Math Faculty Research Interests
Our faculty have research expertise in many different areas. You can see their interests and descriptions of some of their work listed below.
Visit the Meet the Faculty page to see bios for each of the faculty members.
Current students can find more information on research at the College on the Math Student Hub.
Stacie Baumann: Discrete Mathematics
My research area is discrete mathematics, more specifically, design theory and graph theory. Most of my work can be found in the intersection of these areas. Often, I reduce design theory questions, such as those involving graph decompositions or completing partial latin squares, to graph theory questions. Then, I use a variety of techniques from graph theory to solve the problems. Recently, I have also become interested in extremal problems in graph theory. Specifically, I am interested in Turán-type problems involving finding the maximum or minimum number of edges a graph with a certain property can contain.
Annalisa Calini: Integrable PDEs and dynamical systems
I am interested in completely integrable partial differential equations: infinite dimensional counterparts of integrable Hamiltonian systems of Classical Mechanics. As many of them arise in the description of the asymptotic behaviour of various physical systems, these equations exhibit a number of universal properties: large families of exact solutions (among them the solitons described by Alex Kasman), an infinite number of conservation laws, an infinite sequence of commuting flows which allows the dynamics to be linearised, and a solution space with a particularly rich topology. In recent years, mathematicians have discovered (and rediscovered) deep connections between integrable equations and the geometry and topology of curves and surfaces. Exploring these connections is my main interest: I have used tools from Algebraic and Differential Geometry, Topology, Dynamical Systems, Symplectic and Contact Geometry in my explorations. If one perturbs an integrable equation, very complicated motions may arise, due to the underlying presence of instabilities. I use Dynamical Systems methods generalized to an infinite-dimensional setting in order to understand the mechanism for the onset of irregular and chaotic dynamics.
Nick Davidson: Representation Theory
I study representation theory, which is a branch of mathematics using techniques from linear algebra (a well-understood subject!) to learn more about abstract algebra, a more complicated area of mathematics that arose from the study of symmetry.
Behrang Forghani: Random walks on groups, probability, Ergodic theory, geometric group theory
Dr. Forghani works in the wide area of probability and statistics that involves random walks on groups (including compact or discrete examples), along with applications to geometric group theory, combinatorics, and graph theory.
Tom Ivey: Differential geometry and differential equations
I'm interested in several topics that lie at the interface between differential equations and differential geometry. These include using differential equations to construct geometrically significant objects (such as special curves and surfaces), and using geometry to study--and more clearly understand--differential equations. I have published work on geometric evolution equations (including the Ricci flow and the vortex filament flow), and my areas of expertise include integrable systems, exterior differential systems, the calculus of variations, and Cartan's method of equivalence.
Renling Jin: Mathematical logic
My primary research interest is mathematical logic, including set theory, model theory and nonstandard analysis. Set theory is the foundation of mathematics. Theoretically, almost all mathematics can be developed under a system of basic axioms called ZFC (Zermelo-Fraekel Set Theory with the Axiom of Choice). I have been working on set theory with an emphasis on the independence proofs. Using various methods, I have proved that some mathematical statements in infinite combinatorics and in measure theory are neither provable nor disprovable under ZFC. I am also interested in applying set theoretic results to point-set topology. Model theory discusses the relationships between a set of sentences and a model, i.e. a mathematical entity, in which all sentences from the set are true. I have been working on the construction of models using ultrapower construction method and on exploring how the properties of an ultrafilter used in the construction affects the properties of the model. Nonstandard analysis allows us to use "infinitely large" numbers and non-zero numbers which are "infinitesimally close to" zero, in an enlarged universe called nonstandard universe. Taking this advantage, one can derive a result in the nonstandard universe and then push down the result to the standard world to get an interesting theorem. I have been working on both foundational side and applied side of this subject. In the foundational side, I have been exploring the better alternatives of the nonstandard universe. On the applied side, I found a successful application of nonstandard methods in additive number theory, especially for dealing with density problems of sequences of natural numbers. I am also interested in applying nonstandard analysis to probability theory and measure theory.
Liz Jurisich: Representation Theory of Lie algebras
The focus of my current research is a class of algebras called Lie algebras. Specifically, I study the structure and representation theory of infinite-dimensional Lie algebras. The algebras that I study arise ''in nature'' as symmetries of quantum mechanical systems, and are also of interest to group theorists, number theorists and other mathematicians. Research into the type of algebras which I study is relatively new (algebras defined from symmetrizable matrices were introduced by V.Kac and R. Moody in 1968), and there are many unaswered questions.
Bo Kai: Statistics
My research interests are in the areas of high-dimensional data analysis, semiparametric methods, robust modeling and variable selection. Nowadays, researchers are able to collect huge amount of data without too much cost. How to provide effective and efficient ways to analyze high-dimensional data becomes one of the most important research topics in modern statistics. Analysis of high-dimensional data is very challenging. One challenge is that there are too many variables in the datasets. Another one is that high dimensional data particularly likely contain outliers. Motivated by these two challenges, my research aims to develop new statistical methodology and inference procedures for analysis of high dimensional data in the presence of outliers and/or contamination.
Alex Kasman: Algebraic analysis and mathematical physics
Even though I don't think of myself as an "applied mathematician", my research has been published in biology and physics journals as well as in mathematics journals. Algebraic analysis is a very broad area that studies the algebro-geometric structure underlying calculus. Although most of my work is "pure mathematics", much of my research involves an unexpected interplay between this area of math and the dynamics of particles, waves, quantum mechanical systems and biological systems. In my papers you can read about waves (especially solitons), commutative rings of differential operators, Grassmannian manifolds, Jacobian varieties, systems of interacting particles, viral infection of bacterial systems, the bispectral property and quantum integrable systems.
Tom Kunkle: Approximation theory
If you've seen tangent line approximation, the trapezoidal rule, Taylor polynomials, numerical analysis, or linear regression, then you've seen some approximation theory. Without it, calculators wouldn't do much beyond +-*/. My research has included exponential box splines and multivariate divided differences. A box spline is a compactly-supported multivariate piecewise polynomial whose integer translates are used in surface fitting. Divided differences are discretizations of differential operators, and they arise in the study of polynomial interpolation. Most recently, I've been working on following multivariate interpolation problem. Is it possible to extend a function defined on a discrete set of points to all of R^d is such a way that the nth derivatives of the extension are no more than some constant times the nth divided difference of the data?
Stephane Lafortune: Nonlinear wave theory, integrable systems
My research interests include nonlinear wave theory and integrable systems. I am particularly interested in the stability of coherent structures in partial differential equations (PDE). I make use of Hamiltonian structures and the Evans function technique to study the stability of solutions to PDEs appearing in the description of elastic materials, flame propagation and other applications. As far as integrable systems are concerned, I am mainly interested in integrability detectors for continuous, discrete, and ultra-discrete equations.
Amy Langville: Information retrieval, numerical linear algebra, mathematical modeling
I am interested in the mathematics of information retrieval. The results of search engines, such as the Addlestone Library engine or the giant web engines of Google and Yahoo! , are produced by an information retrieval system. Such systems depend heavily on mathematical techniques from fields such as numerical linear algebra, optimization, operations research, and computer science. My recent work focuses on various factorizations of large sparse matrices that improve the speed and accuracy of information retrieval systems. This work involves a great deal of mathematical modeling and numerical experimentation, and as a result, is very applied. Most of my experimental data comes from companies such as The SAS Institute, Yahoo! Research, and Google.
Brenton LeMesurier: Nonlinear wave phenomena and scientific computing
My main research area is self-attractive nonlinear interactions in wave propagation, such as arises with self-focusing of lasers, electromagnetic waves in super-heated ionized gasses (plasmas), vibrations in molecules and thin molecular films, and in dense collections of atoms at extremely low temperatures (Bose-Einstein condensates). Most recently, I have been interested in energetic pulses traveling along protein molecules. This subject is pursued through a mixture of (a) developing numerical simulation methods that deal with the extremely fine spatial and temporal scales that develop in these phenomena, and (b) theoretical analysis guided by observation in numerical simulations.
Jiexiang Li: Statistics
My research focuses on nonparametric estimation for weakly dependent random fields. In classical analysis, we assume our observations are independent and identically distributed. In reality, the assumption does not always hold. For example, the life lengths of a certain species are not independent because the species tend to live in the similar environment. Random fields are stochastic processes indexed by lattice points. Difficulties arise since lattice points in higher dimensional space cannot be linearly ordered. Bernstein Blocking argument is applied to divide the observations of interest into independent blocks and negligible blocks. I am particularly interested in the asymptotic behavior such as asymptotic distribution and convergence rate of the proposed nonparametric estimators.
Dan Maroncelli: Functional Analysis
My general research is in the area of functional analysis, ordinary and partial differential equations, and difference equations.
Robert Mignone: Logic/Foundations of Mathematics, Set Theory
My research is Set Theory, which is a part of the broader area of Logic/Foundations of Mathematics. I study the higher infinite called large cardinal hypothesis, such as super compact cardinals and huge cardinal hypothesis. Lately, my interest has focused on the Ultimate L conjecture, which seeks an inner model type structure so robust and near the boundary of consistency and inconsistency rendering forcing extensions impotent at adding things of interest, yet nice properties of inner models are preserved such as the Continuum Hypothesis, yielding an axiom for the universe of all sets, V = Ultimate L, that could comfortably align with mathematical platonism.
Garrett Mitchener: Dynamical Systems and Probability
I study probability, dynamical systems, and data science, with applications to biology and linguistics. I use artificial life simulations to understand the evolution of gene regulatory networks and natural computation. I also experiment with ranking and explainability in data science.
Kate Owens: Mathematics education; universal algebra; mathematical logic. Nonfinite axiomatizability of equational theories of finite algebras.
Jin-Hong Park: Statistics
I have conducted research about the fundamental problem of modeling time series data, a time ordered sequence of observation, which is radically different from those of conventional time series analyses. Over the years, the scientific community has witnessed the development of many useful parametric and nonparametric methods for analyzing time series data. Nevertheless, there is a never-ending quest to build new and modern methodologies to analyze time series data, which arise in a variety of fields such as economics, meteorology, engineering, geophysics, social and environmental science. My research interests include dimension reduction in time series as well as applied time series modeling, time series regression model such as intervention analysis, and financial/econometric applications in statistics.
Kathryn Pedings-Behling: Ranking, rankabililty; mathematics education
Kathryn's applied mathematics research interests include problems in the field of ranking and the emerging field of rankability and, in particular, their use to ensure fairness in different processes, such as hiring. She also enjoys action research in the field of mathematics education, working with reluctant postsecondary math learners in their terminal math course. She is studying the changes in their attitudes toward mathematics, particularly with respect to a curricular intervention.
Daniel Poll: Mathematical Biology
Dr. Poll works in computational and theoretical neuroscience. He is an expert in nonlinear dynamics, control theory and stochastic processes, reduced models, and parallel computing. Applications include modeling synaptic plasticity and working memory.
Andrew Przeworski: Topology
Dr. Przeworski's area of interest is the geometry and topology of hyperbolic 3-manifolds, in particular how these topics relate to packing problems in hyperbolic space.
Sandi Shields: Low-dimensional topology
To study the topology of 3-manifolds it is often beneficial to look at lower dimensional objects imbedded in the manifold. For example, Haken 3-manifolds contain an imbedded imcompressible surface, that is a surface which sits in the manifold in a way that is topologically significant, and this surface has been used to extract a wealth of information about the 3-manifold. More recently, the topological structure of foliations and laminations has been used to study the topology of 3-manifolds. (A foliation is a decomposition of the manifold into imbedded surfaces, called "leaves" which locally stack up like a trivial product of planes. A lamination is a partial foliation whose complement is incompressible.) The focus of my research is low-dimensional topology; in particular, the study of foliations and laminations of 3-manifolds. For this I use branched surfaces constructed from foliations to find conditions that guarantee certain topological properties such as the existence of a compact leaf of a certain genus, the R-covered property, or finite branching when the foliation is lifted to the universal cover. These conditions often guarantee stability of the topological structure under sufficiently small perturbations of the tangent bundle to the leaves.
Oleg Smirnov: Linear algebras
My research area is Algebra, specifically, the study of linear algebras. A linear algebra is a set of objects with addition and multiplication. Integer, real and complex numbers are classical examples of algebras. Another important examples are matrices, functions, and operators. Linear algebras play important role in geometry, topology, number theory, differential equations as well as in physics, genetics, economics.
Tingting Tong: Statistical inference; Parameters estimation; Statistical models; Model averaging
James Young: Probability; stochastic processes; statistics
Paul Young: Number theory, p-adic analysis
Number theory has been around for thousands of years and has been a source of the most beautiful problems, methods, and results in all of mathematics. P-adic analysis was developed around 1900 to attack problems in number theory and finite field theory, and in this regard has been brilliantly successful. One of its major triumphs has been the proof of the Weil conjectures for algebraic varieties in prime characteristic, especially the proof of the Riemann Hypothesis in prime characteristic. P-adic techniques also played a vital role in Andrew Wiles' celebrated proof of the Fermat conjecture. My research involves p-adic differential equations, p-adic special functions, and p-adic integration, and builds bridges between number theory, algebraic geometry, and formal group theory.