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

摘要Miller和Keyfitz建议资助美国数学家和培训数学家参加将于2009年1月至6月在菲尔兹数学科学研究所（多伦多，安大略，加拿大）举办的主题项目“O-极小结构和真实的解析几何”。所谓经典数学的许多结果都是非常普遍的：可以说，它们适用于大范围的输入，因此往往会产生大范围的输出。但是，如果输入在某些方面表现得特别好，那么输出也会表现得同样好。这在许多重要的情况下都是正确的，但通常需要对经典结果进行新的、更有建设性的证明，以及更深入地理解哪些输入应该被视为行为良好。真实的域上的o-极小结构理论是数理逻辑的一个分支学科，它的发展在很大程度上就是为了解决这个问题。这是一个迅速发展的领域，在过去的25年里，有许多贡献，并合作，研究人员从几个分支的数学和逻辑，特别是模型理论和真实的解析几何。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领域都有应用。该项目的广泛影响主要如下。这是该计划的一个重要目标，介绍或阐述该计划的主要主题，以美国数学家和数学家的培训，作为结合的o-极小和真正的解析几何是目前没有广泛实行在美国的NSF资助的直接影响将是培训和研究的一个显着数量的美国研究人员的进步-特别是初级教师，博士后研究员和研究生-谁将获得参与该计划的机会。

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.