### Algebra, Combinatorics, Geometry and Topology Accordion Closed

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

People active in this area include Dana Ernst, Michael Falk, Guenther Huck, Sudipta Mallik, Jeff Rushall, Nandor Sieben and Bahattin Yildiz.

#### Applied mathematics Accordion Closed

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, Ye Chen, John Neuberger, Nandor Sieben and Jim Swift.

#### Mathematics education Accordion Closed

People active in this area include Brian Beaudrie, Barbara Boschmans, Terry Crites, Shannon Guerrero, Angie Hodge-Zickerman and Jeff Hovermill.

#### Statistics Accordion Closed

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

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

#### Measure Theory and Functional Analysis Accordion Closed

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

People active in this area include Nandor Sieben.

### Learn about NAU’s research faculty

The research faculty of the Department of Mathematics & Statistics each bring a unique background and perspective to their area of expertise.

#### Terence R. Blows—nonlinear differential equations Accordion Closed

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 Accordion Closed

Received PhD from Colorado State University in 1996. 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.

#### Robert Buscaglia—statistics Accordion Closed

Received PhD Applied Mathematics in 2018 from Arizona State University and PhD Biochemistry and Molecular Biology in 2012 from University of Louisville. Research is focused on the interdisciplinary application of statistical modeling and model building and the effective use of computational techniques.

Research on curve discrimination using the fields of Functional Data Analysis, Nonparametric Classifiers, and Ensemble Learning is being used to aid the identification of disease through a Biohphysical assay known as clinical DSC. Clinical DSC has potential applications for autoimmune disorders, cancer, and mycordial infarction.

Statistical research is focused on algorithms capable of identifying disease based on plasma thermograms, and developing models that mix information from multiple sources and variables types, including the mixing of functional and conventional (scalar) covariates.

Research goals include further development and understanding of ensemble modeling, improved curve discrimination based on functional classifiers, and effective programming for practitioner ease-of-use.

#### Dana Ernst—algebraic combinatorics and inquiry-based learning (IBL) Accordion Closed

Received PhD from University of Colorado Boulder in 2008. His primary mathematical research interests are in the interplay between combinatorics and algebraic structures. More specifically, he studies the combinatorics of Coxeter groups and their associated algebras, as well as combinatorial game theory. The combinatorial nature of his research naturally lends itself to collaborations with undergraduate students, and one of his goals is to incorporate undergraduates in his research as much as possible.

Ernst is also a Project NExT national fellow and Special Projects Coordinator for the Academy of Inquiry-Based Learning. His interests also include the scholarship of teaching and learning (SoTL). In particular, he is passionate about inquiry-based learning (IBL) as an approach to teaching/exploring mathematics. Moreover, he actively give talks and organizes workshops on the benefits of IBL as well as the nuts and bolts of how to implement this approach in the mathematics classroom. Together with Angie Hodge, he is a coauthor for “Math Ed Matters”, which is a (roughly) monthly column sponsored by the Mathematical Association of America. The column explores topics and current events related to undergraduate mathematics education.

#### Michael J. Falk—topology and combinatorics Accordion Closed

Received PhD from University of Wisconsin – Madison in 1983. 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.

#### Jeffrey Hovermill—mathematics education Accordion Closed

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.

#### Sudipta Mallik—algebraic graph theory and combinatorial matrix theory Accordion Closed

Received PhD from University of Wyoming in 2014. His research interests focus on Combinatorics and Linear Algebra, in particular the interplay between graphs and matrices. He studies the interrelation between the structure of a graph and the properties of the associated matrix, such as rank and eigenvalues.

His results include a spectral characterization of matchings in a graph, a sharp linear algebraic upper bound of the number of odd cycles in a graph and Matrix Tree Theorem for Signless Laplacians of graphs. He has also contributed to the study of minimum rank problems of graphs and structured inverse eigenvalue problems.

He has recently developed interests on linear coding theory and statistics.

#### John M. Neuberger—partial differential equations and numerical analysis Accordion Closed

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.

#### Jeff L. Rushall—algebraic number theory Accordion Closed

Received M.S. from University of Nebraska – Lincoln in 1992. His formal research area is Algebraic Number Theory, specifically p-adic analysis of various spectra, but he also focuses on research projects that are accessible and tractable for mathematics majors. These include open problems involving irreducible numerical semigroups, generalizations of Hadamard matrices, and graph labelings. His recent undergraduate students that have publications to their credit include Brooke Fox; Ben Lantz and Michael Zowada; Taryn Laird and Jose Martinez; Alessandra Graf; and Nathan Diefenderfer, Michael Hastings, Levi Heath, Hannah Prawzinsky, Briahna Preston, Emily White, and Alyssa Whittemore.

#### Roy T. St. Laurent—statistics Accordion Closed

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 Accordion Closed

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 Accordion Closed

Received PhD in physics from UC Berkeley in 1984. His research is in dynamical systems, with emphasis on symmetric systems and bifurcation with symmetry.

### Publications

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