Mathematical Sciences: Model Theory and Rigid Analytic Geometry

数学科学：模型论和刚性解析几何

• 批准号: 9704981
• 负责人: Leonard Lipshitz
• 金额: $ 15.9万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704981&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Model; Theory; Rigid

Lipshitz and Robinson propose to continue their collaborative investigation in two distinct but related directions: analytic geometry over fields with a non-Archimedean (ultrametric) absolute value, and the model theory of ultrametric fields with analytic structure (i.e., the theory of subanalytic sets). Classical affinoid rigid analytic geometry is based on rings of strictly convergent power series-power series convergent on products of "closed" discs. The proposers have introduced new rings of power series convergent on products of "closed" and "open" discs. These rings of separated power series share many of the desirable algebraic properties of the smaller rings of strictly convergent power series. Lipshitz and Robinson propose to develop local and global rigid geometry over these rings, in analogy to the classical case. This will represent an extension of the classical theory to the case of relative rigid geometry over "open" polydiscs. On the model theory side, Lipshitz and Robinson propose to complete their development of the theory of rigid subanalytic sets based on the rings of separated power series by proving a Uniformization and a Singular Locus Theorem. They also propose to continue their development of the theory of rigid subanalytic sets based on the rings of strictly convergent power series. The methods employed come from model theory, commutative algebra and algebraic geometry. The sets of points in real space defined by systems of equations and inequalities among analytic functions are called analytic sets. The projection (i.e., the shadow) of such a set on a lower dimensional subspace is called subanalytic. Real subanalytic sets arise in several branches of mathematics such as differential equations and geometry. Similar classes of sets arise naturally, for example in number theory, over fields different from the real numbers, where the notion of distance has rather different properties. These sets, however, share many of the nice properties of their real cousins. Lipshitz, Robinson and their students will continue their investigation of the properties of these sets, using methods from mathematical logic, commutative algebra and algebraic geometry.

Lipshitz和Robinson提议以两个不同但相关的方向继续其协作研究：具有非架构的绝对价值（超级）绝对价值的田地上的分析几何形状，以及具有分析结构的超级领域的模型理论（即，亚分析集的理论，是亚分析集的理论）。经典的affinoid刚性分析几何形状基于严格收敛功率系列功率系列的环，收敛于“封闭”盘的产品。 提议者已经在“封闭”和“开放”碟片的产品上推出了电源系列的新戒指。 这些分离的功率系列的环具有严格收敛功率系列的较小环的许多理想代数性能。 Lipshitz和Robinson提议与经典案例相比，在这些环上开发局部和全球刚性几何形状。 这将代表经典理论的扩展到“开放” polydiscs的相对刚性几何形状的情况。 在模型理论方面，Lipshitz和Robinson提议通过证明统一和奇异的基因座定理来基于分离功率序列的环的刚性亚分析集理论的发展。 他们还建议根据严格收敛的功率系列的环继续发展刚性亚分析集的理论。所采用的方法来自模型理论，交换代数和代数几何形状。 由方程系统定义的实际空间中的点集和分析功能之间的不平等现象称为分析集。 在较低维子空间上的这种集合的投影（即阴影）称为亚分析。 实际的亚分析集发生在数学的几个分支，例如微分方程和几何形状。 类似的集合类似的类别自然出现，例如在数字理论中，与实数不同的字段，距离的概念具有相当不同的属性。 但是，这些集合分享了他们真正表亲的许多不错的特性。 Lipshitz，Robinson及其学生将使用数学逻辑，交换代数和代数几何形状中的方法继续研究这些集合的特性。