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和罗宾逊建议在两个不同但相关的方向上继续他们的合作研究：具有非阿基米德（超度量）绝对值的场的解析几何，以及具有解析结构的超度量场的模型理论（即，次解析集理论（Theory of Subanalytic Sets）。经典的仿射刚性解析几何是基于严格收敛的幂级数环-幂级数收敛于“封闭”圆盘的乘积。 提出者引入了收敛于“闭”和“开”圆盘的乘积的幂级数的新环。 这些环的分离幂级数共享许多理想的代数性质的较小的环严格收敛的幂级数。 Lipshitz和罗宾逊提出在这些环上发展局部和全局刚性几何，类似于经典的情况。 这将代表经典理论在“开放”多圆盘上的相对刚性几何的情况下的延伸。 在模型理论方面，Lipshitz和罗宾逊提出通过证明一个均匀化和一个奇异轨迹定理来完成他们基于分离幂级数环的刚性次解析集理论的发展。 他们还建议继续发展基于严格收敛幂级数环的刚性子分析集理论。所用的方法来自于模型论、交换代数和代数几何。 由解析函数之间的方程组和不等式定义的真实的空间中的点的集合称为解析集。 投影（即，这样的集合在低维子空间上的阴影被称为次解析的。 真实的次解析集出现在数学的几个分支中，如微分方程和几何。 类似的集合类自然出现，例如在数论中，在不同于真实的数的域上，距离的概念具有相当不同的性质。 然而，这些集合与它们的真实的表亲有许多共同的好特性。 Lipshitz，罗宾逊和他们的学生将继续他们的调查性质，这些集，使用的方法，从数理逻辑，交换代数和代数几何。