Research Interests: Statistics

Dr. Chen's research mainly focuses on bioinformatics, especially in developing statistical methods and algorithms for functional genomic data. Dr. Chen's past publications involved with the development of mixture models for clustering high dimensional sequences, its related theoretical justifications and applications. She also published papers in neuron spike trend studies, data mining, analysis of election data and DNA sequences' matching probability.

Research interests: Lie theory, Representation theory

My main research interest lies in Lie theory and representation theory. Suppose that W is a Coxeter group, e.g., a finite group generated by reflections on a Euclidean space. Then W can be decomposed into a disjoint union of left cells or two-sided cells. These cell structures can be applied to construct the representations of W and the representations of the Hecke algebra of W explicitly. When W is a Weyl group or an affine Weyl group, the two-sided cells of W are also closely related to the unipotent classes in the corresponding linear algebraic group. Algebraists try to find an explicit description of the left cells and two-sided cells in each W, especially in the case when W is an affine Weyl group.

I am also interested in applied mathematics, e.g., the theory of asset pricing in mathematical finance.

Research interests: Applied Statistics

I am an applied statistician focused on collaboration and consulting with scientists. I often assist in data analysis when colleagues in other fields (geosciences, biology, etc.) have difficult or unusual problems. Some of the problems I have worked on recently involve meta-analysis, discriminant analysis with messy data, partial least squares (projection to latent variables), geographically weighted regression to explore spatial patterns and possible causes of prostate and breast cancer, and the use of LiDAR remote sensing to estimate landscape characteristics in semi-arid climates. When appropriate, information criteria plays a role in model selection and assessment. I intend to use the right tool for the job, so I must teach myself new techniques as needed.

I also have a continuing interest in statistics education, including statistical literacy. I am developing a collection of cases for a case-based approach to applied statistics similar to the Statistical Sleuth by Ramsey and Schafer, but aimed at the undergraduate level.

Research interests: Differential Geometry, Geometric Analysis, and Complex Manifolds.

My main research interest lies in Differential Geometry. My most recent publication with H. T. Laquer is titled Hyperplane Envelopes and the Clairaut Equation, Journal of Geometric Analysis, Vol. 20, Issue 3 (2010), Pages 609-650. The paper brings a modern perspective to the classical problem of envelopes of families of affine hyperplanes. In the process, the classical results are generalized and unified.

A key step in the work is the use of "generalized immersions". Briefly, every classical immersion defines a generalized immersion in a canonical way so that generalized immersions can be understood as ordinary immersions "with singularities." Next, the concept of an envelope is given a modern definition, namely, an envelope is a generalized immersion "solving the family" that has a universal mapping property relative to all other full rank "solutions". The beauty of this approach becomes apparent in the "Envelope Theorem". With one mild assumption, namely that the associated family of linear hyperplanes is immersed, it is proven that a family of affine hyperplanes always has an envelope, and that envelope is essentially unique.

Research interests: Numerical Analysis, Inverse problems for PDE, CFD.

The main research interests of Dr. Gryazin lie in the area of Numerical Analysis and Scientific Computation. More specifically, he focuses on the development of computational approaches to the solution of applied mathematical problems arising from wide range of applications including computational electromagnetics, medical imaging, inverse problems, computational fluid dynamics, and computational finance. The results of his research recently appearing in internationally recognized publications were related to Krylov subspace based numerical methods for large sparse nonsymmetric algebraic systems and regularized stochastic optimization algorithms in risk portfolio management.

Research Interests: 

The main focus of my current research is mathematical modeling and solving associated statistical problems in biomedical sciences including cell biology, molecular biology, biochemistry, radiation biology, bioinformatics, cancer biology and epidemiology, and clinical oncology. The mathematical basis of this work is probability models, stochastic processes and differential equations. I am also working on mathematical problems of heat transfer. 

Research interests: Representation theory, Hecke algebras, Lie theory.

My research concerns algebras and graphs related to groups generated by reflections of a vector space. Recent research projects involve deformation algebras for symmetric groups, as well as Cayley graphs of complex reflection groups, specifically their spectra (eigenvalues) and traversability (paths and cycles).  Past research has involved Hecke algebras for noncrystallographic real reflection groups, their representations, and related hyperplane arrangements.

Note to students interested in research opportunities: Graphs (collections of vertices and some edges between them), provide a rich source of accessible open problems, are widely applicable to modeling relationships in computer science, chemistry, biology, physics, and the social sciences, and provide an area in which to apply techniques from representation theory. Come ask about possible projects.

Research interests: Differential Geometry.

Research interests: Differential geometry, Calculus of variations

My research involves applications of variational calculus to problems in differential geometry. The shape of surface interfaces is found by minimizing a certain surface energy. We are particularly interested in anisotropic interfacial energies. This means that the energy depends on the direction of the surface, as in a crystal.

Research interests: Lie groups, geometry, and dynamical systems

Dr. Payne's research is on geometric and dynamical problems related to Lie groups and Lie algebras. Recently she has been interested in the Ricci flow for homogeneous spaces, soliton metrics on nilpotent Lie algebras, and Anosov maps on nilmanifolds.

Research Interests: My training is in algebraic geometry (PhD, Brown, 1981), but I also have significant background in computer science and subjects related to aviation. I have a strong interest in applications of algebraic geometry and representation theory in coding and cryptography.

Current projects

• Computational Complexity of Quadrature Rules. I am applying concepts from information theory to develop new algorithms for estimating the Riemann integral of a function defined by a table of values.
• Information Theoretic Schottky Problem. I am studying the statistical properties of period matrices of complex algebraic curves to determine to what extent the distribution of the periods determines (1) whether the matrix is in fact a period matrix [Schottky Problem] and (2) properties of the curve [Torelli Problem].
• Theta Divisors in Moduli Spaces of Vector Bundles and Automorphism Groups of Curves. I have several results in this direction, but the project is not mature enough to be written up.
• Using Turbulence to Fly Faster. A development of and analysis of "relative dolphin flight" for powered aircraft.

Other Recent projects

• Linear Codes from Schubert Varieties. Much of this is joint work with Sudhir Ghorpade.
• Analytic Computation of Some Automorphism Groups of Riemann Surfaces, Kodai Mathematical Journal, 30 (2007), 394-408.

Research interests:  Integro-differential equations, dynamical systems and their applications in biology and ecology.

Recent research projects involve the differential equations with nonlocal diffusion operators, which can be considered as a nonlocal analog of the classical diffusion operators (the Laplacian operator). Among others, I'm interested in establishing results for nonlocal differential equations analogous to those of classical differential equations from the perspectives of spectral theory (eigenvalue problems), symptotic dynamics (long time behaviors), as well as applications to population models.

There are some undergoing research projects for undergraduates. Analyze the ordinary differential equation from the geometric point of view and use MATLAB or other techniques to illustrate the asymptotic behavior of solutions qualitatively. There are a variety of applications in Engineering, Chemistry, Physics, and many other
scientific subjects. For example, the bifurcation of a system when some control parameters are changing, and the steady state of a system when the time goes to infinity.

Research interests: Scientific computing

Numerical solutions of differential equations and optimal control problems with partial differential equation constraints. Particularly interested in phase field approaches to optimal control problems.

Research interests:

Dr. Zhu's research is in the area of numerical analysis. The main focus is on developing and analyzing numerical approximation and efficient solvers for both linear and nonlinear partial differential equations arising from many physics, engineering and biochemistry problems, such as groundwater simulation, electromagnetic, electrostatic interactions, and density functional theory, etc.