Appendix 2


In assessing the subfields of the mathematical sciences, we were greatly aided by the staff of the Division of Mathematical Sciences, which made an internal assessment based on information from leading researchers and on results from the peer review system. Individual members of the Panel assumed responsibility for assessing their own areas of expertise; they consulted with experts in the United States and abroad and used bibliographic and peer review data provided by staff of the Division. Although the resulting assessments are fundamentally subjective and qualitative, we have confidence in the data on which they are based and on the results.

We do, however, warn potential readers that the Appendix is more technical than the main body of the Report. It was compiled primarily for the use of the Panel in the short term and the Division of Mathematical Sciences in the medium term.

For assessment purposes the mathematical sciences were divided into nine subfields which mirror the program structure of the Division of Mathematical Sciences of NSF. Each subfield assessment is intentionally rather general; more specific evaluations within subfields are provided by the peer review system, which is regarded as very effective for this purpose.


Foundations, or mathematical logic, is here subdivided into four areas: set theory, model theory, recursion theory, and proof theory. The United States has notable activities in this field, but Israel and Europe are, as a whole, dominant.

In set theory the United States shares leadership with Israel, with exciting results across the areas of determinacy, large cardinals, and combinatorics. However, the research community in set theory is aging, and younger mathematicians and graduate students are few and primarily foreign. The area of model theory is flourishing worldwide. The United States is a participant, but its activities are overshadowed by those in England, France, and Israel. There are notable interactions between this area and other fields of mathematics and with computer science. Recursion (or computability) theory is quiescent, with a substantial body of completed work. Barring a major breakthrough, or the further exploitation of connections with computational mathematics and computer science, the next decade is not expected to be very active. England plays a leading role, with the United States as a contributor, but the aging research population is not being replenished. In the United States, proof theory has, in large measure, moved to computer science. The United States is a minor contributor; research leadership is concentrated in France, Germany, Russia, and Israel. Major advances in computational complexity are expected to continue.

It is notable that some of the acknowledged leaders in Foundations, both in the United States and abroad, are also active in other fields of mathematics or computer science. Several dynamic areas of model theory exemplify how a field can thrive both within its own core and in relation to other fields. Potential interactions exist for set theory and recursion theory. There is concern, however, that Foundations in general has not attracted enough young mathematicians in the United States during the past two decades. This has led to fragility in the field and concerns that the U.S. community cannot respond promptly to advances in other countries. In addition, there is uneasiness over the insularity of several key areas of Foundations, which results in the failure to explore opportunities arising both within mathematics and in science and technology, notably in computer and computational science.

Algebraic Geometry and Number Theory

The United States and Western Europe dominate research in these subfields, although Europe has had a much longer tradition in these areas. These fields have seen spectacular achievements over the last 50 years, as old, even ancient problems have been solved, culminating in the last decade with the solution of Fermat’s problem and the Mordell conjecture. In addition, these subfields have had significant impacts on and interactions with physics, cryptography, and other areas of mathematics.

Algebraic geometry is flourishing. Current problems of significance are likely to be solved in the next few years and to be replaced by others of equal significance. There are notable interactions with other areas of mathematics and theoretical physics. Leadership is shared by the United States, Japan, and Western Europe, with the United States having the most active researchers. In computational algebraic geometry, the United States lacks depth, and leadership is held by Europeans.

Number theory is dominated by the three "grand challenges": the Riemann Hypothesis (RH), the Langlands Program, and the Block-Kato-Beilinson conjectures. There has been recent notable progress on the last two, but additional progress on the Langlands Program awaits a breakthrough. There is renewed activity in RH, but probably not at the depth needed. The United States, and Western Europe are the dominant centers, with Canada quite strong; at one time the Soviet Union was among the leaders.

Arithmetic geometry has experienced recent spectacular breakthroughs, notably the work on Fermat's Last Theorem. The United States has great strength, but Europe is probably stronger. Analytic number theory is relatively quiet at this time, needing a new major line of attack. The United States is the clear leader, with great depth but very few young researchers. Computational number theory is an area of very high activity, driven by increasing access to powerful computers and the link to cryptography. Europeans are major contributors to the open literature; there is also significant classified and proprietary work where the United States is regarded as the leader.

The entire subfield is considered to have notable opportunities for both internal development and for impact external to the area. In spite of this, the number of young Americans entering it has decreased significantly. A large proportion of current major U.S. contributors were educated abroad, which was not the case 20 years ago, and researchers move freely between the United States and Europe.

Algebra and Combinatorics

Algebra has undergone significant developments in the past decade. It is a very active subfield, with significant interaction with topology, geometry, and theoretical physics. The United States is regarded as the leader, with Western Europe a close second; both have benefited from the emigration of mathematicians from the former Soviet Union. There are also major centers of activity in Russia, Japan, Israel, and Australia.

In algebraic representation theory, research is enhanced by interactions with geometry, combinatorial methods, and theoretical physics. The United States is a leader, and Western Europe and Japan have significant strengths. U.S. leadership also holds in finite and combinatorial group theory, where the infusion of geometric ideas is leading to novel approaches and results. There are also strong centers of activity in Western Europe, Russia, and Israel. Rather exciting developments are underway in noncommutative geometry and Lie theory, with connections to algebraic geometry and to theoretical physics (quantum groups). The United States and Western Europe share research leadership. The United States is the clear leader in ring theory, where a breakthrough is needed for further major advances. Computational algebra, although still in its infancy, holds great promise. Europe has decidedly more depth and breath than the United States, and Australia is a strong participant.

Graph theory serves as a bridge between mathematics and areas of applications. The very strong researchers are in the United States, working not in mathematics departments but in computer science, electrical engineering, and industry. In Europe there is greater connectivity between mathematicians, computer scientists, and engineers; consequently, Europe is stronger in applications. The United States has pioneered the area of graph minors and remains the leader in this small subfield.

The United States has clear leadership and great depth in algebraic combinatorics, where significant challenges (MacDonald conjectures) and notable interactions with quantum cohomology are providing exciting results. In probabilistic combinatorics the United States has many research leaders but little depth. Recent breakthroughs have led to solutions of classical extremal problems. Western Europe and Israel are the other major centers of activity.

The subfields of algebra and combinatorics are central to the mathematical sciences and they attract good students and young researchers. In the last ten years, there has been a concerted effort to apply combinatorics in a wide set of areas such as crystallography, robotics, computational efficiency, DNA sequencing, and computer networks. In the United States, this has most frequently been done by theoretical computer scientists. Europe is quite strong in discrete mathematics. There is a paucity of interactions between mainline U.S. algebraists and other disciplines.

Topology and Geometric Analysis

Topology and Geometric Analysis have flourished in the last decade. This subfield is central to the mathematical sciences as an area of specific mathematical investigation, as a mode of thought that uses geometric and topological concepts in other branches of mathematics, and in the analysis of geometric patterns that arise in computing and the natural sciences. Perhaps the most exciting recent development is the manner in which geometry, topology, analysis, and theoretical physics have become intertwined and mutually reinforcing. The United States is regarded as the leader, with substantial strength in Western Europe and some strength in Japan.

Seiberg-Witten theory, originating in theoretical physics, has proven to be an effective tool in symplectic topology/geometry, stimulating much recent activity and leading to the solution of long-standing problems. There are active groups worldwide, most notably in the United States and Western Europe. The theory of algebraic invariants, especially the study of invariants for low-dimensional manifolds and knots in three-manifolds, has been exceptionally fast-moving, stimulated by its interaction with physics. The United States is especially dominant in this area, aided by recent emigration from the United Kingdom; Russia also has strength. Work continues on the classification of three-dimensional manifolds, driven by the Thurston Geometrization Conjecture, with the possibility of near-term success. Strength in this area is concentrated in the United States. Homotopy theory is playing an increasingly important role in algebraic geometry, but is generally mature; the United States is strong in this area. Progress has been made in providing powerful computations in algebraic K-theory, with major contributions from the United States, France, and Norway.

Riemannian geometry has experienced several major developments in the last decade; both the United States and Western Europe have significant depth in this area. Regularity theory for differential equations related to geometric objects has been an important area of research, with activity primarily in the United States. Recent work in the United States and Western Europe on harmonic maps has implications for super-rigidity and representation theory. Noncommutative geometry, involving a synthesis of geometry, analysis, algebra and topology in a quantization of mathematical entities, could lead to significant breakthroughs in the near future. Both the United States and Western Europe are leaders in this area. There has been steady, if not spectacular, progress in geometric measure theory and minimal surfaces over the last decade, with some applications to problems in materials research; this work has occurred mainly in the United States.

There is a strongly felt need, both in the United States and abroad, to actively stimulate interactions between topologists and geometers and members of the other sciences and technology in order to disseminate geometric ideas to prospective users and to stimulate new ideas in the subfield. A limited number of geometry and topology researchers are currently collaborating with specialists on DNA and polymers, control of mechanical systems, robotics, and image processing. A substantial number of young people are entering this subfield, although (as with other subfields) much activity in the United States is the product of the immigration of researchers trained abroad.


Analysis is a subfield where theory and usage meet. The United States is regarded as a leader in this subfield, with very strong activities in Western Europe, Russia, and Japan. The recent past has seen rapid advances in broad areas of analysis, reflected by the award of six of the last eight Fields Medals in this subfield (three to U.S. residents). The international character of mathematics is well reflected in analysis, which features a very high level of international collaboration.

The area of differential equations is most important for its impact on other sciences and on technology. In ordinary differential equations, with related activities in numerical analysis and dynamical systems, the United States has a long tradition and very active groups which are challenged for leadership by groups in Western Europe. In partial differential equations, linear theory has reached maturity and nonlinear theory is developing very rapidly. The United States played a leading role in its early development, a role which is now shared with extremely strong groups in Western Europe, most notably in France. Unless more young researchers in the United States are attracted to this area, the United States will not be able to sustain its present position.

There is high promise for continuing major advances in nonlinear partial differential equations, operator algebras, dynamical systems, representation theory and solvable models of mathematical physics, and harmonic analysis (and applications). These advances are occurring throughout the world, with the United States playing a leading role. As in other areas, a high proportion of U.S. leading researchers, young faculty, and graduate students are recent immigrants. The subfield of analysis continues to have good interaction with applications arising in the physical sciences and engineering; many of its problems have been motivated by the study of phenomena from those fields. There is a felt need for closer contact in the future, with the biological sciences; a paucity of contacts signifies lost opportunities of significance.


Probability arose from the study of gambling choices is relatively new as a rigorous discipline. Modern probability provides the foundation for statistical inference, and it is intimately associated with measure theory, a branch of analysis. Nowadays, the emphasis is on randomness and on indeterminate phenomena. Many new developments in probability are motivated by problems outside mathematics.

The United States is dominant in all aspects of probability, including theory, applications, and computational approaches. Bibliometric data indicate that approximately half the literature in probability theory is produced by U.S.-based researchers. Other centers of activity are France, the United Kingdom, Canada, and Japan. Activities in the U.S. community feature both breadth and depth, whereas activities abroad tend to be more narrowly specialized. Research in the former Soviet Union, once very strong, is now weak.

In general, probability theory is very vital today, both in the development of fundamental theory and in interactions with other branches of mathematics and the other sciences (areas of interaction within mathematics include stochastic partial differential equations, superprocesses, percolation, Yang-Mills equations, turbulence, statistical physics, and critical phenomena). A second strength is that U.S. probability has maintained close contact with a diverse set of areas of applications. Applied probability in the United States is profoundly influencing and drawing inspiration from problems in the biological sciences (genetics, DNA structure, competition processes), medicine (epidemiology), and the environmental sciences (hydrology, environmetrics). Contributions by applied probabilists underpin much applied work in operations research and management, stochastic networks in communications, and financial engineering. In all these areas, U.S. probabilists are at the forefront; the United Kingdom, Canada, and France are also active. There is also strong U.S. activity in computational probability, with a secondary strong center in the United Kingdom.

Probability permeates the sciences and technology, with notable activities in engineering, computer science, physics, management, and finance. Workers in these fields have close contacts with the academic probability community, resulting in substantial accomplishments in both theoretical and applied areas, many of which are stimulated by novel technological developments. Computation and simulation play an increasingly crucial role.

Applied Mathematics

Applied mathematics is the name given to the subfield of mathematics that is motivated by practical problems whose formulation and study is mathematical in method and spirit. Traditionally the term has been associated with applications of analysis to problems in the physical sciences. Nowadays all mathematics is being applied, so the term applied mathematics should be viewed as a different cross cut of the discipline.

The United States has a leadership position in some areas of applied mathematics, notably in the areas of computer vision, financial engineering, most aspects of materials, and some aspects of mathematical biology. The U.S. contingent of invited participants at the last Congress of Industrial and Applied Mathematics was the largest of any country.

Close interaction between applied mathematicians and practitioners in science and engineering is critical. In the United States, while those working in applied mathematicians often work alone, they are more involved in interdisciplinary research than mathematicians in other subfields, but much more needs to be done. The European community is making very large investments to this end. The United Kingdom is more adept in establishing close ties between industry and universities. And in France, engineering has closer relations with mathematics.

The U.S. research community has responded rapidly to opportunities in areas such as fluid mechanics and materials science, but the mathematics of these areas is still in its infancy. Researchers have responded much more slowly to problems arising from chemistry, the biological sciences, manufacturing, and design. Continuum mechanics, constructive gauge theory, and other aspects of theoretical/mathematical physics have long been active fields of research, more so in Europe than in the United States. Optimization is a very active field, with many applications, where the United States is very strong. In the United States it is often found in departments of industrial engineering and computer science. Control theory is another area where the United States has great strength, with activity in every field of engineering.

In the future, applied mathematics will be closer to computer modeling and simulation and farther from analytical theory. There is a need for more vigorous interactions with other fields of science and stronger contact with the industrial community.

Computational Mathematics

Computational Mathematics is the area of mathematics concerned with reliable and effective solutions to mathematical problems using the computer. Numerical analysis, and approximation theory, which are closely linked to applied mathematics, as well as algorithms and data analysis, are generally included, but the area also includes computational modeling and simulation of phenomena. Some would include symbolic manipulation and even the use of computers in an exhaustive delineation of cases in mathematical proofs. As has been noted, many subareas of the mathematical sciences now have a computational component. Because of the impact of computer architecture on effective computation (particularly various forms of parallelism) there is a strong link to computer science.

Factors that have affected the growth of Computational Mathematics include:

Computational mathematics has become a mainstay in industry, finance, and public policy. The best example is the computational design of Boeing airplanes, which requires mathematicians well-trained in computational mathematics. Regretfully, good computational techniques, well studied by mathematicians, are seldom implemented in standard packages and, conversely, important ideas arising in applications are often not refined mathematically.

The literature in this area has grown enormously in the past decade, and its importance and impact continues to increase. However, the field as a whole remains fragmented and there is not enough synthesis and refinement of new techniques developed in industry and by other scientists.

The United States is the acknowledged leader in computational mathematics (especially in its commercial aspects), but not in all areas. The United States trails Western Europe in certain areas of numerical analysis and in symbolic computation, but the United States is the clear leader in providing commercial products.

The current strengths in computational mathematics draw on the widespread acceptance of computational modeling as the replacement for physical tests in a broad number of fields. Significant work in optimization has moved computer modeling close to the heart of analytic technique, thanks also to the easy availability of inexpensive, high-powered computers.

There are also weaknesses in computational mathematics, notably a general failure to synthesize new mathematics drawn from computational modeling using problem characteristics from various fields. As a result the field is more fragmented, and applications fields have not gained the mathematical expertise they need. This fragmentation has led some mathematicians to conclude that the area is in decline (e.g., the comment by a European review that "Nobody dominates, nobody is much interested anymore.").


The statistical sciences are very healthy across all subareas in the United States, which is the clear world leader. Statistics traditionally has been strong in the United Kingdom. It is now developing rapidly in continental Europe, so that the U.S. lead is shrinking. There are centers of significance in Australia and Japan.

Statistics has always been tied to applications, and the significance of results, even in theoretical statistics, is strongly dependent on the class of applications to which the results are relevant. In this aspect it strongly differs from all other subdisciplines of the mathematical sciences except computational mathematics.

The United States has both a high level of activity and a leadership role in theoretical statistics. Other centers are being developed in continental Europe and Australia. In applied statistics the United States is also the leader, with the United Kingdom in a very strong position, centers of excellence in Japan and Australia, and developing ones in Western Europe. In both of these areas, U.S. journals dominate the field. The United States is the clear leader in computational statistics, with the United Kingdom in a very strong position. Very rapid advances in this field have considerable significance to applications.

The interaction between the academic community and users in industry and government is highly developed, and hence there is rapid dissemination of theoretical ideas and of challenging problems from applications, as well as a tradition of involvement in multidisciplinary work. Both in applications and in multidisciplinary projects, however, there exist serious problems in the misuse of statistical models and in the quality of education of scientists, engineers, social scientists, and other users of statistical methods. As observations generate more data, it will be essential to resolve this problem, perhaps by routinely including statisticians on research teams.

There are great opportunities for impact in data mining and in the analysis of very large data sets that information technology now demands. While data analysis is the essence of statistics, challenges in data mining demand new techniques that in all probability will need to come from mainstream mathematics. For example, concepts from quantum mechanics seem to provide promising tools.

There is ample professional opportunity for young people in statistics, both in academia, industry, and government. A very high proportion of graduate students are foreign-born and many remain in the United States upon graduation.