Faculty Research

Topology and group theory have enjoyed a symbiotic relationship since the late nineteenth century with particularly close ties being affirmed and reformed in recent decades. Bogley's research focuses on geometric and homotopical properties of cell complexes and their applications in group theory. Bogley's published works include results in these areas:

1. Asphericity of cell complexes and cohomology of groups
2. Relative homotopy groups and equations over groups
3. Actions on cell complexes and finiteness properties of groups
4. Combinatorial geometry and algorithmic/decidability properties of groups
5. Wild metric cell complexes and omega-group structures, in which certain infinite products can be formed subject to suitable generalizations of the associativity axiom.

Bokil's primary research is in the area of computational electromagnetics. For her doctoral work, she studied ficititious domain methods, operator splitting schemes and perfectly matched layer models for wave propagation problems on unbounded domains. She is interested in finite difference and finite element methods and the analysis of their stability and dispersive properties for modeling electromagnetic materials presenting dielectric dispersion. Recently she has also worked on constructing size and class age structured population models for modeling the spread of diseases in shrimp populations.

Burton has contributions to ergodic theory and dynamical systems and has worked in various parts of pure and applied mathematics. These include probability theory, percolation and interacting partical systems, neural networks and data mining, number theory and its interactions with dynamics such continued fractions, point processes and random measures, algorithms and coding theory, substitution process, symbolic systems in one and higher dimensions. His current projects include work with the Center for Gene Research and Biotechnology, symbolic and physical models, and coding and search algorithms.

The study of Harmonic Analysis provides the foundation for the theory of partial differential equations, several complex variables, and harmonic analysis related to semisimple Lie groups and symmetric spaces. I am interested in four topics in harmonic analysis: (1) singular integral operators; (2) multiplier operators; (3) weighted inequalities for operators; (4) the convergence of the Fourier series in higher dimensional Euclidean spaces. In control theory, I study the stability of nonlinear systems. Results are general and include time variant parameters. While the theorems do not generally provide necessary conditions, they are relatively simple to apply and do not require the development of a Liapunov function. Some of these theorems have been applied to study the spread of tuberculosis. Currently, I am interested in the existence, uniqueness and regularity of solutions to the Navier-Stokes equations.

Dick's research interests include the study of factors related to mathematics achievement and participation, cognitive science as applied to the learning of advanced mathematics, uses of technology in the learning of mathematics, and mathematical discourse. He has worked extensively in the calculus curriculum reform movement. He has served on the joint AMS/MAA Committee on Research in Undergraduate Mathematics Education, the National Council of Teachers of Mathematics Research Advisory Committee, and the Advanced Placement Calculus Development Committee. He is a past co-editor of Connecting Research to Teaching for NCTM's Mathematics Teacher journal, associate editor for School Science and Mathematics, and editorial panel member for the Journal for Research in Mathematics Education.

Past work has emphasized General Relativity, studying model spacetimes and their properties, as well as the interface between relativity and quantum physics. Current work investigates applications of the Octonions, the unique non-associative division algebra, to the physics of fundamental particles. Other past and present interests include Algebraic Computing, Asymptotic Structure, Non-Euclidean Geometry, Quantum Field Theory in Curved Space, and Signature Change.

Work in Mathematics Education includes directing the Vector Calculus Bridge Project, co-directing the High Desert Mathematics Partnership and the Paradigms in Physics Project, and designing course content for the Oregon Mathematics Leadership Institute.

Edwards' research interests are in advanced mathemtical thinking and issues surrounding the teaching of collegiate mathematics. She is currently working in the area of undergraduate mathematics students' understanding and use of formal mathematical definitions. She is also involved in research on teacher change and factors influencing pedagogical innovation at the collegiate level.

Escher's current work lies in the interaction between algebraic topology and differential geometry, in particular the use of surgery theory to classify spaces of positive sectional curvature. Her previous work is connected to the field of minimal submanifolds; in particular existence and uniqueness questions of minimal isometric immersions of spherical space forms into spheres.

Faridani conducts research in numerical analysis, investigating problems arising in signal processing and tomography. In signal processing he is interested in non-equidistant sampling of bandlimited functions in several variables. His research in tomography comprises questions of optimal sampling and resolution; error estimates for reconstruction algorithms in two and three dimensions; and theory and implementation of local tomography.

Finch works on inverse problems, particularly those arising in medical imaging. He has worked on x-ray computed tomography, and is currently working on several problems arising in the medical imaging technique called thermoacoustic tomography. Mathematically, the problems can be phrased as the recovery of initial data for the wave equation from the values of the solution on a subset of the boundary.

Dr. Flahive's work in number theory is principally in Diophantine approximation, with techniques from the geometry of numbers. She has also been investigating pseudo-random number generation schemes.

Garity is involved in the study of the relationship between topological manifolds and homology manifolds. He introduced a new way of measuring the complexity of homology manifolds and developed constructions producing new kinds of homology manifolds. More recently, he has been investigating properties of infinite dimensional spaces. He has developed constructions in the Hilbert cube which parallel certain finite dimensional constructions. He is currently working on infinite dimensional examples analogous to his earlier examples of homology manifolds.

Gibson's primary research interests are computational electromagnetics, finite element and finite difference methods, and inverse problems. Research topics coinciding with primary interests include: wave propagation modeling, direct and indirect (sparse) linear solvers, optimization and regularization techniques, high performance and parallel computing, parameter identification and sensitivity analysis, and modeling uncertainty in coefficients of PDE's. His current work with NASA Langley Research Center involves inverse problem techniques related to non-destructive evaluation (NDE) of Space Shuttle foam.

Higdon has worked on open boundary conditions for wave propagation problems and on issues related to the well-posedness of hyperbolic initial-boundary value problems and the stability of their numerical approximations. He is currently working on some mathematical and computational problems related to large-scale, high-resolution numerical modeling of ocean circulation. This modeling involves the solution of a system of partial differential equations that describes fluid flow. Of particular interest are some problems with stability and efficiency that arise from the multiple time scales that are contained in the system.

Yevgeniy Kovchegov works in the field of probability and stochastic processes. His research interests include interacting particle systems, coupling method, mixing times, occupation times, reinforced processes, percolation, self-avoiding walks, other probabilistic and statistical mechanical models, and applications of probability theory in chemistry and ecology.

Lee has made contributions to the existence, uniqueness, and continuous dependence theory for solutions to nonlinear boundary value problems. This work, which continues, is joint with R.B. Guenther of Oregon State and A. Granas of the University of Montreal. Lee has also worked on the numerical calculation of solutions to such problems. He has helped develop the extension of Sturm oscillation theory and the properties of Sturm-Liouville eigenvalue problems to higher order equations. This work is closely related to the branch of approximation theory which deals with Tchebycheff systems and with positive operator theory. It led to related work on best quadrature formulas.

Murphy is studying the dynamics of age-structured populations. The basic model consists of a first order partial differential equation coupled with an integral equation. Population models are often used to describe renewable resources. When a biological resource is exploited, it is of both practical and theoretical interest to determine the harvesting policy which maximizes the yield. She has found the optimal policy for some versions of the model. Another problem posed by natural populations is the scarcity of age-structured data. Murphy has developed a model based on a physiological, observable quantity (such as size) rather than age. Her model incorporates a commonly observed feedback effect--an individual growth depends upon the state of the entire population, while the dynamics of the population as a whole depend upon the sizes of the individual members.

One focus of Ossiander's research is the development of central limit theorems for sums of random functions. Results in this area are intimately connected with the properties of continuous Gaussian and product-Gaussian processes. The exploration of central limit theory in this general setting has involved the development of exponential bounds for the tail probabilities of sums of random variables. Interesting applications include the calculation of rates of convergence of classes of statistical estimators and the construction of stochastic models for physical systems.

Parks has developed and implemented a computational technique for computing parametric area minimizing surfaces. He derived an existence and regularity theory for a class of constrained variational problems. Parks has discovered, and fully characterized, a new type of minimal surface, with surprising properties, defined in terms of the well-known Jacobi functions.

Peszynska's research interests and accomplishments are in mathematical and computational modeling of of flow and transport in porous media and other phenomena of similar character described by nonlinear coupled PDEs with highly heterogeneous and multiscale data. She is interested in analysis of solutions as well as in development and analysis of numerical algorithms. Most recently she has worked on adaptive techniques for coupled, multiscale, and nonlocal models. In addition, she has been involved in various high-performance computing projects involving domain decomposition, parallel computing, and data-intensive grid-based simulations and visualization.
Currently, she is involved in two funded projects: Model Adaptivity for Porous Media and Multiscale Mathematics Research Group.

These four fields are very large and currently are being intensively developed. Petersen's main interests are in those aspects of these areas where Fourier analysis is the principle technique. Thus his research area may be described as the intersection of harmonic analysis with the union of the four fields. Petersen's book, Introduction to the Fourier Transform and Pseudo-Differential Operators, Pitman, Boston, London, Melbourne, 1983, describes the operational calculus aspects.

Pohjanpelto works on the theory and applications of generalized symmetries of differential equations. He has studied the structure of symmetries of the electromagnetic field and applied symmetries in the construction of conservation laws and classification of group invariant solutions. He has also used variational bicomplexes to study the correspondence of generalized symmetries of equations in physical and potential formulation.

Schmidt is currently most interested in: connections beween the ergodic theory of billards and 1-forms on algebraic curves; Riemann surface geodesics and the Markoff spectrum; and, ergodic theory and arithmetic of generalized continued fractions. Recent results include new examples in the theory of groups of affine diffeomorphisms of flat surfaces (with P. Hubert); and a classification of low height geodesics on a cover of the modular surface (with M. Sheingorn).

The research interests of Showalter include singular or degenerate nonlinear evolution equations and partial differential equations, related variational inequalities and free-boundary problems, and applications to initial-boundary-value problems of mechanics and diffusion. His current work is focused on the development of multiscale mathematical models of coupled fluid-solid dynamics, deformable porous media, and upscaled models of transport and flow in heterogeneous media. He is a member of the Multiscale Mathematics Research Group and the Northwest Consortium in Multiscale Mathematics.

With the advent of modern computers, mathematical techniques can be used to combine information from x-ray pictures of a three-dimensional object, taken from a large number of different directions, and to reconstruct the internal structure of the object. This leads to mathematical questions of practical importance concerning a class of integral transforms known as x-ray and Radon transforms. Solmon has established uniqueness, nonuniqueness, stability and range characterization results for these integral transforms.

Swisher's research interests lie in number theory and combinatorics. In particular, applications of the theory of modular forms to partition theory, combinatorics, algebra, or other areas of number theory.

Thomann's research is primarily in problems in partial differential equations arising from fluid mechanics. He also collaborates with colleagues in other departments, as well as in the Mathematics Department, in the development of mathematical models to problems in Ocean Engineering, Ecology, Oceanography, Hydrology and management of Natural Resources.

Waymire's research concerns applications of probability and stochastic processes to problems of contemporary applied mathematics pertaining to various types of flow and dispersion.

Yi's general research interest is in the area of numerical analysis, including the mathematical and computational modeling of problems motivated by physical processes and phenomena in fluid mechanics and material science and the analysis and implementation of the resulting numerical methods. Her recent work has been focused on Mixed Finite Element Methods for Linear Elasticity and Eulerian-Lagrangian Finite Element Methods for Convection-Diffusion problems. Currently, she is working on multiscale fluid flow problems in porous media.