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Research Descriptions

Slimane Adjerid

Associate Professor

Ph.D., Rensselaer Polytechnic Institute

Research Areas: Adaptive finite element methods, a posteriori error estimation, materials modeling, numerical analysis

Professor Adjerid's research is in developing efficient numerical algorithms to solve partial differential equations originating in many fields of science and engineering. Most of Professor Adjerid's work is on adaptive finite element methods that combine h-, p-, and r- enrichment techniques leading to robust, efficient, and accurate software for singularly perturbed partial differential equations. In the heart of every adaptive finite element method lies an a-posteriori error estimator or indicator that helps assess the quality of the numerical solution of these hard stationary and time-dependent problems and automate the solution process. Other insterests include developing mathematical models for materials processes such as, chemical vapor deposition (CVD), chemical vapor infiltration (CVI), reactive vapor infiltration (RVI), and crystal growth modeling. These models are solved accurately and efficiently using adaptive software to perform parameter studies and help engineers determine the controlling parameters of the processes. The results of these studies may be used to speed-up existing processing techniques or design better ones.

Joseph A. Ball

Professor

Ph.D., University of Virginia

Research Areas: Operator theory, system and control theory

Professor Ball is interested in operator and function theory and its connections with mathematical system and control theory. The main paradigm of the relatively new area of control theory called H- infinity control is the design of stabilizing compensators for a given input-output system (the plant) in a feedback loop which arranges good performance in the face of uncertainties and outside disturbances; the performance is to be guaranteed in a worst case rather than average or statistical sense. Applications being developed by engineers now run the gamut from aircraft design and chemical process control to design of automobile suspension systems. One approach to the theory is through factorization and interpolation for rational matrix functions; the physical problem of constructing a stabilizing compensator can be translated directly to the math ematical problem of constructing a rational matrix function satisfying certain interpolation conditions. Professor Ball's current interests are in extensions of the operator theory and the practical, explicit formulas for solutions of these problems to more realistic scenarios, where the plant is either time-variant or nonlinear.

Christopher A. Beattie

Professor

Ph.D., Johns Hopkins

Research Areas: Numerical analysis, spectral theory, computational linear algebra.

Oftentimes the successful analysis of physical phenomena rests on the ability to closely approximate the eigenvalues and eigenfunctions of differential operators. Frequently encountered examples range from the prediction of resonant frequencies, vibrational mode shapes, and buckling loads of elastic structures; through the determination of bound state energy levels and associated electronic configurations for atoms and molecules; to the computation of critical values of Reynolds numbers in viscous fluid flows. Professor Beattie studies both the theoretical and practical issues involved in constructing computational methods for approaching problems such as these.

Jeffrey T. Borggaard

Associate Professor

Ph.D., Virginia Tech

Research Areas: Numerical Analysis, Optimal Design and Control, Scientific Computation

Professor Borggaard's research interests are in the general areas of numerical analysis and computational science/scientific computation. In particular, he looks at optimization and control problems for systems described by partial differential equations. This includes numerical methods for approximating PDEs such as boundary element, finite difference and finite element methods; gradient-based optimization algorithms including efficient gradient computation (sensitivity analysis, automatic differentiation, etc.); and the interplay between these two processes. Sensitivity analysis has a number of practical applications, for instance, one can use it to propagate uncertainty through PDE simulations. He is currently looking at modeling/sensitivity analysis for large-eddy simulation, using optimization to address optimal sensor/actuator placement in control problems, efficient computation of feedback control laws, optimization under noise and uncertainty, and reduced-order modeling.

Ezra A. Brown

Professor

Ph.D., Louisiana State University

Research Areas: Number theory, cryptography, combinatorics, history of mathematics, expository mathematics

Professor Brown's research interests include number theory (especially quadratic forms, algebraic number theory, Diophantine equations, and most recently, elliptic curves), discrete mathematics (especially combinatorial designs and discrete dynamical systems), history of mathematics (he made an English translation of a German biography of the fifteenth-century mathematician Regiomontanus), and expository mathematics.

John A. Burns

Hatcher Professor

Ph.D., University of Oklahoma

Research Areas:Applied and Computational Control, Ordinary, Functional and Partial Differential Equations, Approximation and Computational Methods for Control, Identification and Optimization of Distributed Parameter Systems, Optimization and Optimal Design of Infinite Dimensional Systems, Parameter Estimation and Sensitivity Analysis.

Professor Burns' research interests are in the general areas of applied and computational control. In particular, he looks at approximation schemes for design, optimization and control of infinite dimensional systems defined by delay, integral and partial differential equations. The goal of his research is to develop practical and efficient computational methods for these problems and to provide a rigorous theoretical foundation for convergence and error estimation. Professor Burns' research is motivated by applications such as controlling large inflatable space structures, designing wind tunnel test facilities, estimating parameters in mathematical models in the biological and life sciences, optimal control of fluid flows and nano-technology. He is currently working on sensitivity analysis methods for optimal parameter estimation and design in structural and biological systems. This work has application to optimal design of experiments and is particularly useful in the life sciences where experimental data collection may be expensive and limited. Professor Burns' research makes use of several areas of mathematics including distributed parameter control theory, functional analysis, semi-group theory, dynamical systems, differential equations, numerical analysis and optimization.

Martin V. Day

Professor

Ph.D., University of Colorado, Boulder

Research Areas: Probability and stochastic analysis of small random perturbations of dynamical systems

There are many physical systems (e.g., mechanical/electronic devices, biological or communications systems) which can be described by what mathematicians call a "dynamical system". Such a mathematical description is usually only an idealization. It typically oversimplifies some aspects of the real situation, and hence fails to predict some features of its behavior. In particular, the mathematical description may not account for the small unpredictable (i.e., random) distur bances that are present in any real situation, such as noise in a radio transmission, or small variations from the usual pattern of incoming calls to the telephone system. Professor Day's research has been concerned with mathematical issues involved in studying how the behavior of a dynamical system is altered by the presence of such small random influences. He has been particularly interested in situations in which the random influences cause the occurrence of some phenomenon that would never occur in the idealized dynamical system itself.

The effect of introducing random influences into a dynamical system produces what is called a stochastic process. (Ex amples are Markov chains, queueing processes or solution of stochastic differential equations.) The study of such stochas tic processes is a branch of probability theory that has some very close connections to parts of analysis, including partial and ordinary differential equations. The problems of small random perturbations in particular have important connections to the calculus of variations and classical mechanics. Professor Day's work is motivated by the insight these connections can bring to the mathematical problems, as well as the implications for various situations in the applied sciences.

William J. Floyd

Professor

Ph.D, Princeton University

Research Areas: Geometric group theory, discrete conformal geometry

Most of Professor Floyd's research is in geometric group theory, an area that has grown out of the interplay between combinatorial group theory and geometry. While many of the techniques come from combinatorial group theory, the prime examples and many of the important questions come from geometry. If G is a group and S is a finite generating set of G, then there is a graph, the Cayley graph, naturally associated to the pair (G,S). In geometric group theory, groups are often studied in terms of asymptotic properties of a Cayley graph of the group. A central task is to classify groups in terms of the spaces on which they can act geometrically.

David Y. Gao

Associate Professor

Ph.D. Tsinghua University, Beijing

Research Areas: Applied mathematics and mechanics, computational mechanics and structural optimization

Symmetry in nature systems is amazingly beautiful. Modeling equations in nonlinear systems, establishing the duality theory and computational methods, and applications to engineering problems have been Professor Gao's main research interests. Theory of nonsmooth analysis and nonlinear finite element methods are applied to study these problems. Re cently, his research has concentrated on the duality theory and numerical methods in nonlinear buckling problems and analysis on the manifolds. These studies are motivated by questions arising in the areas of engineering structural analysis and optimal design.

It is known that direct methods for solving nonlinear variational boundary values problems provide only the upper bound approach to solutions. However, the dual problem will give the lower bound solution. In nonlinear buckling problems and structural limit analysis lower bound solutions are important for engineers to design stable structures. In collaboration with Professor G. Strang at MIT, Professor Gao developed a duality theory in geometrical nonlinear systems. Now he is interested in generalizing this theory to the problems of nonlinear bifurcation. The effort concentrates on obtaining the lower bound of the first eigenvalue and effective computational methods in the stability analysis of nonlinear structures, such as elastic plates and shells subjected to large deformation.

Optimal design of smart structures is one of the current active research fields in applied mathematics and mechanics. Professor Gao's interest in this area lies in the formulation of, and numerical study of, differential-equation models in optimal shape design. Many new phenomena which occurred in this field provide us challenging problems for both theo retical analysis and numerical simulation.

Edward L. Green

Professor

Ph.D. Brandeis University

Research Areas: Representation theory and computational algebra

Professor Green's research interests center on the study of the structure of modules over finite dimensional algebras. In recent years, he has been especially interested in the homological properties of such algebras. This has led him to compu tational aspects of the area.

In 1972, there was a major change in how finite dimensional algebras were studied. To each algebra one can associate a finite directed graph such that the study of modules can be translated to linear algebra problems on such a graph. Although the linear algebra questions that arise in this way are moderately difficult, the problems are nonetheless concrete and allow new methods of attack.

Professor Green has been studying the projective dimensions of modules and for this he has been investigating projective resolutions of modules. In this investigation, he uses a computer program that was developed here at Virginia Tech that employs noncommutative Groebner bases. The theory of noncommutative Groebner bases is young and there are many interesting questions that need to be investigated. With the aid of the computer, new examples and structures can be studied that were inaccessible in the past.

Finally, Professor Green has been investigating the structure of quantum groups and their representations (i.e., modules). Quantum groups are related to knot invariants, physics and many different areas of mathematics come into play in studying these objects.

William Greenberg

Professor

Ph.D., Harvard University

Research Areas: Mathematical physics, statistical mechanics

Professor Greenberg's research has concentrated on the equations of non-equilibrium statistical mechanics, both linear and non-lienar theory. Most recently, he has been studying the nonlinear equations which describe moderately dense gases in order to obtain information on the existence and stability of exact solutions.

Serkan Gugercin

Assistant Professor

Ph.D.,

Research Areas:Dynamical Systems and Numerical Computation

George A. Hagedorn

Professor

Ph.D., Princeton University

Research Areas: Non-relativistic quantum mechanics, mathematical physics

Professor Hagedorn works on mathematical problems that arise in quantum mechanics. His papers deal with many body scattering theory, semiclassical approximations, adiabatic approximations, and molecular physics. He is currently concen trating on studying molecular dynamics in

situations where the standard time-dependent Born-Oppenheimer theory cannot be applied.

Kenneth B. Hannsgen

Professor

Ph.D., University of Wisconsin - Madison

Research Areas: Volterra integral equations

Professor Hannsgen works in a branch of classical analysis with applications to the study of mechanical or other systems that "remember" what has happened to them in the past. Many of the techniques used for differential equations can be extended to this setting, but in a different or more complicated form. The most current work concerns the suppression of vibrations in viscoelastic structures by means of forces applied at the edges.

Peter Haskell

Professor

Ph.D., Brown University

Research Areas: Index theory

Index theory is the study of the relationships between topology and geometry, on the one hand, and analysis. An early example is associated with the vector calculus theorems of Green and Stokes: an irrotational vector field (closed differential one-form) on a region D is guaranteed to be conservative (exact) on D only if D is simply connected (i.e., every loop in D can be contracted to a point in D). Examples of topological invariants that have analytic realizations include the winding number and the Euler characteristic. Index theory has developed from its roots in analysis on manifolds (global analysis) to include the study of singular spaces, with part of this development based on functional analytic tools such as the K-theory of algebras of linear operators on Hilbert space.

Terry L. Herdman

Professor

Ph.D., University of Oklahoma

Research Areas: Applied mathematics, neutral functional differential equations, optimization

Professor Herdman is interested in all aspects (modeling, analysis, optimal design, parameter identification, control, com putational methods, etc.) of systems governed by functional differential equations. The mathematical theory is motivated by applications to aeroelastic systems. Such systems fit into a class of functional differential equations of neutral type, where the difference operator does not have an atom at zero. The development of state space theory for hereditary systems in a product space setting is a goal of this research. This provides an excellent framework for a study of computational methods for singular neutral systems.

Traian Iliescu

Assistant Professor

Ph.D., University of Pittsburgh

Research Areas: Numerical Analysis, Scientific Computation, Computational Fluid Dynamics

Jong U. Kim

Professor

Ph.D., Brown University

Research Areas: Classical applied mathematics

Professor Kim's present research interest lies in the application of microlocal analysis to continuum mechanics with empha sis on wave propagation.

Martin Klaus

Professor

Ph.D., University of Zurich

Research Areas: Inverse problems in scattering theory, spectral theory of operators arising in quantum mechanics

Professor Klaus's current main area of interest is inverse problems in scattering theory. There are several different kinds of inverse problems. The prototype problem is the classical inverse problem of quantum scattering theory. There one tries to find the force field of an atomic particle from an appropriate set of scattering data. Some of the theoretical questions that arise in this context and which Professor Klaus has studied, concern the characterization of the scattering data and the question of unique solvability of the inverse

problem. For technical reasons these questions have so far mainly been studied in one dimension, but one hopes that some of the methods might also prove useful in the study of the three-dimensional problem. Mathematically, the inverse problem can be formulated in several ways. The analysis typically requires tools from the theory of integral equations, Riemann -Hilbert problems and Wiener-Hopf factorizations.

Professor Klaus is also interested in the spectral theory of operators that arise in quantum mechanics. This includes the existence of eigenvalues, in particular of eigenvalues

embedded in the continuous spectrum, and problems in which eigenvalues move as certain parameters are varied.

Werner E. Kohler

Professor

Ph.D., Rensselaer Polytechnic Institute

Research Areas: Applied probability and asymptotic techniques applied to the study of wave propagation in complex media

Wave propagation problems arise in a number of applications, e.g., waves in seismic problems, acoustic waves in oceanic underwater sound propagation, ionospheric electromagnetic waves in atmospheric communication problems and quantum mechanical waves in electron mobility problems. Quite often, the propagation medium is complicated by the presence of inhomogeneities, such as layering and undulations, atmospheric turbulence and rain, or crystalline disorder. These depar tures from an ideal environment are typically too complicated to describe in anything other than a probabilistic or statistical sense. Given such a description, one is then interested in describing the statistics of the resulting waves. The study of such problems requires an understanding of the underlying physical science (e.g., acoustics, electromagnetic theory, quantum mechanics), ordinary and partial differential equations, probability theory and techniques of applied mathematics, such as asymptotic techniques and numerical methods.

Reinhard Laubenbacher

Professor

Ph.D.,

Research Areas:

Tao Lin

Professor

Ph.D., University of Wyoming

Research Areas: Numerical methods for the forward and/or inverse problems of partial differential equations, integral equations, and integro-differential equations

According to pertinent physical laws, many procedures in nature and industry can be described by partial differential equations, integral equations, and integro-differential equations. Unless we make enough simplifications, it is usually difficult, if not impossible, to find analytical solutions for these equations. Solving them numerically on computers turns out to be a practical alternative. Professor Lin's research involves developing both finite difference and finite element methods for these equations. Stability analysis, error estimate, and assessing the actual error in the numerical solution form the core of his research.

Peter A. Linnell

Professor

Ph.D., Cambridge University, England

Research Areas: Group rings, homological algebra, von Neumann algebras

The study of group rings uses a mixture of group theory and ring theory. This subject, as well as homological algebra has topological applications. Another topic included in this research is L p-cohomology, which uses functional analysis.

Gwendolyn M. Lloyd

Associate Professor

Ph.D., University of Michigan

Research Areas: Mathematics Education

Robert A. McCoy

Professor

Ph.D., Iowa State University

Research Areas: General topology, spaces of continuous functions

Professor McCoy has interests in all aspects of general topology, but he has especially been studying the topological properties of spaces of continuous functions on the many topological spaces which arise in general topology and analysis. There are a number of different naturally occurring topologies on these function spaces, and there are many interesting relationships between these topologies. This is one of the topics in McCoy's book on function spaces listed below which was written with a former Ph.D. student of his.

Another of Professor McCoy's interests in topology is hyperspaces of closed (or compact) subsets of topological spaces. Hyperspaces offer a general setting for working with the convergence of a sequence (or net) of subsets of a topological space. This has been used for example in studying attractors for iterated function systems, a topic in the general area of fractals. The list of papers below indicate other topics of interest.

Anderson Norton

Assistant Professor

Ph.D., University of Georgia

Research Area: psychological development of mathematics in K-12 students

Professor Norton researches students' mathematical development, especially development that results from conjecturing activity. Although most of his research occurs in the context of fractions learning, he plans to extend findings into research on the development of algebraic and geometric reasoning. Students' mathematics might thus be understood as a sequence of operational reorganizations from counting to whole- number knowledge, to fractional reasoning, and into more advanced ways of reasoning/operating. This research ties into Professor Norton's teaching as he engages future teachers in making sense of students' ways of operating. Teachers need strong content knowledge to do this--the kind of content knowledge that is "unpackable" from abstract concepts to concrete experiences.

Charles J. Parry

Professor

Ph.D., Michigan State University

Research Area: Algebraic number theory

Professor Parry has conducted research on problems related to factorization in algebraic number fields. Each algebraic number field contains a ring of integers which, in general, does not possess the unique factorization property. The ideal class group and its order, the class number tend to measure how close the ring comes to satisfying the unique factorization property. There are many interesting problems related to the determination of the class number and structure of the class group which he has worked on.

Carl L. Prather

Professor

Ph.D., Northwestern University

Research Areas: Complex Analysys, Wavelets, q

Frank Quinn

Professor

Ph.D., Princeton

Research Areas: Topology of manifolds and CW complexes

Professor Quinn is currently working in the interface between low- dimensional topology, representation theory, and theo retical quantum mechanics. This work involves computation as well as abstract work, and uses machines at the National Center for Supercomputing Applications in Illinois. Professor Quinn is also active in philosophical issues in mathematics and electronic publication.

Michael Renardy

Professor

Ph.D.,University of Stuttgart, Germany

Research Areas: Nonlinear partial differential equations, fluid mechanics

Professor Renardy's area of research is in nonlinear partial differential equations and applications to fluid mechanics, in particular viscoelastic fluids. Such fluids include polymer melts and solutions, and suspensions. In contrast to "classical" Newtonian fluids, the stresses in these fluids are not determined by the velocity gradient at the current time, but depend on the history of the motion. The equations describing the dynamics of these fluids are nonlinear partial differential or integrodifferential equations of a "composite" type. Until about fifteen years ago, no systematic mathematical study of these equations existed. Professor Renardy has contributed to the investigation of basic issues of existence and unique ness, stability of flow, singularities and inflow boundary conditions. In addition to his work on viscoelastic flows, he has also studied stability and bifurcations in multilayer flows with deformable interfaces.

Yuriko Y. Renardy

Professor

Ph.D.,University of Western Australia

Research Areas: Fluid dynamics

Professor Renardy's research has covered a variety of phenomena in fluid dynamics, and is interdisciplinary. One aspect is an investigation of the stability of interfaces between two fluids, with "stability" meaning the ability to survive distur bances. The ideas which come out of her work are applicable to such industrial processes as the pipeline transport of oil and the design of composite polymeric materials, as well as to natural phenomena. She works to create mathematical models of the stability of such processes. Some techniques of ordinary differential equations, partial differential equations, linear algebra, numerical analysis and computations, asymptotic analysis, and bifurcation theory are combined in the re search.

Robert C. Rogers

Professor

Ph.D.,University of Maryland-College Park

Research Areas: Partial differential equations, calculus of variations, continuum mechanics, electromagnetism, hysteresis

Professor Rogers has conducted research in several areas of applied mathematics, but most of his current projects involve a phenomenon known as "hysteresis". Hysteresis is common in nonlinear mathematical problems, and it becomes apparent in physical processes where the rest state of the process depends on its past history. Such processes are often seen in materials that undergo a change in "phase".

One of the primary difficulties in the mathematical analysis of phase transitions is the need to take account of the highly oscillatory structure that is found in so many of these problems. The physical manifestation of these oscillations is seen, for example, in tiny magnetic domains in ferromagnets, fine twinning patterns in shape-memory crystals, and microscopic vortex structures in superconductors. In counterpoint to this physical complexity, mathematical problems in phase transi tions often show troublesome instabilities that make analysis and computation difficult at best.

Fortunately, recent advances in nonlinear analysis such as homogenization, relaxation, Young-measures, and H-measures have allowed researchers to make fresh approaches to these problems. Professor Rogers has been involved in developing and analyzing mathematical models that incorporate some of these modern mathematical tools in a way that makes them suitable for engineering applications. Professor Rogers' techniques have been applied to models of ferromagnetic materi als, superconductors, and mechanical phase transitions. The new models have proved amenable to both classical analysis and numerical computation.

John F. Rossi

Professor

Ph.D.,University of Hawaii

Research Areas: Complex analysis, potential theory

Professor Rossi has conducted research in classical complex analysis of one variable and its relationship to differential equations, potential theory (linear and nonlinear) and quasiconformal mappings.

Over the last few years Professor Rossi's research has primarily involved the growth and zero distribution of solutions to ordinary differential equations in the complex plane.

David L. Russell

Professor

Ph.D., University of Minnesota

Research Areas: Ordinary and Partial Differential Equations, Systems Theory and Mathematical Modeling

Mark M. Shimozono

Associate Professor

Ph.D., Univeristy of California

Research Areas: Algebraic Combinatorics and Representation Theory

Robert L. Snider

Professor

Ph.D., Miami

Research Areas: Noncommutative rings, group rings

Professor Snider works on noncommutative rings, especially group rings. Recently he has studied endomorphism rings of simple modules of group rings and permutation modules of group rings. In noncommutative rings, he has studied the question of zero divisors for noncommutative regular local rings.

Shu-Ming Sun

Professor

Ph.D.,

Research Areas:

James E. Thomson

Professor

Ph.D., University of North Carolina

Research Areas: Operator theory, function theory

Professor Thomson's research is in the area of interplay between complex function theory and operator theory. On finite -dimensional spaces operator theory is linear algebra and the Jordan canonical form describes the structure of an arbitrary matrix operator. On infinite-dimensional spaces only special classes of operators are understood. For example, a descrip tion of the structure of normal operators follows from the spectral theorem. Classes of operators that are close to normal are currently being studied. The general approach is to find functional models for the operators and to use function theoretic methods to discover the structure of the operators. For example, Joseph Bram found a functional model for cyclic subnor mal operators and Professor Thomson's recent description of the closure of the polynomials in L 2-norm gives the structure of such operators.

Another pursuit in this area is to study operators that arise naturally on classical Banach spaces of analytic functions, e.g., Hardy and Bergman spaces. These operators provide interesting examples and also lead to questions concerning the under lying Banach spaces.

Peter Wapperom

Assistant Professor

Ph.D., Delft University of Technology

Research Areas: mathematical modeling and numerical simulation of viscoelastic fluids

Professor Wapperom works in the field of mathematical modeling and numerical simulation of viscoelastic fluid flow. Viscoelastic fluids include for example polymer melts and solutions. In contrast with Newtonian fluids which react instantaneously, the material response of viscoelastic fluids depends on the deformation history. At low deformation rates these materials behave like viscous fluids (for example water) and at high deformation rates the behavior is elastic (like rubber bands).

The complex material response of polymeric liquids is difficult to model mathematically. Currently developed mathematical models for viscoelastic fluids are either nonlinear partial differential equations, integro-differential equations, or stochastic differential equations. My research comprises the numerical simulation of all types of viscoelastic models including stochastic models that are used to describe viscoelastic fluids at the molecular level. The time consuming stochastic numerical simulations rely on parallel computing.

For viscoelastic fluid flow problems of practical interest, thin stress boundary layers develop when the material response is highly elastic. In these cases, current numerical techniques fail to predict accurate solutions. He is developing better numerical techniques that can accurately handle the thin stress boundary layers. The numerical techniques are used to evaluate currently developed mathematical models for viscoelastic fluids in flow geometries of practical interest. Results of these simulations are used to improve these mathematical models. On the long term, better numerical techniques will benefit polymer engineers who use numerical simulations to optimize production processes.

Robert L. Wheeler

Professor

Ph.D.,University of Wisconsin

Research Areas: Volterra integral equations, systems theory

Volterra integral and partial-integrodifferential equations describe phenomena where the behavior of a system at the present time is determined by the state of the system at the present time as well as its state at past times. The behavior of viscoelastic materials is an example where the dynamics are governed by Volterra equations. The analysis of control of systems governed by Volterra equations is an area of current interest.