Research of Applied Math Faculty Members

University of Colorado, Boulder

Mark J. Ablowitz
Professor of Applied Mathematics
(Ph.D. MIT)

An essential element in the study of Applied Mathematics is to explain physical phenomena by mathematical models. Frequently such models lead to nonlinear systems and in a surprisingly large number of cases certain prototypical systems are obtained. A central theme in Ablowitz' research is to understand by approximation, numerical and exact methods, solutions to these underlying equations and their properties.

An important method used to solve certain nonlinear wave equations is the so called Inverse Scattering Transform: IST. The IST is conceptually analogous to the Fourier Transform; IST employs methods of direct and inverse scattering. These techniques were originally developed by physicists and mathematicians studying quantum mechanics. The IST allows one to construct general solutions to certain initial-boundary value problems that arise in a variety of physical problems such as nonlinear optics, water waves, plasma physics, lattice vibrations, and relativity.

A special class of solutions are referred to as solitons, which are extremely stable localized waves. Solitons are important in physical application -- especially in nonlinear optics.

Jerrold W. Bebernes
Professor Emeritus of Applied Mathematics
(Ph.D. University of Nebraska)

Professor Bebernes' interests include nonlinear parabolic systems, hyperbolic systems, and reactive-diffusive-convective systems of partial differential equations with primary emphasis on models from combustion theory.

Understanding the formation of singularities is a primary goal of much of his recent research activity.

Gregory Beylkin
Professor of Applied Mathematics
(Ph.D. New York University)

A remarkable number of practical problems involving differential or integral equations cannot be solved in realistic time simply because many of the available numerical algorithms are too slow. A simple example is the straightforward multiplication of two dense N by N matrices which requires order N^3 operations. It is easy to see that progress in computer hardware provides only a partial answer to this challenge since doubling the size of the matrix increases the required resources by the factor of eight.

It is mathematical analysis that usually yields significant improvements. Some of the recent advances, which utilize newly developed orthonormal bases with controlled localization in the time-frequency domain (e.g. wavelets, wavelet-packets, localized cosine transforms) make this point very explicit since for a wide class of matrices, these new methods yield an order N algorithm for the matrix multiplication.

Professor Beylkin's research interests include the analysis and design of fast numerical algorithms, the development of new methods for solving linear and nonlinear equations, applied inverse problems of wave propagation, and mathematical methods of geophysics.

Jem Corcoran
Associate Professor of Applied Mathematics
(Ph.D. Colorado State University-Fort Collins)

Since the mid-1990's, there has been much work on the development and application of algorithms that will enable the simulation of the invariant measure of a Markov chain, either exactly (that is, by drawing a random sample known to be from this distribution) or approximately, but with computable order of accuracy. These were sparked by the introduction of the "coupling-from-the-past" (CFTP) technique of Propp and Wilson, and several variations and extensions of this idea have since appeared in the literature, and have proven effective in areas such as statistical physics, operations research, and the study of spatial point processes, where they provide simple and powerful alternatives to existing CPU consuming, often unsettlingly inaccurate, simulational methods.

James H. Curry
Department Chair, Professor of Applied Mathematics
(Ph.D. University of California at Berkeley)

An essential element in the study of applied mathematics is the development of computational strategies for solving nonlinear equations. Frequently such equations are embedded in larger problems, e.g. solving either two point boundary value problems or free boundary problems.

In their simplest form, nonlinear equations are solved using iterative methods. Within this context they can be viewed as dynamical systems and may exhibit chaos, sensitivity to initial conditions, and the like.

A component of Professor Curry's research is concerned with both the success and failure of iterative methods for solving equations.

Additionally, Professor Curry's research is concerned with mathematical questions which are of interest to Atmospheric and Oceanic Scientists.

Robert W. Easton
Professor Emeritus of Applied Mathematics
(Ph.D. University of Wisconsin)

A dynamical system is a mathematical model of the time evolution of a physical system. The model consists of a set "states" of the system and a function describing the time evolution of these states.

Professor Easton is interested in the study of dynamical systems of all kinds typically expressed as a differential or difference equation. The mathematical methods involved are geometric and topological. H. Poincaré was the pioneer of the field and Professor Easton's work is deeply rooted in these studies.

Professor Easton has worked on the three body problem of celestial mechanics, looked for chaos in area preserving maps of the plane and investigated transport in phase space. Computer simulations are a useful tool and allow the results to be displayed graphically.

Bengt Fornberg
Professor of Applied Mathematics
(Ph.D. Uppsala University, Sweden)

Professor Fornberg's research focuses on the development of computational techniques in areas such as wave motions and fluid dynamics, with applications most recently to electromagnetic waves and to solar physics. The numerical techniques that are developed mainly build on finite differences and pseudospectral methods, often used in conjunction with asymptotic analysis. A primary current focus area is the development of radial basis functions - a new mesh-free approach for high order computations in irregular domains.

Keith Julien
Associate Professor of Applied Mathematics
(Ph.D. University of Cambridge, England)

Dr. Julien's primary area of research are in the dynamics and instability processes in geophysical and astrophysical fluid dynamics. These include turbulent fluid convection and the effects of stratification, rotation and magnetic fields which are all ubiquitous features of geophysical and astrophysical fluid objects. Particular emphasis has been placed on the characteristics of coherent structures, their transport and organization of large-scale flows, and mean-flow generation. Integral parts of Julient's research include high performance computer simulations, dynamical systems theory, and asymptotics and perturbation techniques in applied mathematics.

Manuel Lladser
Assistant Professor of Applied Mathematics
(Ph.D. Ohio State University-Columbus)

My area of specialization is probability theory with a later emphasis in problems related to discrete probability, asymptotic enumeration, analytic combinatorics and generating functions. I have a variety of other interests (some of them quite interdisciplinary) in problems related to percolation and random graphs, patterns in random strings, random walks on random environments and more recently graphical models and math biology.
  • A state machine approach to study pattern frequencies in Markovian sequences.
       M.Lladser, R.Knight, M.D.Betterton. In preparation.
  • Uniform formulae for coefficients of meromorphic functions in two variables. Part II.
       M.Lladser. In preparation.
  • Multiple pattern matching: A Markov chain approach.
       M.Lladser, R.Knight, M.D.Betterton. Submitted. 2005.
  • Uniform formulae for coefficients of meromorphic functions in two variables. Part I.
       M.Lladser. Submitted. 2004.
  • Asymptotic enumeration via singularity analysis.
       M.Lladser. Ph.D. dissertation, The Ohio State University, 2003.
  • Domain of attraction of the quasi-stationary distributions of the Ornstein-Uhlenbeck process.
       M.Lladser and J.San Martin. JAP Vol 37 N 2, 2000.

    • Congming Li
    Associate Professor of Applied Mathematics
    (Ph.D. New York University)

    Frequently physical problems can be modeled in terms of partial differential equations. Elliptic partial differential equations are a subarea in which there is a well developed theory, linear and nonlinear, which allows us to understand the qualitative structure of the associated solutions. These results give satisfactory answers to many of the original physical problems. Professor Li's interests are to use and to further develop the theory of elliptic partial differential equations in order to understand the solutions to a more varied and rich set of problems.

    Tom Manteuffel
    Professor of Applied Mathematics
    (Ph.D. University of Illinois at Urbana)

    Complex physical phenomena, such as the flow of air over the wing of an airplane or the deformation of a car fender in a crash, are modeled by systems partial differential equations. The solutions to these systems can only be achieved approximately through numerical simulation. The process of creating the correct system of equations, developing algorithms for the solution and developing software that utilizes advanced computing environments is known as computational mathematics.

    Professor Manteuffel's research is mainly focused on formulation of such systems of partial differential equations and creation of algorithms for approximating the solution of these systems. He is especially interested in iterative algorithms including multigrid and multilevel algorithms. This involves developing approximations at different scales to resolve complex fine scale features as well as global features. His research involves applications that include aerodynamics, electromagnetics, meteorology, medical imaging, petroleum engineering, ground water contaminate flow, and particle transport.

    Steve McCormick
    Professor of Applied Mathematics
    (Ph.D. University of Southern California)

    At the core of many problems in applied mathematics, and science in general, are partial differential equations. These equations are used to model such diverse phenomena as fluid flow, heat transfer, particle transport, chemical interactions, electromagnetic processes, and biomedical systems. Most of these equations describe complicated physical characteristics that include multidimensional complex geometry, temporal phenomena, nonlinear ties, and discontinuities. This modeling usually comes with wide variations in scale of the important physical features. Such realistic simulations are seldom solved analytically.

    Professor McCormick's central research focus is on multigrid and multilevel methods for numerical solution of such equations. This involves developing schemes that use a spectrum of discrete approximations at different physical scales to resolve the relevant features of the partial differential equation. This research is driven by applications that include aerodynamics, meteorology, oceanography, medical imaging, petroleum engineering, ground water and contaminant flow, and electromagnetics.

    James D. Meiss
    Professor of Applied Mathematics
    (Ph.D. University of California at Berkeley)

    All of the fundamental equations of physics are Hamiltonian dynamical systems; yet most of our understanding of these systems is restricted to the simplest, linear case. The solutions of nonlinear Hamiltonian systems can exhibit structure of extreme beauty and complexity. While Poincaré anticipated some of this complexity, only recently has research led to a deep understanding of these systems.

    At the extremes, Hamiltonian motion can be either completely integrable or completely chaotic. The former corresponds to the existence of special symmetries, and the latter to the hypotheses required for statistical mechanics. Typically Hamiltonian motion is neither integrable nor chaotic, but an intricate mixture of the two.

    The mathematics of this subject has benefited to an extraordinary degree from a close relationship with computers. Computer visualization suggests theorems, and some theorems require computation for proof.

    Recently, Professor Meiss has concentrated on developing an understanding of chaotic motion, symmetries, bifurcations, and the geometry of the dynamics in multi-dimensional conservative systems.

    Philippe Naveau
    Adjunct Professor of Applied Mathematics
    (Ph.D. Colorado State University)

    My work interests are focused on theoretical and applied statistics, with an emphasis on applications related to environmental and climatic data sets. I am interested in extreme value theory, time series analysis and spatial statistics.

    Harvey Segur
    Professor of Applied Mathematics
    (Ph.D. University of California at Berkeley)

    Many of the outstanding problems in physics and in engineering applications are described by nonlinear partial differential equations. Typically these equations cannot be solved exactly, but some insights can be gained from physical experiments, numerical experiments, or the study of special limiting cases in which the equations simplify. Asymptotic analysis describes the systematic methods by which one approximates the general system by simpler models obtained from limiting cases. It is one of the fundamental tools of applied mathematics, and of mathematical physics.

    On rare occasions the equations that arise can be solved completely. In these cases, the equations are said to be completely integrable or exactly solvable; it is in these special cases that solitons appear. Much of our current knowledge of the general structure of nonlinear differential equations derives from study of these special cases.

    Professor Segur's present research concerns the study of water waves, an interesting physical problem in which both of these approaches are needed. This problem exhibits many of the phenomena seen in more exotic systems, but the experiments for water waves are simpler and more accessible.

    John Williamson
    Professor Emeritus of Applied Mathematics
    (Ph.D. University of Minnesota)

    Traits that "run in families" are not difficult to identify. In terms of the level of family concern, these traits can range from red hair to particular forms of cancer. A more difficult task is to identify which of these family traits are a result of a shared family environment, which are a result of shared genes, and which are a result of gene-environment interaction. To attempt to accomplish this task it is necessary to first construct a probability model for what is being observed. The parameters of this model will, typically, correspond to the forementioned situations: determined by genes; determined by environment; determined by gene-environment interaction. Professor Williamson's recent research concerns the development of models and of statistical tests based on these models so that, following the collection of data, inferences can be made about the trait or disease under study.