**Appendix 2**

**ASSESSMENT OF SUBFIELDS**

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__

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__

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__

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:

- Changes in computing capability and architecture;

- The emergence of standard packages from mathematical subroutines to structural analysis codes, making computation accessible to non-experts;

- The observation that underlying structure from particular disciplines may offer computational capability to broad classes of problems; and

- The desire of users of mathematics to more accurately simulate more detailed physical phenomena so as to replace costly testing.

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.").

__Statistics__

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.