Miller and Keyfitz propose to help fund participation by U.S. mathematicians, and mathematicians in training, in the thematic program "O-minimal Structures and Real Analytic Geometry" to be held at the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Cananda) from January through June of 2009.
The intellectual merit of the project arises as follows. Many results of so-called classical mathematics are very general: They apply to a wide range of input, so to speak, and thus tend to produce a wide range of output. But one could hope that if the input is particularly well behaved in some respect, then the output would be similarly well behaved. This turns out to be true in many important cases, but usually requires new, more constructive, proofs of classical results, as well as a deeper understanding of which inputs should be regarded as well behaved. The theory of o-minimal structures on the real field, a sub-discipline of mathematical logic, has been developed in large part to deal with this issue. This has been a rapidly developing area for the last two and a half decades, with many contributions from, and cooperation between, researchers from several branches of mathematics and logic, in particular, model theory and real analytic geometry. Applications have been found in areas as diverse as theoretical economics, neural-net learning theory, and hybrid control systems, as well as in pure mathematics.
The broader impact of the project arises primarily as follows. It is an important goal of the program to introduce or elaborate the main themes of the program to U.S. mathematicians, and mathematicians in training, as the combination of o-minimality and real-analytic geometry is not currently widely practiced in the U.S. The direct impact of NSF funding will be the training and advancement of research of a significant number of U.S. researchers---especially junior faculty, postdoctoral fellows and graduate students---who will gain the opportunity to participate in the program.
