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

米勒和凯菲茨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月在加拿大安大略省多伦多的菲尔兹数学科学研究所举行的主题项目“O-极小结构和真实解析几何”。该项目的智力价值如下：所谓古典数学的许多结果是非常普遍的：可以说，它们适用于广泛的输入，因此往往会产生广泛的输出。但人们可能希望，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。事实证明，这在许多重要情况下都是正确的，但通常需要对经典结果进行新的、更具建设性的证明，以及对哪些投入应该被视为良好行为的更深层次的理解。作为数理逻辑的一个分支，实域上的o-极小结构理论在很大程度上就是为了解决这个问题而发展起来的。在过去的25年里，这是一个迅速发展的领域，来自数学和逻辑，特别是模型理论和真正的解析几何的几个分支的研究人员做出了许多贡献，并进行了合作。在理论经济学、神经网络学习理论和混合控制系统以及纯数学等领域都有应用。该项目的更广泛影响主要如下。该计划的一个重要目标是向美国数学家和正在接受培训的数学家介绍或阐述该计划的主要主题，因为o-极小和实解析几何的结合在美国目前并未得到广泛应用。NSF资金的直接影响将是培训和推进大量美国研究人员--特别是初级教师、博士后研究员和研究生--他们将有机会参与该计划。

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.