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From page 93... ...
Recommendations on the necessary adjustments are included as appropriate. INCREASING IMPORTANCE OF CONNECTIONS FOR MATHEMATICAL SCIENCES RESEARCH Based on testimony received at its meetings, conference calls with leading researchers (see Appendix B)

From page 94... ...
It also suggests what has long been the experience of leading mathematicians: that the various subfields in mathematics depend on one another in ways that are unpredictable but almost inevitable, and so more individuals need to collaborate in order to bring all the necessary expertise to bear on today's problems. While some collaborative work involves mathematical scientists of similar backgrounds joining forces to attack a problem of common interest, in other cases the collaborators bring complementary backgrounds.

From page 95... ...
Then there is mirror symmetry, discovered by physicists, which, in each original formulation, led to the solution of one of the classical enumerative problems in algebraic geometry, the number of rational curves of a given degree in a quintic hypersurface in projective 4space. This has been expanded conjecturally to a vast theory relating complex manifolds and symplectic manifolds.

From page 96... ...
Example 3: Kinetic Theory Kinetic theory is a good example of the interaction between areas of the mathematical sciences that have traditionally been seen as core and those that had been seen as applied. The theory was proposed by M axwell and Boltzmann to describe the evolution of rarefied gases (not dense enough to be considered a "flow," not dispersed enough to be just a system of particles, a dynamical system)

From page 97... ...
Because of these exciting opportunities that span multiple fields of the mathematical sciences, the amount of technical background needed by researchers is increasing. Education is never complete today, and in some areas older mathematicians may make more breakthroughs than in the past because so much additional knowledge is needed to work at the frontier.

From page 98... ...
But to make this happen, improved mechanisms for connecting mathematical scientists with potential collaborators are needed, such as research programs that bring mathematical scientists and collaborators together in joint groups. Such collaborations work best when the entire team shares one primary goal  such as addressing a question from biology  even for the team members who are not biologists per se.

From page 99... ...
In the last 20 years we have seen new institutes appear in Japan, England, Ireland, Canada, and Mexico, to name just a few countries, joining older institutes in France, Germany, and Brazil. Overall, there are now some 50 mathematical science institutes in 24 different countries.3 These institutes have made it easier for mathematical scientists to form and work in collaborative teams that bridge two or more fields or that connect the mathematical sciences to another discipline.

From page 100... ...
Some illustrations of the impact of mathematical science institutes follow.5 To help the mathematical sciences build connections, the IMA reaches out so that some 40 percent of the participants in its programs come from 4 Thesegoals are explicit for the NSFsupported institutes. 5 Thecommittee thanks IMA director Fadil Santosa, IPAM director Russel Caflisch, SAMSI director Richard Smith, and MSRI director Robert Bryant for helpful inputs to this section.

From page 101... ...
In 2004 IPAM held the workshop Automorphic Forms, Group Theory, and Graph Expansion, which was followed by a program on the subject at the Institute for Advanced Study in 2005 and a second IPAM workshop, Expanders in Pure and Applied Mathematics, in 2008. Similarly, the Statistical and Applied Mathematical Sciences Institute (SAMSI)

From page 102... ...
For example, its 2006 program Computational Aspects of Algebraic Topology explored ways in which the techniques of algebraic topology are being applied in various contexts related to data analysis, object recognition, discrete and computational geometry, combinatorics, algorithms, and distributed computing. That program included a workshop focused on application of topology in science and engineering, which brought together people working in problems ranging from protein docking, robotics, highdimensional data sets, and sensor networks.

From page 103... ...
These new tools have profoundly changed both the modes of collaboration and the ease with which mathematical scientists can work across fields. The existence of arXiv has had a major influence on scholarly communication in the mathematical sciences, and it will probably become

From page 104... ...
One striking instance of this globalization of the mathematical sciences is the first "polymath" projects, which were launched in 2009.7 To quote from T erence Tao, these "are massively collaborative mathematical research p rojects, completely open for any interested mathematician to drop in, make some observations on the problem at hand, and discuss them with the other participants."8 Another recent phenomenon is global review of emerging ideas. Not only do such projects contribute to advancing research, but they also serve to locate other researchers with the same interest and with the right kind of expertise; they represent an ideal vehicle for expanding personal collaborative networks.

From page 105... ...
This cycle is extremely healthy. A recent paper9 evaluated an apparent shift in collaborative behavior within the mathematical sciences in the mid1990s.

From page 106... ...
For example, mathematical scientists collaborate with astrophysicists, neuroscientists, or materials scientists to develop new models and their computationally feasible instantiations in software in order to simulate complex phenomenology. The mathematical sciences can obviously contribute to creating mathematical and statistical models.

From page 107... ...
Some mathematical sciences research would benefit from the most advanced computing resources, and not many mathematical scientists are currently exploiting those capabilities. Because the nature and scope of computation is continually changing, there is a need for a mechanism to ensure that mathematical sciences researchers have access to computing power at an appropriate scale.

From page 108... ...
And although many agencies that deal with national security, intelligence, and financial regulation rely on sophisticated computer simulations and complex data analyses, only a fraction of them are closely connected with mathematical scientists. As a result, the mathematical sciences are unable to contribute optimally to the full range of needs within these agencies, and the discipline  especially its core areas  is as a result overly dependent on NSF.

From page 109... ...
Available at http://www.commonfund.org/CommonFundInstitute/HEPI/HEPI%20Documents/2011/ CF_HEPI_2011_FINAL.pdf. Acronyms: AFOSR, Air Force Office of Scientific Research; ARO, Army Research Office; DARPA, Defense Advanced Research Projects Agency; MICS, Mathematical, Information, and Computational Sciences Division; NIBIB, National Institute of Biomedical Imaging and Bioengineering; NIGMS, National Institute of General Medical Sciences; NSA, National Security Agency; ONR, Office of Naval Research; SciDAC, Scientific Discovery Through Advanced Computing.

From page 110... ...
increases in the following 2 years.11 DMS is faced with an innate conflict: As the primary funding unit charged with maintaining the health of the mathematical sciences, it is naturally driven to aid the expansions discussed in Chapter 3; yet it is also the largest of a very few sources whose mission includes supporting the foundations of the discipline, and thus it plays an essential role with respect to those foundations. As noted in Chapter 3, some mathematical scientists receive research support from other parts of NSF and from nonmath units in other federal funding agencies, but there are only anecdotal accounts of this.

From page 111... ...
• The education of future generations of mathematical scientists, and of all who take mathematical sciences coursework as part of their preparation for science, engineering, and teaching careers, should be reassessed in light of the emerging interplay between the math ematical sciences and many other disciplines. • Institutions, for example, the funding mechanisms and reward systems  should be adjusted to enable crossdisciplinary careers when they are appropriate.

From page 112... ...
• Mathematical scientists should be included more often on the p anels that design and award interdisciplinary grant programs. Because so much of today's science and engineering builds on advances in the mathematical sciences, the success and even the validity of many projects depends on the early involvement of mathematical scientists.

From page 113... ...
This is not surprising  when a new idea in core mathematics first sees the light of day, researchers in other fields do not know how to make use of it, and by the time the new idea has made its way into wider use, it is no longer new. The overwhelming impression one comes away with from reading these journals is what an explosively creative golden age of science we are living through and just how central a role the mathematical sciences are playing in making this possible.

From page 114... ...
With the advent of digital images, the question of how to analyze them  to get rid of noise and blurring, to segment them into meaningful pieces, to figure out what objects they contain, to recognize both specific classes of objects such as faces and to identify individual people or places  poses remarkably interesting mathematical and statistical problems. Core mathematicians are aware of the extraordinary work of Fields medalist David Mumford in algebraic geometry, but many may be unaware of his seminal work in image segmentation (the Mumford Shah algorithm, for example)

From page 115... ...
A sampling of these uses, described in nontechnical language, can be found in the companion volume to this report, Fueling Innovation and Discovery: The Mathematical Sci ences in the 21st Century. Whether or not one gets directly involved in these developments, it would be very useful to the profession if core mathematicians were to increase their level of awareness of what is going on out there.

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