# Research Descriptions

## Slimane Adjerid

*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:** Model reduction, computational linear algebra,
spectral estimation, numerical analysis, scientific computing.

Direct numerical simulation of dynamical systems can play a crucial role in studying a wide variety of complex physical phenomena in areas ranging from ocean circulation to microchip design. Many such phenomena involve heterogeneous mixtures of physical processes that evolve on fine time and length scales while system behaviors of interest occupy much coarser time and length scales. As models are refined at increasingly fine time and length scales in order to attain high accuracy, dynamical systems of enormous scale and complexity are often produced, leading to overwhelming demands made on computational resources. One may address this problem in some cases through methods that encode the fine scale dynamic structure of complex systems into compactly-represented high-fidelity reduced models that may then serve as efficient surrogates for the original systems. While there are a variety of approaches that can accomplish this, my principal focus has been on those associated originally with Krylov subspace projection (sometimes called moment-matching), which now is more descriptively termed interpolatory model reduction. This class of approaches is numerically reliable, widely applicable, and can be realized effectively either with direct methods or inexact iterative methods for problem dimensions on the order of millions.

## Jeffrey T. Borggaard

*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.

## Julianne Chung

*Assistant Professor*

*Ph.D., Emory University *

**Research Areas:**
Scientific computing, inverse problems, image processing

Inverse problems arise in many applications such as biomedical imaging, geophysics, and astronomy, where measurements can only be obtained on the exterior of an object (e.g., the human body or the earth's crust), and the goal is to estimate the internal structures. In other systems, signals measured from machines (e.g., cameras) are distorted, and the aim is to recover the original input signal. Most inverse problems are ill-posed, meaning small errors in the data can lead to large changes in the solution. Professor Chung's research is to develop numerical methods and software for computing solutions to large-scale inverse problems.

## Matthias Chung

*Assistant Professor*

*Ph.D., University of Lübeck*

**Research Areas:**
Scientific computing, computational biology, inverse problems,
applied dynamical systems

Scientific computing, mathematical modeling, and inverse problems are crucial to understanding biological and life science systems. The intersection of theory, experiment and computation for scientific investigation is the center of Professor Chung's research interests. His focus is on computational methods using optimization, parameter estimation, and numerical ODE methods. Professor Chung is especially interesed in interdisciplinary approachs for solving problems arising life science and engineering.

## Stanca M. Ciupe

*Assistant Professor*

*Ph.D., University of Michigan*

**Research Areas:**Mathematical modeling, mathematical biology,
dynamical systems

Professor Ciupe is interested in modeling and applications of systems of ordinary and delay differential equations to infectious diseases, in particular the characterization of immune system onset and reaction against viral diseases like Human Immunodeficiency Virus infection, Hepatitis B and Dengue virus infections. The techniques used derive from dynamical systems, information and model selection theory as well as sensitivity, perturbation and numerical analysis.

## Alexander Elgart

*Associate Professor *

*Ph.D, Technion*

**Research Areas:** mathematical physics, analysis, quantum chemistry

## 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.

## Serkan Gugercin

*Professor *

*Ph.D., Rice Univeristy *

**Research Areas:**Dynamical Systems and Numerical Computation

## George A. Hagedorn

*Professor (Emeritus) *

*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.

## 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

*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., Northwestern University*

**Research Areas:**
Mathematical tools for reverse-engineering of
biochemical networks, large-scale microsimulation of human immune
response to
viral pathogens, computational algebra models for Bayesian networks,
mathematical foundation of computer simulation

## 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.

## Nicholas Loehr

*Associate Professor *

*Ph.D, University of California, San Diego*

**Research Areas:** bijective and algebraic combinatorics

Algebraic combinatorics exploits the connections between algebraic structures (group representations, modules, polynomial rings, Lie algebras) and combinatorial objects (tableaux, lattice paths, permutations, partitions) to obtain new information about both fields. Dr. Loehr's research in this area focuses on the combinatorial properties of symmetric polynomials. One can often gain useful knowledge about a mathematical system by introducing a symmetric polynomial to model the system and then finding a combinatorial formula for the symmetric polynomial. For example, Schur's symmetric polynomials translate fundamental problems in representation theory into enumerative questions about tableaux; the Hall-Littlewood symmetric polynomials encode lattice-theoretic properties of finite Abelian p-groups; and the zonal symmetric polynomials can be used to solve problems in Fourier analysis and multivariate statistics.

## Leonardo Mihalcea

*Assistant Professor *

*Ph.D, University of Michigan*

**Research Areas:** algebraic geometry, algebraic combinatorics,
geometric representation theory

Professor Mihalcea's research interests are in the general area of Algebraic Geometry, with emphasis on the study of ordinary and quantum cohomology theories of generalized flag manifolds. More specifically, he studies the algebra and geometry of the quantum K-theory and quantum cohomology ring of flag varieties, Gromov-Witten invariants, geometry and combinatorics of Schubert varieties and Schubert polynomials, and had contributions in the calculation of Chern-Schwartz-MacPherson classes for Schubert varieties.

## Anderson Norton

*Associate 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.

## Carl L. Prather

* Professor (Emeritus) *

*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.

## Mark M. Shimozono

* Associate Professor *

*Ph.D., Univeristy of California *

**Research Areas:** Algebraic Combinatorics and Representation
Theory

## Shu-Ming Sun

*Professor *

*Ph.D., *

**Research Areas:**

## James E. Thomson

*Professor (Emeritus)*

*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

* Associate 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.

## Megan Wawro

*Assistant Professor *

*Ph.D., University of California, San Diego &
San Diego State University*

**Research Areas:**undergraduate mathematics education

Professor Wawro's research investigates the learning and teaching of undergraduate mathematics content courses. Currently she focuses on investigating students' understanding of linear algebra. In general, her research investigates the development of mathematical meaning over time (for both individual students and the classroom as a collective community) and explores ways to coordinate the analyses at the individual and collective levels. Some of her research makes use of Toulmin's Model of Argumentation as an analytical tool, and she is also developing ways to utilize adjacency matrices as a means to analyze students' discourse and understanding of particular mathematical content. She is also interested in students’ development of formal ways of reasoning about mathematical concepts via authentic participation in discipline-specific mathematical activities.

## Pengtao Yue

*Associate Professor *

*Ph.D., University of Science and technology of China *

**Research Areas:** Scientific computing and fluid mechanics

Professor Yue’s research area is in computational fluid dynamics. In the past, he has worked on supersonic flows, shock wave dynamics, multiphase flows, and combustion. Currently, he focuses on the modeling and simulation of flows involving complex fluids, moving interfaces, and moving boundaries.

Complex fluids are those with internal microstructures whose evolution affects the overall rheology, eg., viscoelastic fluids, liquid crystals, and emulsions. Professor Yue developed an energy-based diffuse-interface method which easily incorporates complex rheology and tracks fluid interface at the same time. The differential equations are solved by the spectral method on a Cartesian mesh or the finite element method on an adaptive triangular or tetrahedral mesh. The same methodology is also being used to study the moving contact line problem, which is crucial to a lot of industrial processes such as ink jet printing, curtain coating, and crude oil recovery.

When one side the interface is occupied by inviscid gas or rigid solid, only the liquid phase needs to be calculated and this constitutes a moving boundary problem. Examples of such problems include bubble growth in the foaming process and particle-bubble interaction in the floatation process. Professor Yue has developed an arbitrary-Lagrangian-Eulerian method for polymer foaming. Currently, he is extending the code to include moving particles, aiming at the direct numerical simulation of particle-bubble interactions in liquid media.

## Lizette Zietsman

*Associate Professor *

*Ph.D., University of Pretoria *

**Research Areas:**