Mathematical Sciences

Extramural Basic Research Mathematical Sciences

U.S. Army Research Office
P.O. Box 12211
Research Triangle Park, N.C. 27709-2211

Commercial: (919) 549-4245
DSN: 832-4245
Fax: (919) 549-4310

The Mathematical Sciences have great impact on a wide range of Army systems and doctrine. The objective of the programs of the Mathematical Sciences Division is to respond to the quantitative requirements for enhanced capabilities of the Army in the twenty-first century in technologies related to the physical, informational, biological, and behavioral sciences and engineering. Mathematics plays an essential role in measuring, modeling, analyzing, predicting, and controlling complex phenomena and in understanding the performance of systems and organizations of critical interest to the Army. In particular, mathematical sciences are integral parts of research in network science, decision science, intelligent systems, and computational science. Mathematical Sciences also play an important role in solving Army issues related to materials, information, robotics, networks, C4ISR, testing, evaluation, decision-making, acquisition, training, and logistics. With the advent and subsequent refinement of high-performance computing and large-scale simulation, the mathematical sciences have become an integral part of every scientific and engineering discipline and the foundation for many interdisciplinary research projects. Computing and simulation now complement analysis and experimentation as fundamental means to understand informational, physical, biological, and behavioral phenomena. High-performance computing and advanced simulation have become enabling tools for the Army of the future. Real-time acquisition, representation, reduction, and distribution of vast amounts of battlefield information are key ingredients of the network-centric nature of the modern digital battlefield. The Mathematical Sciences Division creates coherent basic research programs that are revolutionary, innovative, and responsive to the future needs of the Army. Research supported within this Division is organized into four program areas:

  • Modeling of Complex Systems
  • Probability and Statistics
  • Biomathematics
  • Computational Mathematics

Division Chief
Dr. Joseph Myers
(919) 549-4245

The Mathematical Sciences Division supports the following research areas:

Modeling of Complex Systems

Dr. Jay Wilkins
(919) 549-4334

The Modeling of Complex Systems program conducts fundamental mathematical-analysis-oriented research with the objectives to develop quantitative and qualitative models of complex phenomena of interest to the Army, especially for those current models that are not based on first/basic principles. The complex phenomena of interest to the Modeling of Complex Systems Program are largely physical and geometric phenomena (topological data analysis, mathematical information and signal theory, dynamical systems on complex structures, etc.) and human cognitive and social phenomena (strategic group formation, behavioral and cultural modeling, human-computer interaction and social informatics, etc.). The basic research carried out by this program contributes to the Future Force Technologies Command, Control, Communication, Computers, Information, Surveillance and Reconnaissance (C4ISR), survivability, and mobility, situational awareness and decision making, manned/unmanned integration and reduced logistics, and strategy modernization.

The complex systems of interest to the Modeling of Complex Systems program include those in the following four areas:

  • Information and Data —Information superiority is recognized as a key to success in defense efforts, peacekeeping, and humanitarian operations. Areas of prime interest include – but are not limited to – information sensing and collection over geographically large or dispersed domains, multimodal data fusion (optical, infrared, acoustic, electromagnetic, etc.), analysis and modeling of organized or dynamic information networks, and topological data analysis of large, noisy data sets and under-sampled or corrupted signals.
  • Geometric and Topological Modeling—Representations of complex, irregular objects and of complicated, often high-dimensional abstract phenomena and functions are fundamental for Army and other DOD agencies in modeling geophysical features and terrains, urban and man-made settings with vision, mobility, or otherwise topological obstructions, complex data structures and information flows, and biological objects. Extending classical mathematical modeling and geometric analysis techniques to spaces with irregular (i.e. non-smooth) fine structure is also of key interest.
  • Human Cognitive and Behavioral Modeling—Quantitative, analytical models of cognition, behavior, sociolinguistics, and group/network formation are required for training, simulation, mission planning, predictive and causal modeling, and intelligence analysis.
  • Human-Machine Interaction - Models of human-computer interaction at both the small scale (individual/small group and machine) and the large scale (social informatics, large social information networks) are of interest, particularly for information gathering and supervised machine learning on the small scale, and for the influence of social information networks on strategic group formation and social action on the large scale.

Probability & Statistics

Dr. Joseph Myers
(919) 549-4245

The Probability and Statistics program supports research in stochastic analysis, applied probability, and statistical methods in response to the Army's need for real-time decision making under uncertainty and for the test and evaluation of systems in development. Special emphasis is placed on methods for analyzing data obtained from phenomena modeled by such processes. The two major areas of research are described below.

Stochastic Analysis and Applied Probability

Many Army research and development programs are directed toward modeling, analysis, and control of stochastic dynamical systems. Such problems generate a need for research in stochastic processes, random fields, and/or stochastic differential equations in finite or infinite dimensions. The thrust research areas in stochastic analysis and applied probability include but are not limited to the following:

  • Stochastic Delay and Partial Differential Equations—Research on analytical and numerical methods for solving stochastic delay and partial differential equations and their related nonlinear filtering and control problems is one of the program objectives. These equations play an important role in modeling many physical and biological processes in continuum and under noisy environment.
  • Spatial-Temporal Event Pattern Recognition—Developments of theoretical foundation and efficient algorithms for spatial-temporal event pattern recognition in nonlinear and noisy environments are considered keys to winning the war against terrorism.
  • Quantum Stochastics and Quantum Control—With technological advances now allowing the possibility of continuous monitoring and rapid manipulations of systems at the quantum level, there is an increasing awareness of the applications and importance of quantum control in engineering of quantum states, quantum error correction, quantum information, and quantum computation.
  • Other Areas that Require Stochastic Analytical Tools—Research areas of importance to the Army in stochastic analysis and applied probability include (1) stochastic fluid dynamics and turbulence; (2) interacting particle systems and their applications to material science and nanotechnology; and (3) stochastic modeling and analysis of polymers.

Statistical Methods

The following research areas are of interest to the Army and are important in providing solutions to Army problems.

  • Analysis of Very Large or Very Small Datasets—The state of the art in statistical methods is well adapted to elicit information from medium-size data sets collected under reasonable conditions from moderately well understood statistical distributions. However, Army analysts frequently have very large or very small data sets sampled from nonstandard, poorly understood distributions.
  • Reliability and Survivability—This research area is dedicated to the study of the performance and cost of engineered systems. Many of the models and methods developed will have immediate application to problems that face the military. For example, reliability and life-length methodologies are needed for analyzing mechanical and electrical systems, especially those with extremely low-failure rates. To support future operations, the Army needs novel and efficient statistical tools for improving system reliability and survivability, and for analyzing data collected from sensor networks.
  • Data Stream—The Army has pressing research needs in the area of streaming data. Especially, sampling theory methodology or the consideration of data epochs with meta-analysis relating findings across epochs may reduce the need to retain the entire stream of information. Since the information sought may be contained in a very small fraction of the data, useful methods for data reduction may depend on effective modeling of the data stream and the relationship of the relevant information to the overall stream.
  • Bayesian and Nonparametric Statistics—Future emphasis in statistics on "predictive" models verses explanatory models is important. Military operations call for predictive models based on a growing base of sensor-fueled data stores. Increased computational capability is also leading statistics in a new direction, away from using "classical" results that are really approximations to avoid computational issues.


Dr. Virginia Pasour

Biomathematics is an exciting and important new area of activity for ARO. The introduction of Biomathematics as a separate area of basic research recognizes the importance and specialized nature of quantitative methods in the biological sciences. Biology involves a large number of entities that interact with each other and their environment in complex ways and at multiple scales. This complexity makes Biomathematics a highly interdisciplinary field that requires unique and highly specialized mathematical competencies to quantify structure in these relationships. Mathematical techniques currently utilized in the field range from multicompartmental differential equations modeling in physiology, microbiology, epidemiology, and neurobiology, to agent-based and network models involved in understanding ecosystem and human dynamics, to innovative applications of traditionally pure mathematics to a wide range of biological systems.

The goal of the Biomathematics Program focuses on using existing mathematics and creating new mathematical techniques to uncover fundamental relationships in biology, spanning different biological systems as well as multiple spatial and temporal scales. Of special interest are high-risk attempts to use techniques in fields not traditionally brought to bear on biological problems, as well as innovative efforts at handling large amounts of complex data.

Computational Mathematics

Dr. Joseph Myers
(919) 549-4245

The Computational Mathematics program supports the strategic themes of the Mathematical Sciences Division by developing innovative, efficient, and accurate numerical methods and their implementations in scalable scientific software tools. Such methods and tools assure that mathematical models can be translated into effective simulations. Computation has firmly established itself as the third pillar of scientific inquiry, joining Experimentation and Theory. The quantitative predictions of many modern theories can only be derived from extensive computations. As Army problems become more complex, new and better approaches are needed to understand and develop their solutions. The focus of the Computational Mathematics program is on developing algorithmic methods to model new applications and discover general solution methods for large classes of problems.

Numerical computation and simulation have become an essential part of scientific investigation and engineering design. In science, it has become the accepted third component of the scientific method, complementing theory and experiment. In engineering, it enables us to simulate potential designs over wide parameter ranges and over a wide range of operating conditions, and to virtually autopsy failures after they have occurred. Such simulations often require considerable effort to set up, require considerable run time on large-scale parallel systems, and require considerable effort to distill useful information from the resulting massive data sets. In addition, we often operate our simulations in regimes or on scales at which we cannot collect reliable experimental data, and so, it is often not possible to directly quantify the fidelity of these models in all the regimes or scales in which we employ them. This problem is especially acute for simulations of failure processes. Data has become ubiquitous, but mathematically sound methods for incorporating these into simulations are incomplete. Research to improve validation, relevance and completeness in modeling are critical. The Computational Mathematics program enables mathematical research directed toward empowering the Army in these and related areas.


Last Update / Reviewed: August 9, 2018