Opportunities for the Mathematical Sciences

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Table of Contents
Preface
Summary Article
  Introduction
  Models and Simulations
  Computing with Large Data Sets
  Geometrization of Topology and Physics
  Noise and Randomness
  Nonlinearity
  Beyond Fermat
  Mathematics for Biology and Medicine
  Information Technology
Individual Contributions
List of Contributors with Affiliations


Summary Article

Mathematics -- The Science of Patterns and Algorithms

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.

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 disciplines, 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.

 

Last Modified:
Jan 24, 2013
 

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