Research in Mathematics

Mathematics is widely cultivated at UMCP because of the fundamental role it plays in almost all areas of pure and applied physical, biological and social sciences. The center of mathematical activity, the Department of Mathematics, is in the first rank in several branches of mathematics, including important areas of applied analysis, computational mathematics, dynamical systems, geometry and representation theory. The most recent rankings by U.S. News and World Report place the Department at number 16 among all public and private universities in the U.S. and Applied Mathematics is ranked number 11. In addition to research, the Department is also strongly committed to education: we have almost 450 mathematics majors and more than 200 graduate students pursuing PhD's in pure mathematics, applied mathematics and statistics. The University of Maryland thus has one of the largest mathematics graduate programs in the nation. The Department is also in the forefront of significant interdisciplinary activity, both with other units on Campus and with Government agencies including NIH, NIST, the Census Bureau and the Department of Defense. The following is a list of research areas actively pursued in the Department and selected subfields.

Applied and Computational Harmonic Analysis
The focus of this group is the development and real-world application of signal and image processing methods involving harmonic analysis, wavelets, frames, multi-dimensional uniform and non-uniform sampling, uncertainty principle inequalities, and noncommutative spectral techniques.

The Norbert Wiener Center for Harmonic Analysis and Applications comprises the personnel involved in applied and computational harmonic analysis in the Department of Mathematics. Besides cutting-edge research in theoretical and applicable harmonic analysis, the Norbert Wiener Center has educational and international outreach components. A centerpiece of the educational program is the professional masters degree in The Mathematics of Advanced Industrial Technology.
Specific algorithm design and research programs the Norbert Wiener Center include waveform design for radar and communications, sigma-delta encoding and implementable variants, auditory modeling, functional Magnetic Resonance Imaging (fMRI), fast imaging algorithms, periodicity detection methods, global climate modeling, minefield detection, thin film analogue image processing, and all-optical time domain filterbanks.

Methods devised by the group are employed by a wide variety of scientists and technologists, including MRI research centers throughout the world, target recognition groups in the Department of Defense, adaptive radar processing development groups at major Defense contractors, and start-up companies in optical telecommunications. The group has had a significant influence in determining the direction of large-scale government investments in applied mathematics well into the next decade.
Contact: Prof. John J. Benedetto (jjb@math.umd.edu:301-405-5161)

Chaos and Computational Dynamics
Professor Yorke, his collaborators, and his students actively investigate the dynamics of chaotic processes -- that is, of deterministic processes that exhibit irregular oscillations that persist indefinitely. Such processes are predictable for only short periods because uncertainties tend to be amplified. The objective is to describe those robust properties that are common in the dynamics of (among others) physical, biological, and chemical systems. Sometimes the phenomena can be described using rigorous mathematics, and sometimes only phenomenological descriptions can be obtained from intensive numerical studies. Most often, the research is a blend of numerical and rigorous techniques. Professor Yorke, in collaboration with faculty members from Meteorology, Computer Science, Math and Physics, recently was awarded a Keck Foundation grant to use chaos theory to develop better weather prediction algorithms for use with high performance computing.
Contact: Professor Jim Yorke (yorke@ipst.umd.edu)

Geometry and Topology
The Department has one of the leading geometry research groups in the world. Some of the areas of research include: harmonic analysis on trees, geometric structures on manifolds, hyperbolic geometry, applications of topology to mathematical physics, noncommutative geometry, global (metric) Riemannian geometry, integrable systems, K-theory, L2-cohomology, low- dimensional topology, and geometric methods in dynamical systems.
Contact: Bill Goldman (wmg@math.umd.edu: 301--405-5124)

Material Science
This research focuses on the modeling, analysis, and simulation of microscopic processes and macroscopic behaviors in materials that include phase transitions, interface dynamics, defect evolution, and pattern formation. The aim is to characterize fundamental properties of many physical systems, such as martensite in shape-memory alloys, phase separation in steel and alloys, and epitaxial growth of semiconductor thin films. All these materials have emerging applications in medical devices, modern communication technologies, and aerospace and automobile industries. Analytical methods in nonlinear partial differential equations, the calculus of variations, and nonlinear dynamics, together with simulation techniques of multi-scale and fast algorithms, will be developed. Existing collaboration with physicists and materials scientists around the campus and in the region will be capitalized upon.
Contact: Professor Georg Dolzmann (dolzmann@math.umd.edu, 301-405-5066)

Representation Theory
Representation theory plays a fundamental role in modern approaches to number theory, including the Langlands program and the proof of Fermat's Last Theorem. It is also important in mathematical physics, differential geometry and many other areas. The distinguished group here studies representation theory of Lie groups, p-adic groups, C*-algebras, and other related objects such as Lie algebra and quantum groups. The math department is home to a major computational project in Lie theory, the Atlas of Lie Groups. This is a project to compute the unitary dual of semisimple Lie groups, and make the answer readily available to the scientific community, much like the influential Atlas of Finite Groups. See www.math.umd.edu/~jda/atlas.
Contact: Jeff Adams (jda@math.umd.edu: 301--405--5493)

Numerical Analysis and Computation
Numerical analysis is concerned with the construction, analysis and implementation of novel computational algorithms which play a central role in modern applied mathematics. Our world-class group conducts fundamental research on finite-difference, finite-element, finite-volume, boundary-element, spectral-particle and wavelets based discretization methods. Applications include nonlinear waves, computational fluid dynamics, homogenization, free boundary problems, interface dynamics and nonconvex variational problems such as those that arise in fluid and solid mechanics, electromagnetism, materials science, image processing and biological sciences.
Contact: Professor Ricardo Nochetto (rhn@math.umd.edu: 301-405-5145)

Continuum Mechanics
Continuum mechanics lies at the heart of many areas of scientific and technological research. The Department has maintained a strong group that has worked in fluid dynamics for over fifty years. Over the years this group has enlarged its domain of research to include incompressible and compressible fluid dynamics, nonlinear elasticity and viscoelasticity, kinetic theory, transition regimes, solid mechanics, electro-magnetic and thermal effects in deformable bodies, and other areas of continuum mechanics. Their research embraces formal, numerical, and analytical aspects. Applications include semiconductor fabrication and design, smart materials, aerospace, astrophysics, and climate modeling.
Contact: Professor Dave Levermore (lvrmr@math.umd.edu: 301-405-5127)

Bioinformatics and Computational Biology
A new program involving faculty from Computer Science, Biochemistry, Mathematics and Molecular Biology, which will address fundamental issues at the interface between biological and computational sciences. Current areas of emphasis include visualization, mathematical models and algorithms, protein structures, evolution, and analysis and management of large scale biological data.
Contact: Professor Joseph Ja’Ja (joseph@umiacs.umd.edu, 301-405-6722)

Algebra and Number Theory, including Algebraic Geometry
Members’ interests include algebraic number theory, automorphic forms, arithmetical algebraic geometry, motives and cryptography.
Contact: Professor Larry Washington (lcw@math.umd.edu, 301-405-5116)

Dynamical Systems
Members’ interests symbolic dynamics, complex dynamics, smooth dynamics, and applied dynamics, including the subfield of Chaos described separately.
Contact: Professor Michael Boyle (mmb@math.umd.edu, 301-405-5135)

Logic
Members’ interests include model theory and stability theory proof theory and non-classical logics.
Contact: Professor David Kueker (dwk@math.umd.edu, 301-405-5052)

Partial Differential Equations
Members’ interests include non-linear elliptic and hyperbolic PDEs, fluid dynamics, wave equations and conservation laws, as well as computational methods.
Contact: Professor Manoussos Grillakis (mng@math.umd.edu, 301-405-5173)

Probability and Statistics, Including Applied Statistics
Members’ interests include stochastic processes, Markov processes, asymptotics and applications. Applied statistics permeates all branches of physical sciences, engineering, and social sciences. The statistics group has done significant work in applications in cooperation with Government agencies. Examples include survey sampling and small area estimation using federal survey data (bureau of census), space-time modeling/prediction of remotely sensed data (NASA), survival analysis and medical large-population longitudinal studies (NIH), biostatistical modeling/estimation (NIH), acoustic emission in metals and astronomical estimation (NIST). Signal processing and time series research (discrete spectral analysis in noise and discrimination, parametric filtering) was supported for years by both ONR and AFOSR. A significant amount of applied work is done at the Statistics Laboratory that is available for consultation and problem solving involving the applications of statistics. The Lab is open for internal and external clients.
Contact: Professor Paul Smith (pjs@math.umd.edu: 301-405-5061/5104)