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Mathematics - The Science of Patterns and Algorithms

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( * Occasional use of the term "mathematics" in this text should be understood as referring to all the mathematical sciences; similarly "the sciences" stands as a shorthand for the wider class of all the pure and applied sciences as well as engineering.)

MATHEMATICS IS POWERFUL AND EXCITING

picture by S. Gunturk and I. Daubechies

CHALLENGES AND OPPORTUNITIES FOR MATHEMATICS

Models and Simulations

Simulation and computing have become an essential part of modern science and engineering, complementing theory and experimentation. This development, driven by access to ever more powerful computers, poses new challenges to the mathematical sciences.
At the heart of any simulation is a model, a mathematical formulation that captures the structure and form of real-world phenomena. This model is typically surrounded by numerical techniques that produce quantitative information sometimes augmented by estimates of reliability, and by computer visualization that allows results to be manipulated and summarized, leading to qualitative understanding that inspires further analysis. Each of these elements requires sophisticated mathematics.
In modeling complex and poorly understood phenomena, such as weather or the stock market, the modeler encounters data that are uncertain, inaccurate, inconsistent, incomplete, and/or insufficient to determine a solution. [MS] For example, numerous medical conditions produce very similar CAT scan images, yet doctors must decide on the most plausible diagnosis. Typically, the input data, the modeling process and the solution may contain random and systematic errors and thus need statistical and/or probablistic analysis. On the other hand, solving a problem exactly may be computationally intractable; finding approximate solutions that have sufficient accuracy is then another challenging task, requiring a careful mathematical analysis. The resultant uncertainty assessment provides a mechanism to drive confidence in the validity of the computed solution [GL]. For the simplest cases in which phenomena are nearly linear, powerful mathematical and computational tools have been developed over the last 50 years to estimate and deal with these various forms of uncertainty. Genuinely nonlinear problems, however, are still far beyond reach [MS].

picture by J.Glimm

A particular challenge is posed by "model reduction". Technology relies increasingly on models that describe huge systems on fine scales in length and time while data and conclusions may be concentrated at coarser scales only. Unwieldy fine-scale equations need to be replaced with effective coarse-scale descriptions in order to provide more accurate predictions when computation and analysis on the fine scale is totally unfeasible. Stochastic modeling can also be used to provide a huge reduction in complexity. The challenge to the mathematical sciences is to develop systematic approaches to bridge these levels of organization. Important examples include global climate change, environmental remediation, computer-assisted design for manufactured products, neuroscience, and chemical kinetics of drug design [GL]. Similar organization and reduction are needed in settings where the multiple scales do not result from physical scaling in time or length, but are discrete or hierarchical, such as in multilayered descriptions of large networks [MS].
In many of these areas, we have made progress in the past, but we still need many orders of magnitude improvement before we can tackle truly realistic problems. These activities require substantial effort in collaboration between mathematical and other scientists as well as novel mathematical insight to create new techniques that allow qualitative insights needed to make further breakthroughs.

Computing with Large Data Sets

Breakthroughs in sensor technology are leading to generation of unprecedented volumes of high-dimensional data. The amazing continued growth in the raw power of computers and algorithmic advances offer the promise of addressing the tremendous challenges of analyzing such data. Huge data sets, terabytes of data, in very high dimensional spaces, are now collected routinely in almost all sciences [MS]. Examples are images from diverse sources, dynamics of the internet, neural recordings and, in the discrete domain, gene expression arrays. Whereas data sets in 1,2 or 3 dimensions are easily visualized and analyzed, data sets in 1000 dimensions are much harder to understand. Even 10,000 points constitute a very sparse set in 1000-dimensional space and it is easy to "overfit" the data with a model that makes you detect spurious "patterns" that disappear when you acquire more data. A major challenge is to find methods to analyze the structure of such sets, to fit models robustly and identify and validate patterns. Techniques from statistics, harmonic analysis, graph theory and computer science are only beginning to clarify this problem.

Geometrization of Topology and Physics

The interaction between geometry, topology, physics, and cosmology continues to enrich all these disciplines despite (or perhaps because of) their apparently different goals.
One current area of excitement is "mirror symmetry," which relates certain six-dimensional manifolds that arise in string-theoretic physics to each other [TA]. The spaces in question are not new -- their structure and properties were explored by differential geometers some years ago, and this work was essential for the new developments in string theory. However the notion of mirror symmetry is new -- it was revealed by the use of these spaces in physics. Now the task of understanding mirror symmetry is leading to entirely new mathematics. Efforts to determine the origin of this symmetry, its intrinsic structure, its implications, and especially its generalizations are leading to fundamental questions such as: counting the solutions of certain polynomial and differential equations; exploring certain relations between symmetries of the circle, of matrices, and of higher-dimensional manifolds; and exploiting a new correspondence between solutions of certain equations involving few parameters and solutions of other equations involving many parameters.
Another currently exciting area is the study of four-dimensional manifolds [ST, TA]. Here mathematicians have constructed inexorably complicated families of four-dimensional manifolds that are distinguished by invariants derived from the study of the relationship between gravity and the other physical forces. We have only a small glimpse of the richness of four-dimensional manifolds. As this interplay deepens new mathematics and physics will be developed.

Noise

Noise and randomness are ubiquitous. The correspondence between random walks and diffusion differential equations has provided fertile territory for mathematical analysis, and for applications such as stochastic control, filtering, and predicting the likelihood of rare but catastrophic events. We have become adept at dealing with random perturbations of finite dimensional systems, described by ordinary differential equations. In contrast, the analysis of similar issues for infinite dimensional systems (those described by partial differential equations) is in its infancy. Learning how to deal with them is essential to our understanding of the consequences of uncertainty, imperfection, and thermal fluctuations in physical systems [KO,GL]. By analogy with the existing theory, it will involve random walks in and diffusion differential equations on infinite-dimensional spaces.
Related issues of infinite-dimensional analysis arise in the task of putting realistic quantum field theories on a mathematically sound foundation. Fresh insight in this area is emerging from links between string theoretic physics, topology, and geometry.
A different mandate for infinite-dimensional analysis comes from today's massive data sets, which must typically be interpreted using models with large numbers of parameters. Infinite dimensional approximations provide one approach to get a handle on the behavior of statistical methods in the limit of increasingly large data sets and models [BI, BO].
Most of these infinite-dimensional problems defeat us at present; gaining better insight would have extraordinary pay-off. What we can glimpse already has spectacular ramifications.

Nonlinearity

Understanding nonlinearity is a central challenge for modern science. This is a huge task, since there are many sources and many sorts of nonlinearity. Sometimes its origin is the behavior being modeled: physical examples include combustion, phase transformation, and turbulence; biological examples include protein folding and excitable tissues, such as heart muscle and the nervous system. Nonlinearity can also come from other, more structural sources such as feedback or geometry: examples include the optimization of financial decisions; the pinch-off of fluid droplets; and the motion of surfaces under curvature-driven flows.
We have accumulated great insight into nonlinear phenomena, and a rich collection of viewpoints and methods, such as asymptotic analysis, bifurcation, chaos, shock waves, and viscosity solutions, to name but a few. Our powerful and ever-expanding collection of tools provides many opportunities for advancement. For example, new methods from the calculus of variations are giving fresh insight into the design of composites with optimal microstructures, the consequences of polycrystalline structure in shape-memory materials, and the arrangement of domain walls in magnetic materials [KO]. Another example: nonlinear evolution equations adapted from geometry and fluid dynamics are being used for the creation, manipulation, smoothing, and segmentation of visual images [CH,MU,EV]. A third example: dynamical systems methods are providing insight into the sources and consequences of spatio-temporal patterns, including fibrillation of heart tissue and rhythmic electrical activity in the nervous system [KL].
And yet our mastery of nonlinearity has barely begun. Many fundamental questions remain open, for example: Are solutions of the Navier-Stokes for fluid mechanics equations unique in three space dimensions? What about (viscosity) solutions of systems of hyperbolic conservation laws? These questions are mathematical challenges of course, but their importance goes much deeper. They address the adequacy of our standard mathematical models for fluid mechanics and gas dynamics. If solutions were not unique -- which would be a big surprise -- then some effect we usually ignore would actually be crucial for determining the physically correct solution. If, as seems more likely, solutions are unique, then the methods developed to prove this assertion will also give insight concerning qualitative features of flows and the accuracy and stability of numerical solution schemes. Recent breakthroughs on uniqueness for conservation laws makes this topic particularly timely [EV].

Beyond Fermat

One of the most widely publicized mathematical results of the last few years is Wiles' proof of Fermat's last theorem : that the sum of two nth powers can not be another nth power if n>2. This had been an unsolved problem for 350 years, and that reason alone would make it of great interest. But mathematicians see it as one result in a huge web of connected conjectures, some proven and some supported by extensive computations which together link arithmetic, geometry, analysis, group theory and even physics. These conjectures, and the powerful methods that are being developed to deal with them, have been some of the most exciting themes in fundamental mathematics in the last 50 years. Among these is the ABC conjecture which generalizes Fermat's theorem and expresses a fundamental tension between addition and multiplication: If A and B are any two numbers with many repeated large prime factors, then their sum cannot have many repeated large prime factors (here "many" can be made precise). Other conjectures relate to the distribution of prime numbers, 2,3,5,7,11, .... This tantalizingly irregular sequence is encoded in a kind of generating function, the Riemann zeta function, whose analytic properties express the hidden patterns of the primes in a way which is worked out in this complex of ideas. The Riemann zeta function is the paradigm of a class of functions whose analytic properties are expected to shed light on such classical Diophantine questions as: how many rational solutions does a (given) cubic polynomial in two variables have? This vision connects to the theory of group representations, to the theory of transcendental numbers, and to the study and classification of polynomial equations. The study of polynomial equations and the locus of their zeroes is nowadays being energetically furthered by methods from physics, partial differential equations, and differential geometry. It is even expected that work in this area will shed light on the structure of finite simple groups (the "Monster group" in particular).
The breadth and unity of this development is hard to exaggerate; it makes for a particularly interesting time in number theory, with many challenges ahead.

Mathematics for Biology and Medicine

Biology and medicine are poised to climb the twin peaks of understanding the life process and greatly improving human health. Success of these efforts depends on the increasing involvement of the mathematical sciences. Combinatorics and statistics have already made essential contributions to the rapid and successful sequencing of the human genome. Obtaining the DNA sequence is just the first step in computing the secrets of life; decoding the meaning of the sequence poses serious long-term scientific and mathematical challenges. We must design new methods of exploration and analysis that combine the power and precision of both biology and mathematics. At the cellular level, we must find both the genes which encode for currently unknown proteins and the non-coding DNA regions which regulate the expression of these new genes. This step will involve pattern recognition, signal processing, database mining, statistical methods both exploratory and confirmatory, and new biomathematical cryptology techniques many of which have yet to be invented. After discovering a new protein, we must determine its function, regulation, and role in a cascade network of interaction with other proteins. This will involve comparison of characteristics of the unknown protein to those of known proteins, both human and non-human. This comparison involves measurement of similarity of DNA sequences (coding and regulatory), similarity of spatial geometry of native folded states, similarity of regulation patterns using microarray chips, and the establishment of cause and effect relationships between the many actors in these processes. New mathematical and statistical ideas are required for this step: we need new biologically relevant similarity measures of 1D sequences, 3D geometry, clustering algorithms and other statistical analysis methods for expression vectors of very high dimension. At the macroscopic level, a fundamental problem in medicine is to define and understand ranges of parameters which correspond to "normal" anatomy and function for an organ, a necessary first step before understanding and treating the "abnormal" . Due to high variability, comparison of anatomical and functional information across individuals and groups requires new mathematical and statistical models as well as highly sophisticated computational algorithms. A goal in medicine is to have normal organs and organ systems "in silico", allowing non-invasive computational comparison between subject and template, enhancing detection of disease states and design of treatment. The mathematical sciences have played and will continue to play an essential role in the team effort to understand the complexities of biology and harness this understanding to improve human health.

Information Technology

Stunning advances in information technology have not only profoundly changed the way we live and the way we think, but also offer major challenges for the mathematical sciences in partnership with computer science. Because the basic structures in information technology are discrete rather than continuous, new approaches in combinatorics, logic and statistical modeling are necessary to cope with problems that are growing daily in complexity and size. Just as physics was a motivating force for many mathematical fields in the past, information technology will act as an engine driving the development of new mathematics in the future. Much of today's computing is increasingly interactive, distributed, and heterogeneous---properties that exacerbate the inherent difficulties of maintaining security, privacy, and speed. Communication networks, wired and wireless, span the globe and are being built at a dizzying pace; their ability to function depends on understanding very large, complex graphs. Despite progress in analyzing the efficiency, performance, and tractability of algorithms, very little can be said about problems for which only partial information is available or the input sizes far exceed available memory. Software is everywhere, yet mathematical techniques for assessing, improving, and ensuring its reliability fall far short of the needs. There is work on the table!
The consistent spectacular growth in raw computing power during the past 30 years is common knowledge. It is less widely known but equally significant that advances in numerical algorithms have matched, and often exceeded, that pace. Solving larger, more nonlinear, more complex problems will continue to require smarter, more sophisticated mathematics to create new algorithms. For example, large-scale optimization, continuous and discrete, has had a significant impact in a wide range of areas, including manufacturing, scheduling, routing, real-time control of physical systems, and Web caching; however, techniques for handling the important class of nonlinear mixed integer-continuous problems are still in their infancy in terms of analysis and algorithms.

picture from Applegate, Bixby, Chvatal and Cook
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The familiar model of sequential computation has been expanded in concept by parallel, heterogeneous, and distributed computing. But this is not the only conceivable form of computation. The field of quantum computing, the subject of intense recent publicity, is emerging as a rich source of mathematical inquiry into the nature of computation and communication. From another domain, the processing and transformation of information within biological organisms offer a rich opportunity for new mathematical paradigms of computing. The associated mathematics would provide a deeper understanding in biology, including the neurosciences, and molecular- and population-level systems.

These eight paragraphs illustrate the diversity and intellectual excitement characteristic of mathematics. They are representative of a much richer web of mathematical challenges and accomplishments not presented here. Even in this set of examples, linkages among topics indicate the unity and inseparability of this research enterprise. For example, nonlinearity is a profound challenge within simulations. Large data sets and infinite dimensional stochastic processes are intrinsically linked and both are connected to model reduction, inverse problems and uncertainty. Similarly connections between number theoretic questions and the geometrization of topology and quantum physics are impressive.
The many new exciting results within mathematical discipline as well as the striking connections between them, and the important opportunities offered by these developments and by the challenges to mathematics posed by the other sciences, make this an especially exciting time for researchers in the mathematical sciences, who look forward to the adventure.