# Math Faculty Research Interests

**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.

**James Carter**:*Algebraic number theory*

Algebraic number theory is the study of finite extensions of the field of rational numbers. This study basically originated in the attempts to resolve a famous problem known as "Fermat's Last Theorem" some 150 years ago. Since that time algebraic number theory has grown into one of the major areas of mathematical research. The above mentioned problem of Fermat was finally solved in 1995 by Professor Andrew Wiles of Princeton University, but there remain many interesting problems yet unsolved which have been generated along the way. Some of these problems concern classifying various modules over rings within the context of Galois extensions of algebraic number fields. I am currently interested in these problems and related topics.

**Ben Cox**:*Representation theory*

I work on the representation theory of quantum groups and infinite dimensional Lie algebras. The Norwegian mathematician Sophus Lie (1842-1899) developed an brilliant technique using algebra to solve differential equations. Differential equations and their solutions are vital to applications of mathematics to physics, biology, economics, and chemistry. The subject of quantum groups and infinite dimensional Lie algebras plays a major part in the area of theoretical physics called quantum field 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.

**Martin Jones**:*Probability*

Stochastic processes are families of random variables indexed typically by either the integers or the positive real numbers where one thinks of the index as "time" in either a discrete or continuous setting. Stochastic processes have many applications such as analysis of the stock market, the study of population growth, and the spread of epidemics. My particular research interests are in optimal stopping theory. Here the observer may stop observing the process at a particular time hoping to achieve some goal or maximize a reward. Such stopping is done without knowledge of future observations so the objective is to choose wisely rules for stopping the process so as to maximize the expected value of the reward. A related area in which I am also interested is the study of "Bandit" processes. Here the observer observes one of finitely many processes at each of a finite or infinite number of stages. The observer may switch to a different process at any stage using only information from the stages already observed to guide the selections. The objective is to find optimal strategies to aid the observer in making the selections.

**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.

**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.

**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.

**Mukesh Kumar**:*Numerical analysis and Singular perturbation problems*

Mathematics for better society! My research mostly concerns numerical analysis, ordinary and partial differential equations, singular perturbation problems, and the interplay between these fields. In recent years, I have focused on design and analysis of highly efficient and robust numerical methods for solving partial differential equations. In particular, my research work can be characterized by the following keywords: Isogeometric analysis, Finite element methods, Error estimation and adaptivity, Compatible discretization of PDEs, Domain decomposition methods, Computational mechanics, and Singular perturbation problems. Main area of application is Computational mechanics, i.e. both Solid/Structural and Fluid mechanics relevant for Mechanical, Marine, and Petroleum Engineering as well as Geophysics and Renewable energy.

**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**

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.

**Dinesh Sarvate**:*Combinatorics*

Combinatorics or Discrete mathematics is a branch of mathematics which many bright students who like problem solving may appreciate. My particular area of interest is the construction of block designs and related structures, for example, balanced incomplete block designs, group divisible designs, balanced (part) ternary designs, and Bhaskar Rao designs, but I would like to think that I can work in other branches like Graph Theory, Coding Theory and Database Security if some one would like me to. I like to work with students for their undergraduate research experience if they are ready to spend time on the assigned problem and are ready to meet me almost everyday. Other than pursuing above research interests, I would like to obtain or at least understand the future solution of a conjecture in combinatorics by Peter Frankl. The conjecture is called the Union-Closed Sets Conjecture. It has been considered by Peter Winkler as one of the most embarrassing gaps in combinatorial knowledge.

**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.

**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.