**Division of Mathematical Sciences**

**Mathematics and Science**

#### Dr. Margaret Wright

**Prof. Alexandre Chorin**

**April 5, 1999**

**NATIONAL SCIENCE FOUNDATION**

**PREFACE**

Today\'s challenges faced by science and engineering are so complex that
they can only be solved through the help and participation of
mathematical scientists. All three approaches to science, observation
and experiment, theory, and modeling are needed to understand the
complex phenomena investigated today by scientists and engineers, and
each approach requires the mathematical sciences. Currently
observationalists are producing enormous data sets that can only be
mined and patterns discerned by the use of deep statistical and
visualization tools. Indeed, there is a need to fashion new tools and,
at least initially, they will need to be fashioned specifically for the
data involved. Such will require the scientists, engineers, and
mathematical scientists to work closely together.

Scientific theory is always expressed in mathematical language. Modeling
is done via the mathematical formulation using computational algorithms
with the observations providing initial data for the model and serving
as a check on the accuracy of the model. Modeling is used to predict
behavior and in doing so validate the theory or raise new questions as
to the reasonableness of the theory and often suggests the need of
sharper experiments and more focused observations. Thus, observation and
experiment, theory, and modeling reinforce each other and together lead
to our understanding of scientific phenomena. As with data mining, the
other approaches are only successful if there is close collaboration
between mathematical scientists and the other disciplinarians.

Dr. Margaret Wright of Bell Labs and Professor Alexandre Chorin of the
University of California-Berkeley (both past and present members of the
Advisory Committee for the Directorate for Mathematical and Physical
Sciences) volunteered to address the need for this interplay between the
mathematical sciences and other sciences and engineering in a report to
the Division of Mathematical Sciences. Their report identifies six
themes where there is opportunity for interaction between the
mathematical sciences and other sciences and engineering, and goes one
to give examples where these themes are essential for the research.
These examples represent only a few of the many possibilities. Further,
the report addresses the need to rethink how we train future scientists,
engineers, and mathematical scientists.

The report illustrates that some mathematical scientists, through
collaborative efforts in research, will discover new and challenging
problems. In turn, these problems will open whole new areas of research
of interest and challenge to all mathematical scientists. The
fundamental mathematical and statistical development of these new areas
will naturally cycle back and provide new and substantial tools for
attacking scientific and engineering problems.

The report is exciting reading. The Division of Mathematical Sciences is
greatly indebted to Dr. Wright and Professor Chorin for their effort.

Donald J. Lewis

Director (1995-1999)

Division of Mathematical Science

National Science Foundation

Mathematics and Science

# Contents\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...2

## Overview\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....3

## Themes\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....3

Modeling\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...3

Complexity and
Size\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...4

Uncertainty\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...4

Multiple
Scales\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...4

Computation\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...4

Large Data
Sets\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....5

## Examples\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....5

Combustion\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....5

Cosmology\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...7

Finance\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....8

Functional Magnetic Resonance
Imaging\...\...\...\...\...\...\...\...\...\...\...\...\.....10

Hybrid System Theory and Air Traffic
Management\...\...\...\...\...\...\...\....11

Internet Analysis, Reliability, and
Security\...\...\...\...\...\...\...\...\...\...\...\...\....13

Materials
Science\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...14

Mixing in the Oceans and
Atmosphere\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...16

Physiology\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....17

Diagnosis Using Variational Probabilistic
Inference\...\...\...\...\...\...\...\.....19

Iterative Control of Nuclear
Spins\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....21

Moving Boundaries and
Interfaces\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...21

## Education\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....23

## Conclusions\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\.....25

## References and URLs\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...25

## Acknowledgements\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\...\....27 

# 1 Overview

Mathematics and science[^1] have a long and close relationship that is
of crucial and growing importance for both. Mathematics is an intrinsic
component of science, part of its fabric, its universal language and
indispensable source of intellectual tools. Reciprocally, science
inspires and stimulates mathematics, posing new questions, engendering
new ways of thinking, and ultimately conditioning the value system of
mathematics.

Fields such as physics and electrical engineering that have always been
mathematical are becoming even more so. Sciences that have not been
heavily mathematical in the past\-\--for example, biology, physiology,
and medicine\-\--are moving from description and taxonomy to analysis
and explanation; many of their problems involve systems that are only
partially understood and are therefore inherently uncertain, demanding
exploration with new mathematical tools. Outside the traditional spheres
of science and engineering, mathematics is being called upon to analyze
and solve a widening array of problems in communication, finance,
manufacturing, and business. Progress in science, in all its branches,
requires close involvement and strengthening of the mathematical
enterprise; new science and new mathematics go hand in hand.

The present document cannot be an exhaustive survey of the interactions
between mathematics and science. Its purpose is to present examples of
scientific advances made possible by a close interaction between science
and mathematics, and draw conclusions whose validity should transcend
the examples. We have labeled the examples by words that describe their
scientific content; we could have chosen to use mathematical categories
and reached the very same conclusions. A section labeled "partial
differential equations" would have described their roles in combustion,
cosmology, finance, hybrid system theory, Internet analysis, materials
science, mixing, physiology, iterative control, and moving boundaries; a
section on statistics would have described its contributions to the
analysis of the massive data sets associated with cosmology, finance,
functional MRI, and the Internet; and a section on computation would
have conveyed its key role in all areas of science. This alternative
would have highlighted the mathematical virtues of generality and
abstraction; the approach we have taken emphasizes the ubiquity and
centrality of mathematics from the point of view of science.

# 2 Themes

As Section 3 illustrates, certain themes consistently emerge in the
closest relationships between mathematics and science:

-   modeling

-   complexity and size

-   uncertainty

-   multiple scales

-   computation

-   large data sets.

**2.1 Modeling**

Mathematical modeling, the process of describing scientific phenomena in
a mathematical framework, brings the powerful machinery of
mathematics\-\--its ability to generalize, to extract what is common in
diverse problems, and to build effective algorithms\-\--to bear on
characterization, analysis, and prediction in scientific problems.
Mathematical models lead to "virtual experiments" whose real-world
analogues would be expensive, dangerous, or even impossible; they
obviate the need to actually crash an airplane, spread a deadly virus,
or witness the origin of the universe. Mathematical models help to
clarify relationships among a system\'s components as well as their
relative significance. Through modeling, speculations about a system are
given a form that allows them to be examined qualitatively and
quantitatively from many angles; in particular, modeling allows the
detection of discrepancies between theory and reality.

**2.2 Complexity and Size**

Because reality is almost never simple, there is constant demand for
more complex models. However, ever more complex models lead
eventually\-\--sometimes immediately\-\--to problems that are
*fundamentally different*, not just larger and more complicated. It is
impossible to characterize disordered systems with the very same tools
that are perfectly adequate for well-behaved systems. Size can be
regarded as a manifestation of complexity because substantially larger
models seldom behave like expanded versions of smaller models; large
chaotic systems cannot be described in the same terms as
small-dimensional chaotic systems.

**2.3 Uncertainty**

Although uncertainty is unavoidable, ignoring it can be justified when
one is studying isolated, small-scale, well-understood physical
processes. This is not so for large-scale systems with many components,
such as the atmosphere and the oceans, chemical processes where there is
no good way to determine reaction paths exactly, and of course in
biological and medical applications, or in systems that rely on human
participation. Uncertainty cannot be treated properly using *ad hoc*
rules of thumb, but requires serious mathematical study. Issues that
require further analysis include: the correct classification of the
various ways in which uncertainty affects mathematical models; the
sensitivities to uncertainty of both the models and the methods of
analysis; the influence of uncertainty on computing methods; and the
interactions between uncertainty in the models themselves and the added
uncertainty arising from the limitations of computers.

Uncertainty of outcome is not necessarily directly related to
uncertainty in the system or in the model. Very noisy systems can give
rise to reliable outcomes, and in such cases it is desirable to know how
these outcomes arise and how to predict them. Another extreme can occur
with strongly chaotic systems: even if a specific solution of a model
can be found, the probability that it will actually be observed may be
nil; thus it may be necessary to predict the average outcome of
computations or experiments, or the most likely outcome, drawing on as
yet untapped resources of statistics.

**2.4 Multiple Scales**

The need to model or compute on multiple scales arises when occurrences
on vastly disparate scales (in space, time, or both) contribute
simultaneously to an observable outcome. In turbulent combustion, for
example, the shape of the vessel is important and so are the very small
fluctuations in temperature that control the chemical reactions.
Multiple scales are inherent in complex systems, a topic of great
importance across science, whenever entities at microscales and
macrolevels must be considered together.

When it is known in advance that phenomena on different scales are
independent, one may rely on a separate model on each scale; but when
different scales interact, or when the boundaries between scales become
blurred, models are needed that allow interactions between scales
without an undue sacrifice of structure or loss of information at any
scale. A related complication is that the finiteness of computers limits
the range of scales that can be represented in a given calculation; only
mathematical analysis can overcome this built-in restriction.

**2.5 Computation**

Experiment and theory, the two classical elements of the scientific
method, have been joined by computation as a third crucial component.
Computations that were intractable even a few years ago are performed
routinely today, and many people pin their hopes for mastering problem
size and complexity on the continuing advent of faster, larger
computers. This is a vain hope if the appropriate mathematics is
lacking. For more than 40 years, gains in problem-solving power from
better mathematical algorithms have been comparable to the growth of raw
computing speed, and this pattern is likely to continue. In many
situations, especially for multiscale and chaotic problems, fast
hardware alone will never be sufficient; methods and theories must be
developed that can extract the best possible numerical solutions from
whatever computers are available.

It is important to remember that no amount of computing power or storage
can overcome uncertainties in equations and data; computed solutions
cannot be understood properly unless the right mathematical tools are
used. A striking visualization produced over many days of computation is
just a pretty picture if there are flaws in the underlying mathematical
model or numerical methods, or if there are no good ways to represent,
manipulate, and analyze the associated data.

It is also worthy of note that computation has come to permeate even the
traditional core mathematical areas, which allot expanding roles for
computation, both numerical and symbolic.

**2.6 Large Data Sets**

The enormous sets of data that are now being generated in many
scientific areas must be displayed, analyzed, and otherwise "mined" to
exhibit hidden order and patterns. However, large data sets do not all
have similar characteristics, nor are they used in the same way. Their
quality ranges from highly accurate to consistently noisy, sometimes
with wide variations within the same data set. The definition of an
"interesting" pattern is not the same nor even similar in different
scientific fields, and may vary within a given field. Structure emerges
in the small as well as in the large, often with differing mathematical
implications. Large data sets that need to be analyzed in real
time\-\--for instance, in guiding surgery or controlling
aircraft\-\--pose further challenges.

**3 Examples**

The examples in this section, described for a general scientific
audience, illustrate the scientific and technological progress that can
result from genuine, continuing, working relationships between
mathematicians and scientists. Certain well publicized pairings, such as
those between modern geometry and gauge field theory, cryptography and
number theory, wavelets and fingerprint analysis, have been
intentionally omitted\-\--not to slight their remarkable
accomplishments, but rather to demonstrate the breadth and power of
connections between mathematics and science over a wide range of
disparate, often unexpected, scientific applications.

**3.1 Combustion**

Combustion, a critical and ubiquitous technology, is the principal
source of energy for transportation, for electric power production, and
in a variety of industrial processes. Before actually building
combustion systems, it is highly desirable to predict operating
characteristics such as their safety, efficiency, and emissions.
Mathematicians, in collaboration with scientists and engineers, have
played and continue to play a central role in creating the analytical
and computational tools used to model combustion systems. Two
examples\-\--modeling the chemistry of combustion and engineering-scale
simulation\-\--illustrate the ties between mathematics and practical
combustion problems.

**Modeling the chemistry of combustion.** To model combustion it is
necessary to understand the detailed chemical mechanisms by which fuel
and air react to form combustion products. For a complex hydrocarbon
fuel such as gasoline, whose burning involves thousands of distinct
chemical species, one must identify the reactions that are most
important for the combustion process. The rates of reaction, which are
sensitive functions of temperature and pressure, must also be estimated,
along with their energetics, e.g. the heats of formation of the various
species.

For more than twenty years, mathematicians and chemists have worked
together on computational tools that have become critical to the
development of reaction mechanisms. The need for robust and accurate
numerical solvers in combustion modeling was clearly understood as early
as the 1970s. In response to this need, algorithms and software for
solving stiff systems of ordinary differential equations were developed
and combined into integrated packages for chemically reacting systems,
such as the Chemkin package developed at the Sandia National Laboratory.
Given arbitrarily complex chemical reaction mechanisms specified in a
standard format, Chemkin automatically generates an interface to
numerical methods that compute various chemically reacting systems.
These include spatially homogeneous systems as well as a variety of
one-dimensional systems, such as premixed flames, opposed-flow diffusion
flames, and detonation waves.

The mathematical and numerical analysis embodied in Chemkin has been a
key ingredient in designing and evaluating mechanisms, including those
in wide laboratory use. The existence of a reliable and generalizable
mathematical model facilitates the testing of new ideas in mechanism
design, since the effects of modifying a chemical mechanism can be
assessed directly. Finally, the mathematical software is not only
sufficiently robust to model arbitrarily complex chemical reaction
mechanisms, but also accurate enough so that the numerical error is
negligible relative to laboratory measurements.

Chemkin represents an amalgam of mathematical analysis, numerical
methods, and software development. The history of Chemkin illustrates
the fact that in many application areas advanced mathematical ideas are
more likely to be used by scientists and engineers if they are embodied
in software.

**Engineering-scale simulation.** The goal in this area is to represent
the three-dimensional fluid dynamics and other physical processes as
they occur in combustion devices such as internal combustion engines,
industrial and utility burners, and gas turbines. Two issues make these
simulations particularly challenging. The first is the number and
complexity of the physical processes that must be represented, which
include fluid dynamics, heat and mass transport, radiative heat
transfer, chemical kinetics, turbulence and turbulent combustion, and a
variety of multiphase fluid flow phenomena. The second is the enormous
range of length and time scales in such systems. The relevant physical
processes must operate simultaneously on scales ranging from the
smallest turbulent fluctuations (10^-6^ meters) up to a utility boiler
(100 meters).

Mathematicians have consistently been at the forefront in developing
innovative methods for modeling engineering combustion problems. Within
computational fluid dynamics, a huge field that encompasses numerous
applications, many of the mathematical methods have arisen as a direct
response to specific difficulties presented by combustion problems.
Examples include novel discretization techniques, such as high-order
accurate finite-difference methods and vortex methods; adaptive gridding
techniques, which estimate the error as a calculation is running and
locally increase or decrease the grid density to maintain a uniform
level of accuracy; and new methods for problems in complex geometries,
such as the overset grid and embedded boundary methods.

A major mathematical contribution has been asymptotic analysis that
makes possible an understanding of the coupling between different
physical processes in these complex systems; insights from asymptotic
analysis are used to find stable and accurate representations of these
processes in terms of simpler subprocesses. Examples include the use of
low Mach-number asymptotics to eliminate zero-energy acoustic waves
while retaining the bulk effects of compression and expansion due to
heat release, and front-tracking methods based on a separation-of-scales
analysis for thin premixed flames.

Today, packages such as Chemkin are part of the standard toolkit for
combustion researchers and engineers. New numerical methods for
engineering-scale simulations of combustion systems have been
extensively implemented as research codes, and are slowly making their
way into production engineering software.

Looking ahead, the requirements of combustion simulation suggest
promising directions for mathematics research that will make new science
possible. Even with the most powerful computers, it is impossible to
represent directly all of the processes involved at all of the relevant
length scales. Instead, one needs to introduce sub-grid models that
capture the effect on the large scales of all the scales below the
resolution limit of the calculation. In the area of chemical reaction
mechanisms, this corresponds to the development of *reduced* mechanisms,
i.e., reaction mechanisms with a few tens of species that accurately
represent energy release and emissions. The systematic development of
reduced mechanisms will involve a variety of mathematical tools, from
statistical analysis and optimization to dynamical systems.

For engineering-scale simulations, modeling at the sub-grid scale is a
central requirement for future progress. The development of sub-grid
models for turbulent combustion is particularly difficult, since
chemical reactions are sensitive to small-scale fluctuations in
temperature and composition. The effect of these fluctuations must be
separated from the larger-scale dynamics representable on the grid.
There has been renewed progress in turbulence modeling in recent years,
based on ideas from mathematical statistical mechanics, and extension of
these ideas to turbulent combustion represents a substantial
mathematical challenge; any successes will have enormous practical
consequences.

**3.2 Cosmology**

Cosmology, which once consisted of speculations based on extremely
scarce observations, has become a science rich in both data and theory.
The relativistic "hot big bang" model for the expanding universe is
widely accepted today and supported by a substantial body of evidence;
just as significantly, no data are inconsistent with this model. But the
standard cosmology leaves unanswered certain key questions about the
nature and evolution of the universe, including the quantity and
composition of energy and matter, and the origin and nature of the
density perturbations that seeded all the structure in the universe.
While a promising paradigm for extending the standard
cosmology\-\--inflation plus cold dark matter\-\--is being developed and
tested, many fundamental cosmological issues remain to be resolved or
clarified. ("Inflation" refers to the quantum-mechanical fluctuations
occurring during a very early burst of expansion driven by vacuum
energy; cold dark matter consists of slowly moving elementary particles
left over from the earliest fiery moments of the universe.) Mathematical
progress in two broad areas will be essential for cosmology: techniques
for dealing with massive data sets and large-scale, nonlinear,
multiscale modeling and numerical simulation.

**Massive data sets.** As cosmology moves toward becoming an exact
science, major mathematical challenges arise in coping with, displaying,
understanding, and explaining the unprecedented avalanche of
high-quality data expected during the next few years. To mention only a
few sources, NASA\'s MAP and the European Space Agency\'s Planck
Surveyor will map the full sky to an angular resolution of 0.1°,
allowing determination of the mass distribution in the universe before
nonlinear structures formed. The Sloan Digital Sky Survey will obtain
the redshifts of a million galaxies over 25% of the northern sky, and
the Two-Degree Field Survey will collect 250,000 redshifts in many 2°
patches of the southern sky, together covering around 0.1% of the
observable universe and mapping structures well beyond the largest
presently known size. In addition, experiments at accelerators, nuclear
reactors, and large-underground detectors are planned or in place to
search for neutralinos, explore the entire theoretically favored mass
range, and pursue neutrino mass. The quantity, quality, and nature of
the data require connections between mathematics and cosmology. Although
some generic principles of data analysis have emerged, the various
features to be "mined" in cosmological data differ from one another in
ways whose definition remains far from precise. The patterns of interest
change from application to application, and may even vary when several
uses are made of the same data set. In contrast to data from other
scientific areas, the cosmological data are likely to be of very high
quality; thus it will be important to squeeze every possible insight
from each data set.

A further striking feature of cosmological data is the vastness of the
scale ranges in almost every dimension. Data will be gathered not only
on the scale of galaxies, but also from particle physics; the "hot" part
of big bang cosmology implies the need for physics of ever-higher
energies and ever-shorter times.

Finally, astronomical data not only arrive at very high speed, but
patterns detected in real time may be used to control subsequent data
collection adaptively\-\--for example, to concentrate on regions where
something interesting is being observed. Careful mathematical analysis
will be needed because techniques appropriate for "on the fly" data
mining are quite different from those used to examine data at leisure.

**Modeling and simulation.** The mathematical models in cosmology
typically involve highly nonlinear coupled partial differential
equations that cannot conceivably be solved analytically\-\--for
instance, the equations may model turbulence in nuclear explosions that
occur when stars blow themselves apart. Small differences in the
mathematical form of these equations can lead to big variations in the
predicted phenomena. Cosmological models need to be complex enough to
capture all the phenomena reflected in the data, yet amenable to
analysis. Important modeling questions arise in the inverse problem,
reasoning backwards from observations and images to find the laws that
created them. The hope is that, by varying the initial conditions and
the parameters embedded in mathematical models, simulations can reveal
the fundamental parameters that define the universe, such as the mean
density and Einstein\'s cosmological constant Λ.

Like the associated data, cosmological models contain enormous ranges of
scales that pose difficulties for both mathematical analysis and
numerical solution. Creating *a priori* cutoffs that define different
scale regimes is a common tactic, but it breaks down as the ends of the
scales approach each other\-\--when the noise for a large scale becomes
comparable to the signal for the next-smaller scale. Subtle mathematical
modeling is essential to separate the phenomena that can be ignored from
those that count.

Carefully executed large-scale simulations match observations well, and
have become a standard tool in modern astrophysics. Cosmological
calculations consume a large portion of the available supercomputer
cycles in the United States, and worldwide as well. This is because
solving the complex partial differential equations of cosmology over the
wide multidimensional range of scales for problems of realistic size is
a massive undertaking at the edge of current mathematical and
computational capabilities.

To illustrate these points, consider the formation and evolution of
galaxy clusters, the largest objects in the universe. For a simulation
to be credible, enormous dynamic ranges in size and density are required
to resolve individual galaxies within a cluster; the range of mass is
perhaps 10^9^, over a time period of 10 billion years. One approach is
to begin with a "box" (part of the universe) that is initialized with a
large number (say, 10 million) of uniformly distributed particles, and
then to follow the motion of each particle as its position and velocity
are perturbed following theoretical predictions.

This approach poses formidable difficulties for numerical methods in
addition to those arising from the already-mentioned nonlinearities and
ranges of scale: the particles move non-uniformly, model geometries are
highly complex, and there is a demand for ever-finer resolution. A
fruitful arena for mathematical analysis is the effect of decisions
about partition into scales on numerical accuracy; here the recent
mathematical work on particle methods and on fast summation and
multipoles may be of key importance.

Since cosmological calculations will continue to tax the capabilities of
the highest-performance available hardware, further mathematical and
algorithmic ingenuity is needed to make the implementations of these
simulations run efficiently on parallel machines without inordinate
specialization for a particular hardware configuration. Taking advantage
of new computer architectures without unduly compromising generality is
a problem for all applications that strain today\'s high-performance
computers.

**3.3 Finance**

Modern finance, although not a science in the traditional sense, is
intertwined with mathematics, and the connection is not limited to
theory\-\--mathematics is a central feature in the day-to-day
functioning of the world\'s financial markets. Mathematics and finance
are tightly connected in the two areas of derivative securities and risk
management.

**Derivative securities.** In recent years, headlines about business
have repeatedly mentioned "derivatives". A financial derivative is an
instrument that derives its value from other, more fundamental
instruments, such as stocks, bonds, currencies, and commodities (any one
of which is called an *underlying*). Typical derivatives include
options, futures, interest rate swaps, and mortgage-backed securities.
The Nobel-prize-winning papers on option pricing containing the famous
Black-Scholes partial differential equation were published in 1973 as
the Chicago Board of Options Exchange was being established, and within
months the Black-Scholes model became a standard tool on the trading
floor. Worldwide, the volume of trade in derivatives has since grown to
rival the volume of trade in equities. One of the reasons for this
phenomenal growth is the existence of reasonably reliable mathematical
models to guide their pricing and trading.

In theory, derivatives are redundant because they can be synthesized by
dynamic trading in the underlying instruments. Trading in derivatives
thus rests on the possibility of finding the fair price of a derivative.
Under standard assumptions, the unique fair price of an option can be
found from the Black-Scholes equation. However, certain key parameters
need to be determined before this equation can be used in practical
settings.

One of these parameters, the *volatility*, has been the subject of
intense mathematical and algorithmic attention for almost twenty years.
The original Black-Scholes model requires the estimation of a constant
volatility derived from a diffusion model of the underlying\'s price
process. Multiple approaches have been devised to calculate this form of
volatility\-\--for example, using weighted past data, or selecting the
*implied volatility* corresponding to a specific similar traded option
with the same underlying. (The implied volatility of a traded option is
the value that, substituted into the Black-Scholes equation, produces
the known price of the option; implied volatility is calculated by
solving a one-dimensional inverse problem.)

The classical Black-Scholes model has known limitations that are often
displayed through the "smile effect"\-\--the characteristic U-shaped
curve that relates implied volatility for comparable options to the
price associated with buying or selling the underlying. Existing models
of volatility are not completely satisfactory, with *hedging* playing a
major role in the difficulties. Hedging is related to the sensitivity of
the option\'s value to different parameters; the choice of volatility
may have a large effect on the hedging strategy. There is extensive
mathematical research today on formulating stochastic models of
volatility (with, in some cases, widely differing time scales as a key
feature), and on modeling volatility as a two-dimensional surface that
depends on properties of the underlying. In addition, new approaches
involving approximation and optimization are being developed for
calculating volatility.

Today\'s derivative models include heavy doses of continuous-time
martingale theory, changes of measure for stochastic integration, the
fine structure of stochastic processes, supermartingales and potential
theory, stochastic calculus, and partial differential equations. The
continuing creation of more complex derivatives calls for new
mathematics in these areas as well as in simulation, large-scale
optimization, and real-time analysis of large data sets.

In the financial world, a few minutes or even seconds may make a major
difference in profit, so that an ideal financial model should be able to
make accurate predictions in the very short term. However, the
relationship between models and data is very different in finance than
in experimental sciences: the world\'s markets do not lend themselves to
meaningful large-scale experiments designed to test or validate models.
Thus models are of necessity evaluated by their ability to track the
huge quantities of financial data generated throughout the day around
the world. A further contrast to other scientific areas is that neither
the quantity nor quality of financial data is likely to improve, so that
new models must make do with the same forms of data that are available
today.

**Risk management**. The creation of an international financial system
in which large volumes of capital move quickly has led to the new
challenge of *risk management*. Despite some spectacular failures to
manage risk in derivatives markets, such as the 1998 debacle of Long
Term Capital Management, derivative securities are too useful to
disappear. Hence strategies are needed for managing the risks associated
with derivatives and other financial instruments.

The long-standing assumption, originating with Markowitz in the 1950s,
that stock returns are normally distributed is known to be an inadequate
approximation to reality in times of crisis. Indeed, repeated studies
have found stock returns to have "fatter tails" than the normal
distribution and models based on the normal assumption can err by ten to
twenty standard deviations.

In addition, the implicit idea in the Black-Scholes model and its
successors is to synthesize derivative securities or a portfolio of
derivative securities, thereby allowing institutions to hedge the risk
associated with their business by owning financial instruments which
offset this risk. But liquidity can disappear in times of crisis, so
that hedging may become impossible. Even in normal times, some
derivative securities cannot be hedged, for example a security that
offsets the default risk of a corporate bond.

A yet-to-be developed mathematical theory would show how to decouple a
portfolio into two parts, one part whose risk can be hedged and another
part that is "purely unhedgeable". One possible strategy is to project
the space of all portfolios onto the subspace of hedgeable portfolios,
but the complexity and difficulties of such an approach are daunting
from both theoretical and computational perspectives.

The standard portfolio risk measure is *value at risk*, "VaR", which is
the probability that the portfolio will lose money exceeding a specified
threshold within a specified time period. There are several objections,
conceptual as well as practical, to VaR, primarily that it assigns the
same risk measure to all portfolios with the same probability of loss
exceeding the threshold, regardless of the distribution of loss above
the threshold. It also fails to satisfy the desirable mathematical
property of subadditivity, since the sum of the VaRs of two portfolios
can be less than the VaR of their sum; this encourages institutions to
play accounting games, subdividing dangerous positions into smaller ones
entirely for risk-accounting purposes. A further disadvantage is that
VaR assumes normally distributed returns, and hence tends to give
optimistic values in the tail of the loss distribution, which is where
risk matters most.

The mathematics of risk management is in its infancy, building on ideas
such as extreme value theory from actuarial science. The distributions
describing extremal events are well understood, but it is not yet known
how to build extreme-value models based on large numbers of jointly
distributed random variables. A fundamental open problem in this area is
defining how to measure risk in any particular model. An appealing
approach, currently under active exploration, is to devise mathematical
properties that are desirable in a risk measure and then define a set of
risk measures that possess these properties.

The problems of quantifying, computing, and managing risk are likely to
pose substantial mathematical challenges into the foreseeable future.

**3.4 Functional Magnetic Resonance Imaging**

Magnetic resonance imaging (MRI) is a well-established tool for
analyzing the structure of the brain. Starting in the early 1990s,
functional MRI (fMRI; the "f" is by convention lower-case) began to be
used to study brain *dynamics*. The underlying principle of fMRI is
related to the properties of blood within active areas of the brain.
"Blue" blood (deoxygenated hemoglobin) is more paramagnetic than "red"
blood (oxygenated hemoglobin), so that the MR signal from blue blood is
stronger. In the late 1980s, positron emission tomography research
showed that, although active areas of the brain require a larger supply
of blood, the corresponding increase in available oxygen is not used.
Consequently, the blood leaving active regions of the brain contains
relatively more oxygen and interferes less with the local magnetic
field, which means that the MR signal in the vicinity of active regions
shows an apparent gain. By comparing the averages of MR images taken at
close time intervals while a brain function is activated and
deactivated, the active areas can be identified.

To test the brain function of working memory, for example, a subject is
presented with a sequence of letters arriving at one per second, and is
asked to press a button when a letter is repeated. Next, the subject is
asked to press the button when a letter is repeated with a different
letter in between ("one back"); and so on. Most people cannot sustain
"four back" for more than a few minutes, and the work in the brain as
*n* in "*n* back" increases can be measured and correlated with *n*.

In studying brain function through fMRI, the key is to *compare* images
rather than to study an individual image in detail. Images are compared
by determining the voxels that are "significant", i.e., those that have
changed by more than a given tolerance between images. The MRI used to
observe brain structure requires approximately 20 minutes to produce a
single fine, detailed image. By contrast, in fMRI a sequence of images
is collected rapidly, to observe the brain dynamics, and there is an
obvious tradeoff between the time per image and image quality. The role
of mathematics (in the form of statistics) in fMRI is to analyze the
image data. The data sets collected are extremely large (a typical
experiment produces between 0.5 and 500 gigabytes of data), and also
extremely noisy, thus presenting multiple statistical difficulties.

Many sources of noise are present in fMRI data; some are understood,
while others remain mysterious. The signal induced by neural activation
has approximately the same magnitude as noise in the experiments, which
means that many images need to be acquired in each experiment to obtain
a meaningful result. The noise process in fMRI has a complex
distributional structure that is not yet fully understood\-\--for
example, signal variance depends on the mean in an unknown, nonlinear
way, and significant spatial correlations exist that depend on how the
data are collected. Outliers of various forms are frequent, and the
variations between individual brains are enormous. Most importantly,
statistical analysis of fMRI data needs to be built on a detailed
understanding of the structure of the noise, which means understanding
the scientific elements of fMRI: physics, MRI technology, and theories
of brain functioning. Consequently, statisticians in this area
necessarily work with teams of physicists, electrical engineers,
psychologists, neurologists, technologists, and computer scientists.

To obtain the best possible images, the data need to be corrected to
reduce the effects of noise, which arises from at least two sources: the
hardware and the subject. In the hardware, there may be a lack of
uniformity in the main magnetic field, or a lack of linearity in the
gradient field. In addition, the analog-to-digital converter may be
miscalibrated, or mistimings of resonant gradients may cause "ghosts" in
the images. The main source of variation originating in the subject is
movement of the brain, which can result from, for example, rigid motion
of the head, the almost-periodic compression and vertical movement
caused by the cardiac cycle, and distortions caused by respiration.

To deal with the noise, two approaches are being taken simultaneously:
removing or reducing the noise at its source through engineering; and,
through mathematics, modeling the data and noise, then adjusting the
predicted variation. With the latter approach, the goal is to develop a
mathematical model that accurately relates the data to parameters of
interest, but this remains a daunting task. Substantial progress has
been made by successively estimating and correcting for each effect
known to cause noise. To date, these effects include analog-to-digital
miscalibration, gradient mistimings, receiver drift, subject head
motion, and shot noise. After these corrections, the images are
reconstructed by a fast Fourier transform and then the (still
unexplained) voxel-wise trend over time is removed. Finally, statistical
methods such as *t*-tests are used to assess the effect of the
experimental paradigm.

New statistical and computational techniques have already contributed
substantially to the quality of fMRI data. It is now possible, for
instance, to estimate and correct for rigid motions of the brain as
small as 50 microns. Statistical models can also account for
differential brain response, and have extended motion correction between
images using a fully three-dimensional method. Incremental task effects
from a variety of administered cognitive tests have been quantified by
novel statistical methods, and statistical methods of spatial growth
curves have been extended to quantify changes in the pattern of
activation over time. More powerful statistical tests are still needed;
*t*-tests are often sufficient, but subtler methods will be called for
as MR techniques and the cognitive questions become more complex.

Contributions from statistics have answered several important questions
about fMRI data\-\--for example, how to make multiple comparisons while
retaining the power of statistical tests, and what happens if the same
experiment is repeated. However, statisticians working on fMRI have
found that every answer leads to a new question, and that substantial
mathematical challenges arise from every new question, with no end in
sight.

There has been tremendous progress not only in conceptual techniques for
modeling and resolving the noisy data, but also in numerical and
computational algorithms. Several years ago, processing the data from a
15-minute experiment required 12 hours of computation; now it takes
three seconds. Concurrently, there have been continuing, rapid gains in
the achievable spatial resolution\-\--for example, an eight-fold
improvement between 1996 and 1997. Most of the gains in speed and
accuracy are attributable to better mathematical algorithms, not to
increased computational power.

The cognitive science driving fMRI has also advanced; one interesting
discovery was that reading more complex sentences causes greater brain
activity in precisely the ways predicted by theory and earlier, more
primitive external measurements of eye movements. In ongoing projects,
fMRI is being used to study the cognitive and brain activity
characteristics of high-functioning autistic subjects, and to examine
brain plasticity and rehabilitation in aphasia therapy.

One final point of interest is that certain aspects of the statistical
techniques developed in the context of fMRI generalize to analysis of
seismic data collected by geophysicists in oil exploration.

**3.5 Hybrid System Theory and Air Traffic Management**

Hybrid system theory, a field of applied mathematics abutting control
theory and computer science, has an enormous potential for impact on
practical problems. Hybrid systems can be loosely defined as systems
that allow the interaction of discrete events and continuous dynamics;
hybrid system theory attempts to prove properties such as reachability
and stability. Discrete event models naturally accommodate linguistic
and qualitative information, and are used to model modes of operation of
a single system, for example an aircraft or the interaction of several
aircraft. The continuous dynamics in a hybrid system model physical
processes, such as the continuous response of an aircraft to changes in
the positions of aileron and throttle.

Hybrid systems are good models of complex reactive systems, in which
physical processes interact with man-made automated environments;
algorithms developed to analyze and control the behavior of hybrid
systems may therefore be used in the design of automatic controls for
these systems. A common real-world example of a hybrid system arises
when advanced automation is introduced into manually operated systems in
order to enhance performance and flexibility while significantly
reducing the workload of human operators. Accompanying this increase in
automation, however, is the necessity of ensuring that the automated
system always performs as expected. This is especially crucial for
*safety-critical* systems: if a telephone switch crashes or a power grid
node goes down, lives are usually not lost; if an error occurs in the
automated avionics in a commercial jet, the results could be disastrous.

Many of today\'s safety-critical systems are growing at a rate that will
make their manual operation extremely difficult if not impossible in the
near future. The air traffic control system is an example of such a
system. Air traffic in the United States is expected to grow by 5%
annually for the next 15 years, and rates of growth across the Pacific
Rim are expected to be more than 15% a year. Even with today\'s traffic,
ground holds and airborne delays in flights due to congestion have
become so common that airlines pad their flight times with built-in
allowances. Aging air traffic control equipment certainly contributes to
these delays: the plan view displays used by controllers to look at
radar tracks and flight information are the very same ones that were
installed in the early 1970s, and they fail regularly. The computer
systems that calculate radar tracks and store flight plans were designed
in the 1980s, using software written in 1972.

The introduction of new computers, display units, and communication
technologies for air traffic controllers will help alleviate the
problems caused by failing equipment, yet the Federal Aviation
Administration admits that any significant improvement will require that
many of the basic practices of air traffic control be automated. For
example, today\'s airspace has a rigid route structure based on altitude
and on ground-based navigational "fixes". The current practice of air
traffic controllers is to route aircraft along predefined paths
connecting fixes, to manage the complexity of route planning for several
aircraft at once. The rigid structure puts strict constraints on
aircraft trajectories, which could otherwise follow wind-optimal or
"user-preferred" routes (routes that are shorter or involve lower fuel
consumption because of tailwinds). Also, while a data link between
aircraft and ground is being considered as a replacement for the current
voice communication over radio channels between pilot and controller,
there is a limit to the amount of information processing that a
controller can perform with these data. Recent studies indicate that, if
there is no change to the structure of air traffic control, then by the
year 2015 there could be a major accident every 7 to 10 days; obviously
this cannot be permitted to happen.

The main goal of air traffic control is to maintain safe separation
between aircraft while guiding them to their destinations. However, its
tight control over the motion of every aircraft in the system frequently
causes bottlenecks to develop. Uncertainties in positions, velocities,
and wind speeds, along with the inability of a single controller to
handle large numbers of aircraft at once, lead to overly conservative
controller actions and procedures. An example is the set of methods used
by air traffic controllers to predict and avoid conflicts between
aircraft. If a controller predicts that the separation between two
aircraft will become less than the regulatory separation, the controller
will issue a directive to one or both of the pilots to alter their
paths, speed, or both. Often the resolution is not needed, and usually
it is too drastic. Also, user-preferred routes are disallowed because of
the requirement that prescribed jetways be used.

As a result of all these difficulties, there is a widely perceived need
in the air traffic, airline, and avionics communities for an
architecture that integrates data storage, processing, communications,
and display into a safe and efficient air traffic management system; a
new air traffic system has been proposed that involves the Global
Positioning System and a datalink communication protocol called
Automatic Dependent Surveillance for aircraft-aircraft and
aircraft-ground communication. While the degree of decentralization and
level of automation in such a system are still under debate, the
integrity of any automated functionality in a new air traffic management
system depends on a *provably safe* design as well as high confidence
that the control actions will not fail.

This level of reliability requires accurate models, techniques for
verifying that the design is safe to within the accuracy of these
models, and procedures for synthesizing the system\'s control actions.
Hybrid system researchers have designed models and control laws for two
systems: a provably safe algorithm for resolving trajectory conflicts
between aircraft, and a provably safe algorithm for a single aircraft to
switch between different flight modes. A rigorous notion of "safety" in
each case is crucial. In the conflict resolution problem, the system is
safe if the aircraft always maintain minimum separation from each other.
In the mode-switching problem, system safety means that the state of the
aircraft remains within minimum and maximum bounds imposed on its
velocities, angles, etc., so that the aircraft does not stall and plunge
out of the sky.

The hybrid system associated with air traffic control models the
discrete dynamics with finite-state automata whose transition functions
describe the mode-switching logic, and uses nonlinear ordinary
differential equations to model the continuous dynamics. The system
includes continuous as well as discrete variables to model parameters
that the designer may manipulate (such as a flight mode switch in an
on-board flight management system) and disturbance parameters that the
designer must control against (such as an aircraft entering the
five-mile-radius protected zone around another aircraft). Using analysis
based on traditional discrete and continuous optimal control techniques,
and on two-person zero-sum game theory for automata and continuous
dynamical systems, partial differential equations can be derived whose
solution describes exactly those states (aircraft positions, velocities,
accelerations, and modes of operation) that the system may reach from a
given initial state. By analyzing these reachable states, it is possible
to determine automatically those configurations that the system must be
prevented from entering if safety is to maintained.

Ten years ago such a method would have been prohibitively
computationally expensive, but advances in computational power and new
fast methods for integrating partial differential equations have made
such solutions feasible even for real-time applications such as on-board
autopilots and computer-aided tools for air traffic controllers. The
same approach has been applied to design conflict resolution maneuvers
for multiple aircraft and to verify the mode-switching logic for
vertical flight modes in an aircraft\'s flight management system.

**3.6 Internet Analysis, Reliability, and Security**

The Internet is one of the most talked-about and written-about phenomena
of the late twentieth century. Data traffic on the Internet has grown
exponentially since the early 1980s\-\--there were 235 IP hosts on the
Internet in 1982, 100,000 in 1989, and more than 30 million in 1998. The
most optimistic extrapolations

have consistently underpredicted the continuing expansion of the Web,
which is known within the Internet research community as a "success
disaster"; because the Internet has succeeded beyond anyone\'s
expectations, it is not prepared or able to cope with the consequences.
Problems with the Internet are likely to escalate as popularity of the
Web spreads; the efficiency, reliability, and security of the Internet
are becoming important to an increasing fraction of the population. All
of these areas are obvious candidates for new connections between
mathematics and communications technology.

There is a long history of involvement by mathematics in the development
of existing voice communication networks\-\--in fact, traditional
teletraffic theory is widely regarded as one of the most successful
applications of mathematical techniques in industry. Mathematical models
of voice traffic and call arrivals at network links have been available
for at least 50 years. These models typically involve only a few
parameters, they are associated with intuitively satisfying physical
interpretations, and their predictions have consistently matched
measured data. Their well understood mathematical structure has led to
further applications of mathematics in telecommunications\-\--for
example, in designing highly optimized network management and control
systems.

But any expectation that the known mathematics of teletraffic theory can
be generalized to Internet traffic is doomed to disappointment. In
almost every dimension, Internet traffic is *completely different* from
voice traffic. Because computers do not communicate with other computers
in the same way as humans speaking on the telephone, the old
mathematical properties no longer apply. Most strikingly, both length
and transmission rates for data traffic range across scales that are
unimaginable for voice connections: data connections may last for days,
and high-end users are already transmitting data at hundreds of megabits
per second, with higher rates regularly becoming available. Furthermore,
data network traffic displays *multiscale burstiness*\-\--it arrives in
fits and starts, interspersed with gaps, and this burstiness persists
over three orders of magnitude in time scales. Standard voice traffic,
by contrast, is bursty when observed over short time intervals such as
100 milliseconds, but is essentially smoothed out over longer periods of
(say) one hour.

Existing networks, designed for voice traffic, are under stress.
Information on the Internet is sent using the Internet Protocol (IP);
when too many data packets arrive, routers keep them in buffers until
traffic is reduced. If traffic is heavy for a sustained period, buffers
fill up and packets are "dropped". From an engineering perspective,
Internet traffic plays havoc with standard voice network design: there
is a need for big buffers in routers and switches to avoid loss of data
packets when buffers overflow; links may be saturated without warning at
any time, so that safe operating points must be chosen conservatively;
and individual users may experience poor response even though overall
network performance is satisfactory. Internet users today routinely
encounter delays in access and sluggish performance that are essentially
unknown in voice communication, and these problems are likely to become
more severe as Internet traffic grows.

Many networking experts argue that the mathematics needed to model the
Internet will be radically different from traditional teletraffic
theory, and the topic of Internet-based mathematics is in a state of
lively ferment. For example, the multiple time and rate scales observed
in Internet traffic have led to work on scaling phenomena, a
multidisciplinary field that includes mathematicians, network engineers,
physicists, and control theorists. Much press has been devoted to the
idea that Internet traffic processes can be modeled effectively in terms
of fractal and multifractal scaling behavior\-\--ideas that have been
embraced by some but rejected by others. Approaching the problem from
another angle, work on Internet analysis has been done using
renormalization group techniques and mean-field theory. For the problem
of controlling data networks, mathematicians have begun looking at
paradigms of pattern formation, self-organization, and adaptation.

Irrespective of the mathematical constructs used, a universal theme in
modeling and analysis of the Internet is the importance of *data*.
Because the Internet\'s behavior is emphatically not captured by
standard teletraffic models, researchers in this area rely on large
quantities of multidimensional data gathered over wide-ranging time
scales. The size and complexity of these data sets create a further
mathematical challenge of devising methods that can meaningfully manage,
manipulate, represent, and visualize the data. An important issue in
handling these particular large data sets is their inherent extreme
variability in scale.

Additional mathematical questions related to the Internet arise from
concerns about reliability and security. As Internet connectivity
expands, there are more and more opportunities for damage by malicious
users\-\--for example, targeted sites can be and have been crippled by
deliberate saturation. The Internet\'s history of functioning without
regulation means that systems are needed to *detect* attacks across the
network in real time. Network intrusion detection is being approached by
designing monitors that can be added to a network without modifying the
hosts; such a property is essential when dealing with several thousand
heterogeneous, individually administered hosts.

Of course, any network monitor can itself be subject to attacks intended
to subvert the monitoring; hackers attempting to break in might well
attack the monitor also. Such attacks may take several forms, each
progressively more subtle and difficult to detect: overload attacks,
where the strategy is to overburden the monitor so that it cannot keep
up with the data stream; crash attacks, in which the goal is to knock
the monitor out of service; and subterfuge attacks, in which the
attacker tries to mislead the monitor about the meaning of the traffic
that the monitor is analyzing. Each of these forms of attack calls for a
different mathematical model that allows the attack to be detected in
real time and then protects against it.

Mathematics is also needed to define and verify protective techniques
such as congestion control. The end-to-end congestion control techniques
of the Transmission Control Protocol (TCP) have been critical in the
robustness of the Internet, but the Internet has ceased to be a small,
close user community. Hence it is no longer possible to rely on
end-nodes to cooperate in achieving end-to-end congestion control, nor
on developers to include congestion control in their Internet
applications.

Several distinct varieties of congestion arise from *unresponsive flows*
that do not use end-to-end congestion control, implying that they do not
reduce their load on the network when subjected to packet drops. Without
congestion control, well-behaved traffic will reduce its sending rates
in response to congestion, leading to a situation in which the
uncooperative flows shut out the responsive traffic. In addition to this
kind of unfairness, congestion collapse\-\--a decrease in useful work by
the network because of an increase in load\-\--may occur in various
forms. For example, "classical" congestion collapse occurs when there is
unnecessary retransmission of packets. Undelivered packets can cause
congestion collapse when bandwidth is wasted by transmitting packets
that are dropped before they reach their destination; the latter
situation is exacerbated by applications that willfully raise their
sending rate as more packets are dropped. Research on congestion control
involves queueing theory, scheduling algorithms, and fairness metrics.
Inevitably, further mathematical complexity will be needed to blend
modeling and network measurements as well as (eventually) policy issues.

**3.7 Materials Science**

Mathematical and computational techniques are assuming an increasing
role in materials science, as illustrated by two areas\-\--new materials
and multiscale phenomena.

**The search for new materials.** Since prehistoric times, the
*systematic method*\-\--changing synthesis or processing variables over
a limited range and then measuring properties of the resulting
samples\-\--has been the main tool in materials science. The classical
variables are composition, heat treatment time and temperature, and
quantities that influence formation of a specimen into a certain shape.
With modern methods of synthesis, this process encompasses a wide range
of controllable variables associated with thin films, composites,
microscale and nanoscale devices, and electronic, magnetic, and
dielectric materials.

Despite its successes, the systematic method can be inefficient or
inappropriate in some situations. A common example occurs when the
tolerances that define a new material are tight relative to the possible
range of variable values. Consider shape memory materials, a class of
materials that can undergo large plastic deformations, but recover their
original shape upon heating. These materials are part of a revolution in
biomedical technology, the \$500 million, two-year-old technology of
*stents*. Stents, placed in the coronary artery using a guidewire (often
made out of shape memory material as well), in many cases allow an
outpatient procedure rather than difficult bypass operations. The most
important shape memory material in stents, an alloy of nickel and
titanium, shows crucial differences in behavior as its composition
varies from 50.2% to 50.6% nickel; furthermore, there are interesting
alloys of nickel, titanium, and copper, and even quaternary alloys. If
0.1% is conservatively regarded as an acceptable tolerance, then it
becomes extremely difficult to make many samples of slightly different
composition and test their properties. If the parameters involved in
heat treatment are varied as well, the systematic method is simply not
practical.

The systematic method is also unlikely to discover entirely unexpected
behavior\-\--for example, a previously unknown microelectronic property
that occurs in a film having a certain precise thickness, configuration,
orientation or defect structure. The special behavior could not be
inferred by looking at a trend based on small changes from a known
sample; in such circumstances, the only path to new materials is through
mathematics.

In cases where the systematic method cannot be used to find new
materials, mathematical theory is playing an ever-growing role on two
fronts separated by huge length and time scales. The first stems from
improvements in continuum theories of materials. There is an emerging
understanding of how to model and simulate accurately the growth of a
new phase, including its complex geometrical shape and topology. An
instance of this work is the development of materials with large
magnetostriction. (Magnetostrictive materials convert magnetic energy to
mechanical energy, and vice versa.) In the 1960s, the class of "giant"
magnetostrictive materials was discovered using an ingenious strategy
that relied on the inherently large magnetostriction of some rare earth
metals. Recently, guided by gains in understanding of the theory of
micromagnetics, predictions were made of a new class of materials with
even larger magnetostriction. The mathematical theory not only directly
predicted the mechanism of magnetostriction, but also guided the alloy
development and subsequent experiments that revealed the effect. The
resulting class of materials shows a magnetostrictive effect 50 times
that of giant magnetostrictive materials.

The other development, perhaps more spectacular in the long run, is the
use of density functional theory. Density functional theory, based on
the observation of W. Kohn (the 1998 co-winner of the Nobel Prize in
chemistry) and his colleagues that the density of electrons in quantum
mechanics is subject to a variational principle, has as input only
fundamental atomic information, in particular the locations of nuclei
and their charges. Thus density functional theory can directly predict
material properties from knowledge of the composition. Recently,
unexpected new phases of even common materials have been predicted
theoretically using density functional theory, and have subsequently
been observed at the predicted temperature, pressure and composition.
The key mathematical problems for density functional theory are
understanding the quantum mechanical foundations, passing to simpler
models of atomic forces, improving methods for including statistical
mechanics to produce predictions at finite temperature. The future
integration of density functional theory and materials science is likely
to lead to major advances in "new materials from theory", and may one
day surpass even the time-honored systematic method.

**Multiscale phenomena.** Quantum mechanics cannot deal effectively
today with some of the most interesting and useful characteristics of
materials\-\--properties that are *structure-sensitive*, meaning that
they are affected, often profoundly, by the microstructure of the
material. Examples of structure-sensitive properties are strength,
plasticity, magnetic hysteresis, dielectric constant, optical properties
of liquid crystals, superconductivity, and almost any property
associated with a phase transformation. The relevant microstructural
features are, for example, a precipitate produced by a diffusional phase
transition, a magnetic domain, a vortex, a point or line defect, or a
dislocation tangle.

Unfortunately, the smallest microstructural features of general interest
in materials science are much too small for the application of density
functional theory. Furthermore, these microstructural features
frequently act collectively in a way that cannot be discovered by
analyzing only one of them. The gap in the time scales is even wider:
kinetic processes have time scales that range between milliseconds,
seconds, and days, yet the analysis of even a microsecond event is far
beyond the capability of first-principles computations with only a few
atoms.

Despite these difficulties, there is hope because of the recent
appearance of mathematical methods suited to the passage from one scale
to another. When properties exhibit large spatial or temporal
fluctuations on one scale governed by a certain set of partial
differential equations, it is now becoming understood how to derive
equations appropriate to a larger scale, using weak convergence methods,
homogenization, Young measures, and various notions of convergence of
operators. While these methods have mainly been applied to derive one
continuum theory from another, they could well serve more generally for
problems of change of scale, such as the direct passage from density
functional theory to continuum theory. The dream of researchers in this
area is to have the coefficients of macroscale differential equations
evaluated directly by atomic-scale computations with an input of only
fundamental constants.

The other opportunity for multiscale methods comes because it is
becoming possible to reproducibly synthesize structures with an
atomic-scale dimension. The subsequent investigation of the unexpected
properties and possible applications of these nanoscale structures has
given rise to the huge, dynamic field of nanotechnology. Properties that
are known to be structure-sensitive on the macroscale are susceptible to
unusual behavior at the microscale or nanoscale. Qualitatively,
something strange is expected when the size of the structure is
decreased below the typical size of the feature that gives rise to the
structural sensitivity. But, quantitatively, there is a conspicuous
absence of mathematical theory that can be used to predict the behavior
of such structures; when this theory becomes available, important
breakthroughs are likely.

**3.8 Mixing in the Oceans and Atmospheres**

At first blush it would appear that mixing in the atmosphere or ocean is
straightforward and of little mathematical interest. After all, children
who make chocolate milk from a powder quickly learn that the longer and
more energetically they stir, the more evenly the chocolate powder is
spread and dissolved in the milk. While that common-sense lesson is
valid, the oceans and atmosphere are, in some sense, less vigorously
stirred, so that the mixing is incomplete.

A careful look at mixing in oceans, atmospheres, and laboratory
experiments reveals "islands" of unmixed fluid that nothing from the
outside seems capable of penetrating; thus there are clearly demarked
contours that act as barriers to mixing. While this phenomenon results
in pretty pictures of laboratory experiments, the consequences can be a
matter of life or death for fish whose survival depends upon the correct
mixing of nutrients, chemicals, plankton, other fish, and even their own
larvae or juveniles. Similarly, the spread of pollution and greenhouse
gases in the atmosphere depends on the vagaries of natural mixing. When
mixing changes in oceans or atmospheres, there is an immediate and large
impact. For example, the changed mixing of nutrients for anchovies in
Monterey Bay led to the disappearance not only of the anchovies, but
also of the active warehouses and factories of Cannery Row. Our ability
to predict the effects of pollution, global and long-term changes in
climate, and the health of our oceans depends on our ability to
understand and model the vagaries of mixing.

Using sophisticated ideas with origins in classical mechanics, nonlinear
dynamics, and chaos theory, mathematicians have been able to show that
mixing is far more complex than a fast diffusion process (i.e., how ink
spreads in non-moving water). Mixing occurs at unequal rates depending
upon direction and locations. It is profoundly affected by the state of
the fluid and by the locations of eddies and currents. The mathematics
of mixing shows that, while large eddies or vortices make the mixing and
transport of chemicals and pollutants very efficient around and along
their outside edges, the edges themselves act as barriers to the mixing
of chemicals into or out of the vortices.

An interesting example is the "ozone hole" over Antarctica, a region of
the atmosphere where ozone is nearly completely destroyed due to a
chemical reaction in the upper atmosphere\'s clouds. Since the hole is
surrounded by ozone and the atmosphere is highly stirred from
atmospheric turbulence, it is natural to ask why the surrounding ozone
does not mix into the hole. The answer is that the hole is at the center
of a large vortex (the Antarctic stratospheric polar vortex), and
mathematical models correctly predict that its outer edge acts as a
strong barrier to mixing even though the atmosphere is turbulent and the
edge of the vortex is constantly changing position in response to the
turbulence. The vortex is crucial to maintenance of the hole. Each
spring the stratospheric vortex breaks up due to warming of the ground
below; this destroys not only the vortex, but also its edge\-\--the
barrier to mixing. Thus the ozone is replenished in the hole and the
hole goes away.

The effects of barriers to mixing can be appreciated on even a larger
scale. It has long been recognized that the equator hampers mixing in
the atmosphere between the northern and southern hemispheres.
Mathematical analysis is beginning to explain the selective permeability
of this barrier, which is a complex and interesting function of location
(with respect to continents and other topographic features), time of
year, and time with respect to longer-term cycles (such as that
associated with *El Niño)* that occur in the ocean-atmosphere system. An
important question being addressed is how man-made greenhouse gases such
as carbon dioxide, which are predominantly created in the north, spread
south of the equator.

The incomplete mixing caused by stirring has consequences beyond the
distribution of pollutants. Not only are tangible things such as
chemicals mixed by stirring, but so is "vorticity" (the amount of spin
or rotation of a small parcel of fluid). If the fluid is not quiet or
rotating as a solid body, the vorticity changes as a function of
position in the fluid, and is likely to change with time as well. What
makes the mixing of vorticity so fascinating is that it is a highly
nonlinear process in the sense of having substantial feedback onto
itself, since the locations of vortices, especially of their edges,
determine where mixing of vorticity occurs. In particular, if a double
row of oppositely-signed vorticity accumulates in sheets, it produces an
ocean current or jet stream.

Observations show that the complex mixing due to stirring often divides
oceans and atmospheres into separate regions ("patches") such that the
fluid within each region is well mixed, but there is very little mixing
among the regions. Mathematical theories based on statistical mechanics
have recently taken away the mystery from the most visually striking
examples of these patches: Jupiter\'s Great Red Spot and the horizontal,
multi-colored stripes of Saturn and Jupiter.

The Red Spot, seen through some of the first telescopes 360 years ago,
is a very robust vortex that owes its existence to the continuous
accumulation and mixing together of small patches of vorticity. The
stripes of Jupiter and Saturn are alternating jet streams. The east-west
bands of Saturn appear multi-colored because the chemicals within each
band are well mixed but there is little mixing of chemicals from band to
band.

Numerical simulations of mixing on supercomputers have contributed
several insights. If the Red Spot were artificially broken apart, the
pieces would simply mix back together. If Jupiter or Saturn initially
had no motion in its atmosphere with respect to an observer on the
planet, then mixing of vorticity would start spontaneously and create,
after several decades, the jet streams we see today. An atmosphere
initially at rest with respect to an observer on the planet contains a
continuous distribution of vorticity, maximized at the north pole,
decreasing to zero at the equator, and reaching its minimum at the South
pole. Since different fluid elements thus have different values of
vorticity, the distribution can be mixed so that there are alternating
sheets of vorticity with opposite sign (and consequently jet streams).

Saturn and Jupiter both display rings; Saturn\'s are easily seen with a
pair of binoculars, but Jupiter\'s ring is so faint that it was not
detected until the 1970s. These rings are made of small particles, are
extraordinarily thin, and have very sharp, well-defined edges. The rings
of Saturn consist of a nest of several rings with well-defined gaps in
between. Several properties of the rings are explained by the
mathematics of mixing and of nonlinear dynamical systems. As with the
ozone hole over Antarctica, it might appear that the gaps would quickly
fill with particles that are continuously bumped into them via
interactions with other particles. However, gravitation from the moons
controls the mixing in such a way that these narrow gaps are kept free
of particles and the edges of the rings are well-defined.

The mathematical theory of mixing in nonlinear systems allows us to
understand and predict much of what we see in nature. Although
*controlling* nature with these theories is well beyond present
capabilities, the same mathematics is being used in practical
engineering problems. In micro-electrical and mechanical systems, there
is an ever-increasing desire to miniaturize both electronic and
mechanical components. Millions of motors can now be created on the head
of a pin and used as pumps to deliver medicines, carry out chemical
reactions, act as precise carburetors for engines, and so on. For these
applications it is necessary to mix one or more chemicals together.
Early on, researchers believed that everything in these devices would
mix together easily because of the very small distances that the
constituents had to travel, but this belief failed to take into account
the many barriers to mixing. Fortunately, the mathematics reveals how to
destroy these barriers. One such method, known as chaotic mixing,
requires stirring the fluid at two incommensurate frequencies. This can
be done in one of these devices by creating two small heating elements
via the same type of lithographic techniques used to build electronic
chips. When the elements are supplied with voltage, they boil tiny
amounts of fluid, producing two bubbles. Oscillating the voltage makes
the two bubbles oscillate, which in turn provides stirring at any
desired frequency. Thus an application of mathematical mixing that was
originally inspired by the study of natural phenomena solves a critical
problem in state-of-the-art engineering.

**3.9 Physiology**

With a few notable exceptions such as Helmholtz, Frank, Hodgkin, and
Huxley, physiology and the mathematical sciences have not been closely
linked until recently. Many, perhaps most, physiologists have regarded
their science as primarily descriptive, with little scope or need for
mathematics; mathematicians trained in the traditional way almost
invariably "speak physics", and may be reluctant to enter a field in
which competence demands a significant degree of specialized scientific
study of an unfamiliar kind.

But this situation is changing from both directions. Mathematical models
and computational simulation offer means for characterizing and
analyzing processes in physiology that are individually complex to begin
with and whose interactions add further complexity; in return,
physiology provides a rich, fascinating field of science with
opportunities for new applications and new mathematics. A prize-winning
book on mathematical physiology[^2] stresses the importance of increased
connections between mathematics and physiology: "\... teaching
physiology without a mathematical description of the underlying
dynamical processes is like teaching planetary motion to physicists
without mentioning or using Kepler\'s laws; you can observe that there
is a full moon every 28 days, but without mathematics you cannot
determine when the next total lunar or solar eclipse will be nor when
Halley\'s comet will return\'\'.

One area among many in physiology where the mathematical sciences are
beginning to make major contributions is *integrative biology*, the
system-level study of how complex, spatially distributed biological
systems manage to perform their functions. Mathematical models are being
developed that analyze the following aspects of complex physiological
systems, to mention just a few:

-   the macroscopic behavior of lung tissue based on the microstructure
    of respiratory regions;

-   the self-organization of cells and molecules in the immune system
    that underlies responses to attacking pathogens; and

-   the control of cells in a developing system so that they "know"
    where they should go and what to do at their destination.

A pervasive example of integrative behavior is *movement*: living
organisms internally move things such as nutrients, blood, oxygen, and
pigment. Somehow, based on principles that remain unknown, living
creatures self-organize a movement that achieves a prescribed result.
Two instances in cell biology of self-organizing behavior related to
movement involve *centering*: a nucleus spends most of its time at the
center of its cell, yet it cannot directly sense the cell membrane nor
evaluate its distances from points on the membrane; and cell division
requires chromosomes to be aligned along a central plane during cell
division. Current biological models rely on the unrealistic assumption
of non-local dynamics, so an obvious question is whether global behavior
such as centering can be achieved entirely through local interactions.

To answer this question, mathematical models constructed from local
processes can be studied to see whether (and, if so, why) they result in
centering behavior. An illustration of the role of mathematics is
provided by recent work on modeling properties of cells in the black
tetra, a small colorful fish popular in home aquariums. Melanophore
cells create the tetra\'s colors and patterns through self-organizing
behavior that depends on the interactions of microtubules, dynein, and
pigment. *Microtubules* are long tubelike protein polymers with a polar
structure (a difference between the two ends). *Dynein* is a molecular
motor that transforms stored energy into mechanical work. When activated
by adrenaline, dynein moves along microtubules, always in the same
direction\-\-- toward the "minus end", away from the "plus end". Dynein
has an affinity for pigment molecules, and will drag them along as it
moves.

In melanophore cells, microtubules are normally arranged in a radial
pattern with minus ends near the nucleus and plus ends near the
membrane. If the dynein motors are activated, pigment tends to aggregate
in a small region around the cell nucleus. The macroscopic effect of
pigment aggregation is to change the intensity of the cell\'s (and, in
the large, the tetra\'s) color.

Recent experiments were designed to understand the dynamics of how
pigment centers around the nucleus. A fragment of a melanophore cell was
sliced off, separating it from the nucleus. Following the cut, the
dynein was activated in the fragment. Very soon, a pigment aggregate
formed near the "minus" edge of the fragment. A slower process then
occurred in which the pigment aggregate drifted toward and eventually
stopped at the "center" of the fragment. In this final state, the
microtubules had actually rearranged themselves in a radial pattern
within the fragment. Numerous other experiments demonstrated that the
radial array of microtubules did not form unless the dynein was
activated and pigment was present.

A mathematical model of this process begins with several facts and
assumptions. In the absence of pigment, microtubules grow (at the plus
end) and shrink (at the minus end) at the same rate. Dynein, even when
carrying pigment, moves much faster than the growth/shrinkage rate of
microtubules without pigment. Plus ends of microtubules are stabilized
when they reach the cell boundary; minus ends tend to be "caught" in
regions of high pigment concentration. Nucleation (the appearance of new
microtubules) occurs on pigment particles. Together, these assumptions
descriptively explain the fast initial movement of pigment to the cell
boundary followed by the slow centering of the pigment aggregate.

The challenge for mathematical modeling is to translate these
assumptions into a form that captures, both qualitatively and
quantitatively, the observed relationships among microtubules, dynein,
and pigment. Work in this direction has begun with a one-dimensional
version of the problem. Although overly simplistic, it is nonetheless
appropriate for a long thin fragment in which almost all microtubules
run down the long axis and there is little variation in pigment along
the thin axis. The main parameters are the fragment length, the speed of
dynein, the plus end growth rate, and the diffusion coefficient for
pigment. The pigment concentration is treated as a function of position
and time; microtubules are described by their plus and minus ends and
orientation (left- or right-moving).

To define the cell dynamics, the shrinkage and nucleation rates are
described as functions of pigment concentration. Growth at the plus end
of a microtubule is interpreted as moving that end at a particular
velocity, and nucleation is treated as a reaction term. Pigment flux is
determined by diffusion and motion along microtubules. Using
conservation principles, analysis of the flux due to microtubules, and
pigment dynamics, a system of partial differential equations and
boundary conditions has been defined. A crucial feature of the model is
that all relationships are local, as they are in the theory being
represented.

Even with this relatively simple formulation, centering of the pigment
aggregate within the fragment occurs consistently, and simulations have
satisfactorily matched appropriate experimental observations. The
ability to vary the mathematical parameters and initial conditions in
the model allows "virtual experiments" in which essentially every
conceivable combination of pigment distribution and microtubule
orientation can be tried. The next step is, of course, to refine and
extend the model to convey the full set of known properties of the
melanophore cells, with the ultimate goal of understanding centering.

An implicit but crucial general point is that mathematics and physiology
must be intimately connected to succeed in this kind of endeavor. The
example of melanophore in the black tetra clearly illustrates that
serious knowledge of physiology is required to create even an elementary
mathematical model of pigment centering.

**3.10 Diagnosis Using Variational Probabilistic Inference**

The rapid growth of the information sciences is leading to new
challenges for mathematics. Although in many cases entirely new
mathematical theories must be formulated, the reservoir of mathematical
knowledge is vast and what is called for is sometimes the discovery of
appropriate analogies so that old ideas can be applied in new ways. In a
recent success story, a difficult problem in probabilistic diagnosis has
been solved via the use of techniques originally developed for
statistical physics, quantum mechanics, and mechanical engineering.

The problem of diagnosis is an instance of the general problem of
"inductive inference" or, more informally, "reasoning backward".
Consider, for example, the problem of diagnostic reasoning in medicine.
A doctor observes a set of symptoms in a patient and wishes to infer the
disease (or diseases) that could be responsible. In general the doctor
must utilize basic medical knowledge inductively to uncover an
explanation of a pattern of symptoms. Basic medical knowledge consists
of biologically-based, causal theories specifying the way in which
various diseases affect the organism and lead to various symptoms. From
this knowledge, in the form of disease-to-symptom relationships, the
doctor must reason backwards to make predictions about
symptom-to-disease relationships.

Backward reasoning can be complex computationally. A major source of
complexity is that causally unrelated diseases (i.e., with unrelated
biological origins) can become strongly dependent diagnostically.
Suppose that two unrelated diseases have a predicted symptom in common
and that the symptom is in fact observed; then the two diseases compete
to explain it, i.e., additional evidence that one of the diseases is
present tends to reduce our belief in the presence of the other disease.
In general, a disease can "explain away" a symptom, decreasing the need
to posit some other disease as the explanation of the symptom. This
changed belief in a disease can then "flow forward", lowering or raising
the support for other symptoms, which\-\--by the same explaining-away
mechanism\-\--can affect the belief in yet other diseases. The fact that
different diseases have common sets of symptoms can lead to a tangled
web of interdependencies.

Scientists have started to build probabilistic tools for diagnosis not
only in medicine but in many other domains, including manufacturing,
transportation, and communications. Building these tools has improved
understanding of the mathematical issues underlying backward reasoning.
Significant progress has been made in an area known as *graphical
modeling*, where, in the past ten years, a general mathematical theory
has emerged that yields a clear specification of the complexity of
diagnosis in probabilistic systems and allows optimal algorithms to be
defined. Many classical probabilistic tools, including the Kalman filter
(used in control and estimation theory) and the hidden Markov model
(used in speech recognition and molecular biology), are special cases of
this general methodology. But the theory applies much more widely,
providing a general understanding of probabilistic inference in
arbitrary probabilistic networks.

A particularly challenging instance of a complex probabilistic knowledge
base is the "Quick Medical Reference" (QMR) database for diagnosis in
internal medicine. This database, developed at the University of
Pittsburgh with 25 person-years of effort, is one of the largest
probabilistic databases in existence, and contains a significant
fraction of the diseases in internal medicine. The QMR database is
organized as a probabilistic network in which approximately 600
binary-valued nodes representing diseases are linked to approximately
4000 binary-valued nodes representing symptoms.

Unfortunately, when one analyzes the QMR network from the viewpoint of
the recently developed theory of inference, one finds that *exact*
diagnostic reasoning is infeasible computationally. For a set of typical
symptoms, it has been estimated that calculation of the exact
probabilities of diseases would require approximately 50 years on
current computers. Research on QMR and related large-scale diagnostic
systems has consequently lain fallow for want of efficient algorithms.

The general mathematical problem underlying probabilistic inference,
hinted at in the earlier discussion of explaining away, takes the form
of a set of nonlinear equations in which each equation can have an
exponentially large number of terms. Roughly speaking, to determine the
probability of a disease in the QMR network, given a set of symptoms,
one must take the product over the probabilities of the observed
symptoms (a nonlinear operation) and then take the sum over all
configurations of other diseases (a sum involving 2^599^ terms). The
actual computation is not as bad as this, given that the network is not
fully connected (e.g., some diseases have zero probability of producing
certain symptoms), but it is still intractable.

There is a way out of this computational dilemma: viewing the problem as
numerical, with a need to find accurate statistical estimates, rather
than as symbolic, with a need to compute a large number of terms. The
fact that there are so many terms in the sums to be calculated offers
hope that laws of large numbers will come into play, rendering the
system probabilistically simple despite its apparent symbolic
complexity.

This point of view is of course natural in the context of the highly
interacting systems in statistical physics, and one might hope that the
tools developed in physics could be employed in the service of
large-scale diagnostic inference problems. In fact, a number of useful
analogies can be drawn between graphical models and statistical physics
models. The major technical difficulty arises because the graphical
models studied in diagnostic reasoning are generally based on directed
graphs (graphs in which the nodes are linked by arrows), whereas in
statistical physics the graphs tend to be undirected (a consequence of
Newton\'s third law). Once this technical hurdle is overcome, many ideas
from the physics context can be exploited in diagnosis. In particular,
the mean field approach in statistical physics has a natural analogue
for graphical models.

More broadly, mean field theory can be viewed as a variational method in
which a nonlinear system with strong couplings is approximated by a
variational principle. Variational principles are highly successful in
mechanics, where variational finite element methods characterize the
global state of stress or strain of a piece of material. These methods
can also provide useful insight into approximation methods for
diagnostic reasoning.

Researchers have recently developed an approximate approach to
probabilistic inference known as *variational inference*, which is very
similar to mean field theory and finite element analysis. Rather than
performing inference directly on a dense probabilistic network, the
variational approach considers a simplified network in which some of the
links are missing. Roughly speaking, a variational parameter is
introduced for each missing link; this parameter captures in an
approximate way the high-order probabilistic dependencies induced when
that link is present in the network. The simplified network is chosen so
as to obtain bounds on the probabilities of interest rather than exact
values.

The advent of variational methods in probabilistic inference has created
new mathematical problems. Some of these are analogous to problems in
statistical physics and finite element analysis, and solutions in these
domains may prove useful in variational inference. For example,
variational methods can fail when there are deterministic relationships
between nodes in a network. This is conceptually similar to the
difficulty posed by incompressible media in finite element analysis,
where solution methods are available and may be broadly useful.

The variational approach has been highly effective for the QMR database,
where it can yield accurate estimates of disease probabilities within
less than a second of computer time. It has also been applied
successfully to a number of other graphical models in which exact
inference is intractable. Applications to diagnosis, pattern
recognition, statistical genetics, and error-correcting codes are
currently being explored.

A particularly interesting application is to learning theory, where one
would like to find out the parameters of a graph based on data; there
are many interesting relationships between inference and learning that
variational methods may help us to understand.

**3.11 Iterative Control of Nuclear Spin**

Nuclear spins play a central role in nuclear magnetic resonance (NMR),
spectroscopy, and magnetic resonance imaging (MRI). Control of nuclear
spins is tantamount to control of the parameters that determine the
features and information content of NMR spectra and MRI images. Recent
research has led to development and implementation of an approach for
*iterative control* of nuclear spins.

In general terms, control takes a system from a given initial state to a
desired final state under the action of a control propagator. The system
may be a robot, a vehicle or spacecraft, a molecule, or a system of
nuclear spins. Traditional differential control involves the feedback
adjustment of the parameters of an evolving system in order to prevent
deviations from a prescribed trajectory. Such control necessitates
comparison of the actual evolving trajectory with the prescribed
trajectory, i.e. it is necessary to "see where you\'re going".

In the novel iterative schemes, by contrast, the propagator that induces
the desired trajectory is chosen as the stable fixed point in
"propagator space" of the iterative map that is applied between stages
of the system. This choice ensures that any initial propagator,
regardless of errors or perturbations, will always converge to the
desired final state.

With this approach, it is not necessary to "see where you\'re going".
Thus, instead of tailored differential control for each member of an
ensemble that may experience different errors, the same control sequence
can be applied "blindly" to the whole ensemble. There is, of course, a
price to pay for this broadband privilege\-\--the trajectory from
initial to final state may be considerably longer and more complex.
However, convergence to the desired final state with predetermined
precision is assured. Clearly there are circumstances in which
differential control is more appropriate, and there are others where
iterative control is superior.

Systems containing nuclear spins are often well suited to iterative
control because they involve large ensembles with broad ranges of
control parameters and errors. The new stable, indeed "super stable",
fixed points for such systems have been obtained through dynamical
systems theory.

Iterative sequences derived from these mathematical models have been
implemented in NMR and MRI through collaborations between mathematicians
and scientists. With the resulting enhanced instruments, precise and
selective control of the states of nuclear spins can be achieved. On the
microscopic scale, for example, iterative decoupling sequences permit
elimination of the effects of spin-spin interactions. As a result, the
NMR spectra are enormously simplified, allowing the structures of
molecules in solution and in materials to be determined. On the
macroscopic scale, iterative excitation in MRI makes it possible to
elicit and to selectively enhance or suppress signals from particular
regions of the images of organisms, consequently providing spatially
selective biomedical information.

In recent years, NMR has emerged\-\--beyond its role as a diagnostic
analytical tool for molecules, materials, and organisms\-\--as a
potentially powerful environment for implementation of quantum
computing. The nuclear spins are, after all, quantum systems with a
natural binary basis, namely the two quantum states "up" and "down" in a
magnetic field. The spins can therefore function as "qubits" whose
entangled quantum states are manipulated in quantum logic gates by means
of delicately controlled radiofrequency pulses, as in NMR spectroscopy.

Enormous potential advantage of quantum computing over classical
computing is foreseen because quantum algorithms involve participation
of all qubits at the same time. This is a uniquely quantum phenomenon
akin to capitalizing on the simultaneous existence of the alive and dead
quantum "Schrödinger cat". Iterative control schemes currently under
development should make it possible to overcome the effects of
decoherence, thus allowing the implementation of extended quantum
computation algorithms even in the presence of imperfect quantum logic
gates and interactions with the environment.

**3.12 Moving Boundaries and Interfaces**

Many physical problems involve moving boundaries. *Dynamic* boundaries
change position and shape in response to the particular physics at work:
examples are breaking waves in the ocean, dancing flames in the
fireplace, and milk swirling in a cup of tea. *Static* boundaries, such
as tumors in medical scans and cartoon characters against a background
animation, can be just as perplexing: try finding edges in a picture of
a dalmatian lying on a rug with spots! Surprisingly, many other
interesting problems, such as negotiating a robot around obstacles and
finding the shortest path over a mountain range, can also be cast as
evolving boundary problems.

The physics and chemistry that drive a boundary or interface may be
difficult to describe, but even when the speed and direction of a moving
interface are well understood, following its shape can be difficult. The
first concern is what to do when sharp corners appear, as they do in,
for example, the intricate patterns of a snowflake. Second, distant
edges can blend together: the "edge" of a forest fire changes as
separate fires burn together and sparks carried by the wind ignite
distant regions. Finally, in three dimensions (and higher), even finding
a nice way to represent\-\--let alone move\-\--an undulating boundary is
a challenge.

One technologically important example of interface motion involves the
manufacture of computer chips. In the etching and deposition process, a
layer of metal is deposited on a silicon wafer, etched away, and then
the process is repeated numerous times until a final profile is
obtained. As device sizes get smaller and smaller, using trial and error
to obtain the correct design becomes impractical. Instead, one would
like to simulate these processes as accurately as possible in order to
test various layering strategies and resulting device characteristics.
In recent years, the application of new mathematical and numerical
algorithms for interface motion has afforded real breakthroughs in this
area. Before these techniques, complex problems involving the evolution
of profiles in two dimensions were difficult; now, fully
three-dimensional simulations involving a wide range of physical effects
are easily within grasp. The new algorithms have been incorporated into
the simulation packages at many major semiconductor manufacturers in the
United States, and are part of the production environment in various
chip lines today.

These computational techniques, known as level set methods and fast
marching methods, rest on a fundamental shift in how evolving fronts are
viewed. Rather than focus on the evolving front itself, these techniques
discretize the region in which the front moves. Each point in that space
keeps track of either its distance to the front or of the time when the
front passes over it; the accumulation of all this information gives an
accurate portrait of the moving interface. The key is to define
equations for the time at which the front passes over each point and
then to solve these equations.

The equations which keep track of the front at each grid point in the
domain are variants of the Hamilton-Jacobi equations; these equations
have a long history in such areas as optics, wave propagation, and
control theory. While they can be very complex, their derivatives bear a
resemblance to hyperbolic conservation laws and to the equations of
fluid mechanics, allowing use of the knowledge acquired in those
well-developed fields. The main breakthrough in modeling interface
motion was the realization that schemes from fluid mechanics could be
unleashed onto the equations of moving fronts. The result is a wide
range of computational tools for tracking evolving interfaces with sharp
corners and cusps, with topological changes, and in the presence of
three-dimensional complications. These schemes have found their way into
a vast number of applications, including fluid mechanics, dendrite
solidification and the freezing of materials, image processing, medical
imaging, combustion, and robotic navigation.

Some of the most complex interface applications appear in simulating the
manufacture of computer chips. To begin, a single crystal ingot of
silicon is extracted from molten pure silicon. This silicon ingot is
then sliced into several hundred thin wafers, each of which is polished
to a smooth finish. A thin layer of crystalline silicon is oxidized, a
light-sensitive "photoresist" is applied, and the wafer is covered with
a pattern mask that shields part of the photoresist. This pattern mask
contains the layout of the circuit itself. Under exposure to a light or
an electron beam, the unshielded photoresist polymerizes and hardens,
leaving an unexposed material that is etched away in a dry etch process,
revealing a bare silicon dioxide layer. Ionized impurity atoms such as
boron, phosphorus, and argon are implanted into the pattern of the
exposed silicon wafer, and silicon dioxide is deposited at reduced
pressure in a plasma discharge from gas mixtures at a low temperature.
Finally, thin films like aluminum are deposited by processes such as
plasma sputtering, and contacts to the electrical components and
component interconnections are established. The result is a device that
carries the desired electrical properties.

This sequence of events produces considerable changes in the surface
profile as it undergoes various processes of etching and deposition.
Describing these changes is known as the "surface topography problem" in
microfabrication and requires an analysis of the effects of many
factors, such as the visibility of the etching/deposition source from
each point of the evolving profile, surface diffusion along the front,
complex flux laws that produce faceting, shocks and rarefactions,
material-dependent discontinuous etch rates, and masking profiles. The
physics and chemistry that contribute to the motion of the interface are
areas of active research. Once empirical models are formulated, one is
left with the problem of tracking the evolving front.

Here is where level set methods and fast marching methods come into
play: they provide the means to follow the evolving profile as it is
shaped by the etching and deposition process, and they capture some of
the most subtle effects. For example, visibility has a key role; if part
of the evolving surface causes a shadow zone that blocks the effects of
the etching or deposition beam, the motion is reduced. Computing this
shadow zone was formerly a very expensive proposition; however, the fast
marching method yields an elegant and fast way to do it.

Another example is the complex manufacturing process called ion-milling,
in which a beam of reactive ions acts like a sandblaster and etches away
at a surface. The etching rate depends on, among other things, the angle
at which the beam hits the surface. The most effective etching angle is
not always directly straight down; the "yield function" relates how much
material is removed to the incoming angle. Interestingly enough, this
process produces beveled, rounded edges in some areas and sharp cusps in
others. While these are difficult problems to model, they are easily
handled by level set and fast marching methods.

## 4 Education

The importance of strong ties between mathematics and science is
self-evident from the examples presented\-\--which, we stress again, are
only a tiny sample from a very large pool. Unfortunately, there is a
clear shortage of people able to bridge the gap between mathematics and
the sciences, and one of the challenges that must be faced is how to
educate more.

It is obvious to us that students of mathematics should be able to
understand problems in science, and that students of science should
understand the power and roles of mathematics. Each area of science has
its own unique features, but the different areas share common features
that are often of a mathematical nature.

The themes of *modeling*, *computation*, and *problem solving* are
especially relevant to education.

-   **Modeling.** Students in science and mathematics need to be
    educated in modeling far beyond the simple paradigm exemplified by
    \`\`do this experiment, plot the data, and observe that they lie
    almost on a straight line\'\'. Given a physical problem and/or data,
    students should learn to construct a mathematical model, explain why
    the model is appropriate, perform mathematical analysis or a
    computational simulation, devise experiments to check the accuracy
    of their model, and then improve the model and repeat the process.

-   **Computation.** The view that \`\`anyone can compute\'\' is just as
    wrong as the statement that \`\`anyone can build a telescope\'\'.
    One has to learn how. Much of the current teaching of computation is
    flawed; a \`\`cookbook\'\' strategy of using canned programs without
    attention to fundamentals is completely inadequate. At the other
    extreme, scientists should not waste their time implementing
    outmoded methods or reinventing known algorithms and data
    structures. Students in science and mathematics need to be aware of
    the intellectual content and principles of modern computer science.

-   **Problem-solving.** In traditional academic presentations of
    scientific and mathematical problems, the context is stripped away
    and simplified so that students can focus on the essentials. But,
    especially when developing mathematical insights, students must
    learn how to approach ill-defined, poorly formulated problems\-\--an
    area in which education is lacking. There are no shortcuts; the only
    way to learn is by direct experience.

We offer a number of recommendations for education in mathematics and
science. Our primary focus is education for students who specialize in
mathematics or science; we cannot begin to address the national problem
of mathematics and science education for all.

1.  Support curriculum development in areas that are essential for
    connections between mathematics and science. Every
    curriculum-related activity should include production of Web-based
    materials.

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(a) Create modeling courses for high school, undergraduate, and graduate
    students. Unlike many other skills, modeling can be taught (at an
    elementary level) to students in high school. At the undergraduate
    level, there would be enormous benefits if a one-year modeling
    course were part of the core curriculum in science, engineering,
    mathematics, and computer science. Graduate modeling courses would
    deepen the scientific knowledge of mathematics students while
    enriching the mathematical skills of science students.

(b) Support development of courses that tie core computer science to
    science, engineering, and mathematics. Programming, numerical
    analysis, data structures, and algorithms\-\--each of which is a
    topic with serious mathematical content\-\--should be part of the
    education of every scientist and mathematician.

(c) Encourage experiments in activities (courses, summer or short-term
    workshops) that teach scientific and mathematical problem solving.
    Such programs could involve not only techniques and direct
    experience of problem solving, but also group projects that teach
    students how to work collaboratively with others and how to present
    their work.

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2.  Encourage students to undertake programs of study, at both
    undergraduate and graduate levels, which combine mathematics and
    science. That this can be done at the graduate level has been shown
    by the successful Computational Science Graduate Fellowship program
    of the Department of Energy, which requires students to undertake a
    demanding interdisciplinary program in exchange for a generous
    fellowship.

3.  Support summer institutes in (i) mathematical topics that address
    scientific applications and (ii) scientific topics with mathematical
    content.

The NSF Research Experiences for Undergraduates (REU) program has been
extremely successful in exposing students to research at an early stage.
REU and other institutes have become important for top undergraduates
interested in science and mathematics, and it is now common to prepare
for graduate school by attending a summer school or institute. However,
these programs are overwhelmingly devoted to highly specialized
subjects. In part this is understandable; the organizers want to give
the students a taste of research, which is more easily done in a narrow
area. But because those summer institutes often determine the direction
students will take, NSF should ensure that there are high-quality
institute programs with a multidisciplinary emphasis centered on
connections between mathematics and science.

Certain emerging areas (such as mathematical biology) are not yet widely
covered in graduate programs. Carefully designed summer institutes would
help to broaden the education of graduate students whose home
institutions lack offerings in such fields.

4\. Fund research groups that include both (i) a genuine collaboration
between scientists and mathematicians, and (ii) a strong educational
program for graduate students, postdoctoral fellows, and possibly
undergraduates. To be effective, such funding should be as long-term as
possible; if funding is only short-term, researchers are unlikely to
make the huge investment of time needed to develop group structures that
will sustain multidisciplinary collaborations.

5\. Fund postdoctoral fellowships in environments that combine
excellence in science with excellence in mathematics. Efforts to create
industrial postdoc programs could be expanded to create joint
university/national lab postdoctoral fellowships, as well as short-term
fellowships for scientists in mathematics programs with a strong applied
component.

Beyond the postdoctoral level, there should be programs to encourage and
support faculty who would like to become active in collaborations
outside their own discipline. The existing NSF program in this vein,
Interdisciplinary Grants in the Mathematical Sciences (IGMS), is small
and imposes relatively strict requirements on qualification and support
by the home department.

6\. Develop a program of group grants for mathematics and science
departments that encourage the creation of new courses, experimentation
with instructional formats, and coordinated programs of hands-on
experiments, modeling, and computation. Departments that receive such
grants should have substantial science requirements for undergraduate
degrees in mathematics, and substantial mathematics requirements for
undergraduate degrees in science. Many, if not most, U.S. undergraduates
in mathematics take no, or almost no, science courses. In certain areas
of science and engineering, undergraduates take only minimal, and
sometimes outdated, mathematics courses; even worse, those courses may
give students no understanding of the ties between their fields and
mathematics. These unfortunate situations are likely to be corrected
only if there is an incentive for departments to change their basic
programs.

## 5 Conclusions

Strong ties between mathematics and the sciences exist and are thriving,
but there need to be many more. To enhance scientific progress, such
connections should become pervasive, and it is sound scientific policy
to foster them actively.

It is especially important to make connections between mathematics and
the sciences more timely. Scientists and engineers should have access to
the most recent mathematical tools, while mathematicians should be privy
to the latest thinking in the sciences. In an earlier era of small
science, Einstein could use the geometry of Levi-Civita within a few
years of its invention. With today\'s vastly expanded scientific
enterprise and increased specialization, new discoveries in mathematics
may remain unknown to scientists and engineers for extended periods of
time; already the analytical and numerical methods used in several
scientific fields lag well behind current knowledge. Similarly,
collaborations with scientists are essential to make mathematicians
aware of important problems and opportunities.

## 6 References and URLs

### Combustion

\[1\] Information about Chemkin, a registered trademark of Sandia
National Laboratories:

http://stokes.lance.colostate.edu/CHEMKIN_Collection.html

http://www.sandia.gov/1100/CVDwww/chemkin.htm

http://www.sandia.gov/1100/CVDwww/theory.htm

### Cosmology

\[2\] M. S. Turner and J. A. Tyson (1999), *Cosmology at the
Millennium*, working paper.

\[3\] Web sites about mathematical models and numerical simulation:

http://star-www.dur.ac.uk/\~frazerp/virgo/aims.html

http://phobos.astro.uwo.ca/\~thacker/cosmology/

### Finance

\[4\] I. Karatzas and S. E. Shreve (1998), *Methods of Mathematical
Finance*, Springer-Verlag, New York.

\[5\] T. F. Coleman (1999), An inverse problem in finance, *Newsletter
of the SIAM Activity Group on Optimization*.

### Functional Magnetic Resonance Imaging

\[6\] W. F. Eddy (1997), Functional magnetic resonance imaging is a team
sport, *Statistical Computing and Statistical Graphics Newsletter*,
Volume 8, American Statistical Association.

\[7\] Information about functional image analysis software:

http://www.stat.cmu.edu/\~fiasco

### Hybrid System Theory and Air Traffic Management

\[8\] C. Tomlin, G. J. Pappas, and S. Sastry (1998), Conflict resolution
for air traffic management: a case study in multi-agent hybrid systems,
*IEEE Transactions on Automatic Control*, 43, 509\-\--521.

### Internet Analysis, Reliability, and Security

\[9\] Willinger and V.\\ Paxson (1998), Where mathematics meets the
Internet, *Notices of the American Mathematical Society* 45,
961\-\--970.

\[10\] The Web site of the Network Research Group, Lawrence Berkeley
Laboratory:

http://www-nrg.ee.lbl.gov

### Materials Science

\[11\] Research trends in solid mechanics (G. J. Dvorak, ed), United
States National Committee on Theoretical and Applied Mechanics, to
appear in *International Journal of Solids and Structures*, 1999.

\[12\] G. Friesecke and R. D. James (1999), A scheme for the passage
from atomic to continuum theory for thin films, nanotubes and nanorods,
preprint.

### Mixing in the Oceans and Atmospheres

\[13\] P. S. Marcus (1993), Jupiter\'s great red spot and other
vortices, *The Annual Review of Astronomy and Astrophysics* 31,
523\-\--573.

### Physiology

\[14\] J. Keener and J. Sneyd (1998), *Mathematical Physiology*,
Springer-Verlag , Berlin.

\[15\] Details about modeling melanophore in the black tetra (the home
page of Eric Cyntrynbaum, the University of Utah):

http://www.math.utah.edu/\~eric/research

### Diagnosis Using Variational Probabilistic Inference

\[16\] T. S. Jaakkola, T. S. and M. I. Jordan (1999). Variational
methods and the QMR-DT database, submitted to *Journal of Artificial
Intelligence Research*.

\[17\] M. I. Jordan (1998), *Learning in Graphical Models*, MIT Press,
Cambridge, Massachusetts.

### Iterative Control of Nuclear Spins

\[18\] R. Tycko, J. Guckenheimer, and A. Pines (1985), Fixed point
theory of iterative excitation schemes in NMR, *J. Chem. Phys.* 83,
2775\-\--2802.

\[19\] A. Lior, Z. Olejniczak, and A. Pines (1995), Coherent isotropic
averaging in zero-field NMR, *J. Chem. Phys.* 103, 3966\-\--3997.

### Moving Boundaries and Interfaces

\[20\] J. A. Sethian (1996), *Level Set Methods: Evolving Interfaces in
Geometry, Fluid Mechanics, Computer Vision, and Materials Sciences*,
Cambridge University Press.

### Acknowledgements

This document was assembled and written by Alexandre J. Chorin and
Margaret H. Wright. They gratefully acknowledge help from:

Dr. Phillip Colella,

Professor Thomas F. Coleman,

Professor William F. Eddy,

Professor John Guckenheimer,

Professor Richard D. James,

Professor Michael Jordan,

Professor James Keener,

Professor Philip Marcus,

Dr. Andrew M. Odlyzko,

Professor Alexander Pines,

Professor Shankar Sastry,

Professor James Sethian,

Professor Steven E. Shreve,

Professor Claire Tomlin, and

Dr. J. A. Tyson.

[^1]: ^1^ For compactness, throughout this document "mathematics" should
    be interpreted as "the mathematical sciences", and "science" as
    "science, engineering, technology, medicine, business, and other
    applications".

[^2]: ^2^ J. Keener and J. Sneyd, *Mathematical Physiology*,
    Springer-Verlag, Berlin, 1998