U.S. participation in "O-minimal Structures and Real Analytic Geometry"

miller和Keyfitz提议资助美国数学家和正在接受培训的数学家参加将于2009年1月至6月在菲尔兹数学科学研究所（多伦多，安大略省，加拿大）举行的“o -极小结构和真实解析几何”专题项目。这个项目的智力价值如下。许多所谓的经典数学的结果是非常普遍的：它们适用于广泛的输入，可以这么说，因此往往产生广泛的输出。但是我们可以希望，如果输入在某些方面表现得特别好，那么输出也会同样表现得很好。这在许多重要的情况下被证明是正确的，但通常需要对经典结果进行新的、更具建设性的证明，以及对哪些输入应该被视为表现良好的更深入的理解。实域0极小结构理论是数理逻辑的一个分支学科，在很大程度上是为了解决这个问题而发展起来的。在过去的25年里，这是一个快速发展的领域，来自数学和逻辑的几个分支，特别是模型论和实解析几何的研究人员做出了许多贡献和合作。在理论经济学、神经网络学习理论、混合控制系统以及纯数学等不同领域都有应用。项目的广泛影响主要如下。该计划的一个重要目标是向美国数学家和正在培训的数学家介绍或阐述该计划的主要主题，因为o-极小性和实解析几何的结合目前在美国尚未得到广泛实践。NSF资助的直接影响将是培训和推进大量美国研究人员的研究，特别是初级教师。博士后和研究生——将有机会参与该项目。

ABSTRACTMiller 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.