Renewing U.S. Mathematics A Plan for the 1990s (1990) / Chapter Skim
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3 Research Progress and Prospects
3 Research Progress and Prospects
Pages 39-54
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From page 39... ...
Whether motivated by the practical problem of blood flow in the heart or by the abstraction of aspects of number theory, the mathematical scientist seeks patterns in order to describe them, relate them, and extrapolate from them. In part, the quest of mathematics is a quest for simplicity, for distilling patterns to their essence.
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From page 40... ...
The pace was so rapid that specialization increased, and for a while it seemed that topologists, algebraic geometers, analysts, and other groups of mathematical scientists could barely speak to one another. Each was inventing powerful new mathematical structures to unify previously disparate ideas and shed new light on classical problems.
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From page 41... ...
which contains moredetailed, brief descriptions of 27 important research areas that have produced significant accomplishments in recent years and that offer opportunities for further research. The committee emphasizes that the achievements anal opportunities discussed in Appendix B are not intended to be comprehensive, nor are they intended to suggest a specific agenda for funding research in the mathematical sciences.
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From page 42... ...
Computers have profoundly influenced the mathematical sciences themselves, not only in facilitating mathematical research, but also in unearthing challenging new mathematical questions. Many of the research advances described in this chapter and in Appendix B would not have been possible without computers and the associated mathematics that is concurrently being developed.
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From page 43... ...
computer modeling plays a major role in the development of critical technologies such as the fabrication of microelectronic circuits and the understanding of fluid flow. Developing appropriate simulations for a given technology invariably involves a high degree of scientific knowledge as well as sophisticated mathematical tools to describe and evaluate the model.
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From page 44... ...
Recent Advances in Partial Differential Equations 2. Vortices in Fluid Flow 3.
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From page 45... ...
For instance, developments in such core areas as number theory, algebra, geometry, harmonic analysis, dynamical systems, differential equations, and graph theory (see, for instance, Sections 1, 7, 8, 9, 10, 11, 12, and 19 in Appendix By not only have significant applications but also are themselves influenced by developments outside of core mathematics. The Living World The mathematical and the life sciences have a long history of interaction, but in recent years the character of that interaction has seen some fundamental changes.
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From page 46... ...
In particular, computer models of the human heart have led to improved design of artificial heart valves. Mathematical methods were fundamental to the development of medical imaging techniques, including CAT scans, magnetic resonance imaging, and emission tomography (Section 5~.
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From page 47... ...
The equations of fluid dynamics fit into the broader class of partial differential equations, which have historically formed the main tie between mathematics and physics. Global climate change is a topic of intense debate, and greater quantitative understanding through the use of mathematical modeling and spatial statistics techniques would greatly help in assessing the dangers and making reliable predictions (see Sections 6 and 17~.
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From page 48... ...
The study of chaotic dynamics, which employs a range of mathematical tools, has demonstrated that unpredictable behavior can arise from even the simplest deterministic systems and has been used to describe diverse phenomena, such as the interfaces between fluids (see Section 7~. Investigation of quasicrystals, a category of matter combining properties of crystals and glasses, utilizes the mathematical theory of tiling, which describes ways of fitting geometrical figures together to cover space.
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From page 49... ...
Meanwhile, other statisticians are assisting computer science by developing statistical techniques for characterizing and improving software reliability. As the language of computer modeling, mathematics is revolutionizing the practice of science and engineering.
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From page 50... ...
Because communications signals are sometimes modeled as higher-dimensional spheres, this result will help in improving the efficiency and quality of transComputer graphics have opened a whole world of visualization techniques that allow mathematicians to see, rotate, manipulate, and investigate properties of abstract surfaces. In particular, the subject of the mathematical properties of soap films "minimal surfaces" that are visible analogues of the solutions to optimization problems in many fields has witnessed a recent breakthrough and a resurgence of interest because of computer modeling, as described in Section 13.
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From page 51... ...
The discipline must move forward on all of its many fronts, for its strength lies in its diversity. THE PRODUCTION OF NEW MATHEMATICS It is enlightening to note how so many of the research topics mentioned here and in Appendix B for instance, the developments in partial differential equations, vortices, aircraft design, physiology, and global change—have developed out of the mathematical research described in the 1984 Report, both in its body and in Arthur Jaffe's appendix, "Ordering the Universe: The Role of Mathematics." The recent developments in number theory and geometry follow naturally from the Mordell conjecture, which was featured prominently in the 1984 Report.
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From page 52... ...
The continued production of valuable new mathematics requires not only that the number of individual investigators be increased as recommended in the 1984 National Plan, but also that the entire field reach out to the rest of the scientific world. Collaborating with researchers in other fields and making an effort to understand applications and improve the mathematical sophistication of others help the mathematical sciences become increasingly robust and valuable.
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From page 53... ...
These include the BMS advisory panel reports (to the Air Force in 1987 and the Navy in 1987 and 1990) mentioned in Chapter 2; the American Statistical Association report Challenges for the '90s, listed in Chapter 2; and Operations Research: The Next Decade ("CONDOR report")
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