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Directorate for Mathematical and Physical Sciences
Division of Mathematical Sciences

The Division of Mathematical Sciences (DMS) supports a wide range of projects aimed at developing and exploring the properties and applications of mathematical structures. Most of these projects are awarded to single investigators or small groups of investigators working with graduate students and postdoctoral researchers. Programs such as Mathematical Sciences Infrastructure handle activities that fall outside this mode.

Proposals for General Conferences, Workshops, Symposia, Special Years, and Related Activities in DMS

Proposals for general conferences, workshops, symposia, special years, and related activities should be submitted to the appropriate disciplinary program. Proposals should be submitted 1 year before the start of the activity. Contact the division for information on proposal requirements or see program solicitation NSF 00-109.

Specific Types of Grants Supported by DMS

In addition to the usual types of research grants awarded to principal investigators and institutions, DMS supports the following:

  • University/Industry Cooperative Research—DMS feels it is important to provide more opportunities to conduct research and training in an industrial environment and for industrial scientists to return periodically to academia. To facilitate research and training, the division provides Mathematical Sciences University/Industry Postdoctoral Research Fellowships, Senior Research Fellowships, and Industry-Based Graduate Research Assistantships and Cooperative Fellowships in the Mathematical Sciences.
  • Interdisciplinary Grants—These grants enable faculty members to expand their skills and knowledge into areas beyond their disciplinary expertise, to subsequently apply that knowledge to their research, and to enrich the educational experiences and career options for students. These grants support interdisciplinary experiences at the principal investigator's (PI's) institution (outside the PI's department) or at academic, financial, or industrial institutions in a nonmathematical science environment.

Disciplinary Programs

1. Algebra, Number Theory, and Combinatorics

Supports research in algebra, including algebraic structures, general algebra, and linear algebra; number theory, including algebraic and analytic number theory, quadratic forms, and automorphic forms; and combinatorics, graph theory, and algebraic geometry.

2. Analysis

Supports research on properties and behavior of solutions of differential equations; variational methods; approximations and special functions; analysis in several complex variables and singular integrals; harmonic analysis and wavelet theory; Kleinian groups and theory of functions of one complex variable; real analysis; Banach spaces, Banach algebras, and function algebras; Lie groups and their representations; harmonic analysis; ergodic theory and dynamical systems; some aspects of mathematical physics such as Schroedinger operators and quantum field theory; and operators and algebras of operators on Hilbert space.

3. Applied Mathematics

Supports research in any area of mathematics except probability or statistics. Research is expected to be motivated by or have an effect on problems arising in science and engineering, although intrinsic mathematical merit is the most important factor. Areas of interest include partial differential equations that model natural phenomena or that arise from problems in science and engineering, continuum mechanics, reaction-diffusion and wave propagation, dynamical systems, asymptotic methods, numerical analysis, variational methods, control theory, optimization theory, inverse problems, mathematics of biological or geological sciences, and mathematical physics.

4. Computational Mathematics

Supports research in algorithms, numerical and symbolic methods, and research in all areas of the mathematical sciences in which computation plays a central and essential role. The prominence of computation in the research is a key distinction between Applied and Computational Mathematics. Proposals from interdisciplinary teams to develop critical mathematical and computational techniques from modeling and algorithm development through implementation are encouraged. Also encouraged are proposals for innovative computational methods within the mathematical sciences.

5. Geometric Analysis

Supports research on differential geometry and its relation to partial differential equations and variational principles; aspects of global analysis, including the differential geometry of complex manifolds and geometric Lie group theory; geometric methods in modern mathematical physics; and geometry of convex sets, integral geometry, and related geometric topics.

6. Statistics

Supports research for developing and improving statistical theory and methods that are used for the collection, exploration, analysis, and interpretation of data to enable discovery and advancement in virtually all areas of science and engineering. Subfields include parametric and nonparametric inference, multivariate analysis, Bayesian analysis, experimental design, robust statistical methods, time series analysis, spatial analysis, and resampling methods.

7. Probability

Supports research on the theory and applications of probability. Subfields include discrete probability, stochastic processes, limit theory, interacting particle systems, stochastic differential and partial differential equations, and Markov processes. Research in probability, which involves applications to other areas of science and engineering, is especially encouraged.

8. Topology

Supports research on algebraic topology, including homotopy theory, ordinary and extraordinary homology and cohomology, cobordism theory, and K-theory; topological manifolds and cell complexes, fiberings, knots, and links; differential topology and actions of groups of transformations; geometric group theory; and general topology and continua theory.

9. Foundations

Supports research in mathematical logic, including proof theory, recursion theory and model theory, foundations of set theory, and infinitary combinatorics.

Infrastructure Programs and Other Activities

In addition to support in the disciplinary programs, the Division of Mathematical Sciences (DMS) offers activities that differ from the usual type of research projects. A few examples of these programs are included here. For additional programs and further information, visit the DMS home page,

1. Mathematical Sciences Research Institutes and Other Activities

The Division of Mathematical Sciences (DMS) currently funds seven awards given to different mathematical sciences research institutes. These projects stimulate research in all of the mathematical sciences through thematic and residential programs, workshops, and access to distinctive resources. All of the institutes offer visiting opportunities for researchers in every stage of their career, and most offer postdoctoral fellowships for one or more years, with mentoring provided by outstanding scientists. Many of these centers involve new researchers, graduate students, and undergraduates through tutorials related to current programs, mathematical research experiences based on industrial or other problems, and summer schools. Interested parties are encouraged to contact the institutes directly for information on current and future programs, visiting opportunities, and other activities. The seven institutes and their Web sites are:

American Institute of Mathematics
AIM Research Conference Center
360 Portage Ave
Palo Alto, CA 94306-2244
Web address:

Institute for Advanced Study
School of Mathematics
1 Einstein Drive
Princeton, NJ 08540
E-mail address:
Web address:

Institute for Mathematics and its Applications
University of Minnesota
400 Lind Hall, 207 Church Street SE
Minneapolis, MN 55455-0436
Web address:

Institute for Pure and Applied Mathematics
IPAM Building
460 Portola Plaza
Box 957121
Los Angeles, CA 90095-7121 (it's important to include the entire 9-digit Zip Code)
Web address:

Mathematical Biosciences Institute
The Ohio State University
231 W. 18th Avenue
Columbus, OH 43210
Tel: 614-292-3648
Web address:

Mathematical Sciences Research Institute
17 Gauss Way
Berkeley, CA 94720-5070
Web address:

Statistical and Applied Mathematical Sciences Institute
19 T. W. Alexander Drive
P.O. Box 14006
Research Triangle Park, NC 27709-4006
Tel: 919-685-9350 FAX: 919-685-9360
Web address:

In addition to these institutes, DMS contributes to the support of the Banff International Research Station for Mathematical Innovation and Discovery in Banff, Alberta, a joint venture between Canada and the United States (visit the station’s Web site at This site is an international center for workshops, team research, and summer schools for mathematical sciences and mathematical challenges in science and industry.

  • Regional Conferences—Operated by the conference board of the mathematical sciences, these conferences feature a principal speaker who gives 10 1-hour talks on a particular subject during a weeklong session.
  • Scientific Computing Research Environments in the Mathematical Sciences—Offers moderate grants for computing equipment that will benefit groups of outstanding researchers who are highly productive but whose work has been seriously impeded by the lack of computing facilities.
  • Undergraduate Activities—Awards are made in conjunction with NSF-wide undergraduate efforts, including Research Experiences for Undergraduates (REU), cooperative activities with the Directorate for Education and Human Resources (EHR), and other related activities. For more information on REU, visit the NSF Crosscutting Programs home page, Further information about EHR programs and activities can be found in the EHR section in this Guide.
  • Mathematical Sciences Postdoctoral Research Fellowships—Fellowships will be awarded to between 30 and 35 new fellows in 2004. Tenure provides a research instructorship option.

Eligibility Requirements for the Mathematical Sciences Postdoctoral Research Fellowships

Each applicant will be required to submit a research plan for the tenure period requested. The fellowships are not intended to support the preparation of prior research results for publication or the writing of textbooks.

To be eligible for one of these fellowships, an individual must (1) be a citizen, national, or lawfully admitted permanent resident alien of the United States as of January 1, 2004; (2) have earned by the beginning of his or her fellowship tenure a doctoral degree in one of the mathematical sciences listed above, or have research training and experience equivalent to that represented by a Ph.D. in one of those fields; and (3) have held the doctorate for no more than 2 years as of January 1, 2004.

2. Focused Research Groups

The mathematical sciences thrive on sharing ideas and information from various scientific fields and disciplines. Certain research needs can only be met appropriately through the use of investigative teams. The Focused Research Groups (FRG) Program supports these teams, thereby allowing groups of researchers to respond to the scientific needs of pressing importance, take advantage of current scientific opportunities, and prepare the ground for anticipated developments in the mathematical sciences. In addition to mathematical scientists, groups may include researchers from other scientific and engineering disciplines. FRG projects are highly focused scientifically, timely, limited to 3 years’ duration, and substantial in both scope and impact. Projects supported through FRG are essentially collaborative in nature, their success dependent on the interaction of a group of researchers.

3. Enhancing the Mathematical Sciences Workforce in the 21st Century (EMSW21)

The long-range goal of the EMSW21 Program is to increase the number of U.S. citizens, nationals, and permanent residents who are well prepared for and want to pursue careers in the mathematical sciences and in other NSF supported disciplines. EMSW21 builds on the Vertical Integration of Research and Education (VIGRE) Program and now includes a broadened VIGRE activity, an additional component for Research Training Groups in the Mathematical Sciences (RTG), and an additional component for Mentoring through Critical Transition Points in the Mathematical Sciences (MCTP).

  • The Grants for Vertical Integration of Research and Education (VIGRE) component focuses on enhancing the educational experience of all students and postdoctoral associates in a department (or departments). Broad faculty commitment and a team approach to enhancing learning are necessary for the success of this program. A principal element of VIGRE is the increase in interaction among undergraduates, graduate students, postdoctoral associates, and faculty members. Integrating research and education for graduate students and postdoctoral associates, involving undergraduates in substantial learning by discovery, and developing a team approach are keys to successful VIGRE projects. These goals can be accomplished in many ways, and proposers should develop creative approaches that suit their circumstances.
  • The Research Training Groups in the Mathematical Sciences (RTG) component provides groups of researchers who have related research goals in the mathematical sciences with funds to foster research-based training and education. Although the groups may include researchers and students from different departments and institutions, the research-based training and education activities must be based in the mathematical sciences. The RTGs are expected to vary in size, scope, proposed activities, and plans for organization, participation, and operation.
  • The Mentoring Through Critical Transition Points in the Mathematical Sciences (MCTP) component provides a system of mentoring that focuses on points of transition critical for success in a mathematical science career path—from undergraduate studies to the early years in a tenure track position. The program may be a comprehensive department effort or a more focused endeavor involving a few faculty mentors and aimed toward a specific transition point or group of points. However, department-wide programs that include components for undergraduates, graduates, and postdoctorates, may be more appropriate for the VIGRE component. Successful proposals will be those that provide ways to increase the number and the quality of training of U.S. citizens, nationals, or permanent residents entering the scientific workforce with strong mathematical training, including the number of degrees awarded in the mathematical sciences.
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