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, Samuel Harris, Jeff Rushall, Nandor Sieben, and Michele Torielli.

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.

Data Science Accordion Closed

Research in this area focuses on bioinformatics and systems biology data analysis and modeling, data driven epidemiological modeling, statistical inference for dynamical systems, and more.

People active in this area include Robert Buscaglia, Ye Chen, Jaechoul Lee, Ben Lucas, and Rachel Neville.

Mathematics Education Accordion Closed

People active in this area include Brian Beaudrie, Barbara Boschmans, 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, time series, and extreme value analysis.

People active in this area include Brent Burch, Robert Buscaglia, Jaechoul Lee, Ben Lucas, Roy St. Laurent, and Jin Wang.

Analysis Accordion Closed

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

People active in this area include Samuel Harris and 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 (Emeritus) – 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.

Barbara Boschmans – Mathematics Education Accordion Closed

Dr. Barbara Boschmans earned a Doctorate in Curriculum and Instructions with an emphasis in Mathematics and Technology Education in 2003. She has taught for over 25 years and specializes in the mathematics content courses for pre-service K-8 teachers as well as teaching with technology in lower-level mathematics courses. She is a co-author on the college textbook A Problem Solving Approach to Mathematics for Elementary School Teachers (Pearson).

Brent D. Burch (Emeritus) – 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.

Ye Chen – Bioinformatics, Graph Theory, and Systems Biology Accordion Closed

Received PhD from West Virginia University in 2014. Her research interests are graph theory, network science, bioinformatics and systems biology data analysis and modeling, statistical inference for dynamical system.

In addition to her work in graph theory, she has collaborated on research projects in several disciplines including, sequencing (DNA, mRNA, miRNA, scRNA, 16s rRNA, proteomics) data processing and modeling, epidemiological (COVID-19, flu) modeling and prediction, hybrid dynamical systems modeling of biomolecular networks, land carbon cycle modeling.

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. 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.

Ernst is a Project NExT national fellow, a President’s Distinguished Teaching Fellow at NAU, and Co-Director of the Academy of Inquiry-Based Learning.

Michael J. Falk (Emeritus) – 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 University’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.

Shannon Guerrero – Mathematics Education Accordion Closed

Received PhD from UC Davis in 2005. She is a recognized leader in applying research-based approaches in teaching and learning mathematics and has received more than $5.5 million in grants to support her work. Her research interests include K-12 pre-service / in-service professional development, the effective use of technology in support of mathematics teaching and learning, and the process of teacher change.

Guerrero is also a Project NExT national fellow, a President’s Distinguished Teaching Fellow at NAU, and Co-Director of NAUTeach, NAU’s teacher preparation program for secondary mathematics and science education students.

Samuel Harris – Operator Algebras and Quantum Information Theory Accordion Closed

Samuel Harris received his PhD from the University of Waterloo (Canada) in 2019. His research focuses on connections between operator algebras, quantum information theory and (quantum) graphs.

Angie Hodge-Zickerman – Mathematics Education Accordion Closed

Dr. Angie Hodge-Zickerman earned her master’s degree in mathematics in 2005 and her PhD in mathematics education in 2007 both from Purdue University. Her research interests include active learning in the mathematics classroom, technology and its role in active learning, and equity issues in the STEM disciplines.

Jeffrey Hovermill – Mathematics Education Accordion Closed

Dr. Hovermill received his PhD in Instruction and Curriculum in Mathematics Education from the University of Colorado in 2003. His dissertation focused on the interactions between teachers’ understandings and practices regarding content, pedagogy, and technology and their instructional practices.

Most of his research and scholarship has focused on assisting classroom teachers to effectively support all of their students in increasing their efficacy towards, and understanding of, STEM.

Jaechoul Lee – Statistics and Data Science Accordion Closed

Received PhD from the University of Georgia in 2003. His main research interests are time series, extreme value analysis, and applied statistics. Specifically, he is interested in (1) theory and methods for time series analysis, (2) application of extreme value theory in climatology, (3) time-varying coefficient dynamic regression models, (4) algorithms and methods for large and big data, and (5) interdisciplinary research on range management, forestry, epidemiology, health science, and various fields.

Ben Lucas – Data Science Accordion Closed

Received PhD from the Monash University (Australia) in 2021. His research centers around using statistical learning, big data, and high-performance computing to solve real world problems. The applications of his work include environmental sciences and remote sensing; health and epidemiology; and sports analytics.

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.

Rachel Neville – Applied Algebraic Topology and Pattern Forming Systems Accordion Closed

She received her PhD from Colorado State University in 2017. Her research interests are in applied algebraic topology and modeling – specifically pattern forming systems.  From a topological and geometric perspective, she addresses questions like: How can one efficiently characterize and compare spatiotemporally complex patterns?  How can changes in geometric and topological structure inform our understanding of the mechanisms driving the dynamics in the system? Her work includes application of numerical techniques and machine learning algorithms.

These patterns arise in a wide diversity of physical, chemical, biological and ecological systems. Several applications of current interest are snow surface roughness, nano dot formation via ion bombardment, and dryland vegetation patterns.

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; coupled cell networks.

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.

Michele Torielli – Algebra and Combinatorics Accordion Closed

Received PhD from the University of Warwick (UK) in 2012. His research centers around the interplay between algebraic, geometric, and combinatorial properties of arrangements of hyperplanes. In addition, he also has an interest in the study of Groebner basis and in domination problems in graph theory.

Jin Wang – Statistics Accordion Closed

Received PhD in statistics from the University of Texas at Dallas in 2003. His research interests include nonparametric multivariate analysis, asymptotic theory, change-point problems, reliability and probabilistic risk analysis, and biostatistics.

His recent research focused on nonparametric multivariate analysis and its applications. The following are some representative works. Wang and Serfling (2005) introduced a nonparametric multivariate kurtosis measure. The measure is not only robust but also discriminates better among distribution shapes. In 2009, he proposed a family of kurtosis orderings for multivariate distributions, which is a pioneering work on multivariate kurtosis ordering. Various applications of the orderings have appeared in the literature. Furthermore, Wang and Zhou (2012) defined a generalized multivariate kurtosis ordering. Based on the ordering, they developed a two-dimensional visual device to compare two distributions in any dimension with respect to spread and kurtosis. In 2019, he established the uniform strong and weak convergences of generalized depth-based spread processes. Based on the results, he designed a graphical method to compare the spread and the kurtosis of two multivariate data sets and a new graphical method to assess multivariate normality.

Besides the theoretical research, he is also interested in applications of statistics in various fields. He worked in industry for eight years (1991-1999) and currently participated in several health-related projects at NAU.

 

 

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