Title: Report: Mathematics and Science
Date: January 12, 2000
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 science1 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 .
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 109, 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 Ni¤o) 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 physiology2 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 2599 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
"Schr”dinger 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.
(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.
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 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 J. Keener and J. Sneyd, Mathematical Physiology, Springer-
Verlag, Berlin, 1998