# Department Research

#### Shu-chuan Chen, *Associate Professor, Ph.D.*

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.

#### Yu Chen, *Associate Professor, Ph.D.*

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.

#### DeWayne Derryberry, *Assistant Professor, Ph.D.*

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.

#### Robert Fisher, *Professor, Ph.D.*

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.

#### Briana Foster-Greenwood, *, Ph.D.*

My research combines several areas of mathematics: Algebra (Commutative and Noncommutative), Geometry, Representation Theory, Invariant Theory, Combinatorics, and Cohomological Algebra. More specifically, I am interested in complex reflection groups (including symmetry groups of regular complex polytopes) and the deformation theory of algebras arising from group actions.

Given a finite group acting linearly on a finite-dimensional vector space V, one can define a skew group algebra as the semidirect product of the group algebra and the coordinate ring of V. Various algebras of interest (such as graded Hecke algebras obtained by replacing a commutative relation vw-wv=0 with a noncommutative relation of a certain form) are already known to occur as deformations of skew group algebras. What other associative deformations are possible? Current projects invoke invariant theory and partial orderings on groups to determine the Hochschild cohomology governing potential deformations of a skew group algebra.

#### Yury Gryazin, *Associate Professor, Ph.D.*

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.

#### Leonid Hanin, *Professor, Ph.D.*

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.

#### Cathy Kriloff, *Professor, Ph.D.*

I use primarily algebraic and combinatorial methods to study the representation theory of graded (also called degenerate affine) Hecke algebras that are built from the groups of symmetries of

- regular polygons in the plane
- the regular icosahedron (or its dual the dodecahedron) in 3-space, and
- the two related dual regular polyhedra in 4-dimensional space with five-fold symmetry (the 120-cell and 600-cell)

The algebras are infinite-dimensional but their representation theory (the ways they can act as linear transformations on finite-dimensional vector spaces) is tightly controlled by combinatorial properties of the finite symmetry group from which they are built. These non-crystallographic cases are less commonly considered than the crystallographic cases that arise within Lie theory, but the differences and similarities in their representation theory provide intriguing hints at possible new underlying objects and directions for unifying currently separate theories. Recent generalizations of graded Hecke algebras include the specific examples listed above, providing new avenues of exploration and further motivation to study these cases. Various appearances of non-crystallographic objects in the mathematics and physics literature (e.g., in the study of representation theory, in connection with moment graphs in geometry, in several combinatorial contexts, and in quasicrystals, amorphous solids, and wavefronts) also indicate tantalizing connections and directions for further study.

#### Patrick Lang, *Professor, Ph.D.*

Mathematical analysis of statistical methods.

#### Bennett Palmer, *Professor, Ph.D.*

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.

#### Tracy Payne, *Professor, Ph.D.*

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.

#### Dennis Stowe, *Professor, Ph.D.*

Dr. Stowe's recent research addresses the differential geometry of second-order differential equations in the complex plane. It emphasizes using the Schwarzian derivative to deduce properties of solutions and properties of conformal or harmonic mappings.

#### Jim Wolper, *Professor, Ph.D.*

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 incoding 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.

#### Wenxiang Zhu, *Associate Professor, Ph.D.*

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

#### Yunrong Zhu, *Assistant Professor, Ph.D.*

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.