The Model Theory of Valued Fields with Analytic Structure

Lipshitz和罗宾逊建议继续他们的合作研究，研究具有解析结构的值域的模型理论。 经典的刚性解析几何是基于严格收敛的幂级数环，即，幂级数收敛于“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。 此外，它们特别适合模型理论应用。Lipshitz和罗宾逊建议继续发展交换代数环的分离幂级数和相应的刚性几何类似的经典情况。 在模型理论方面，他们建议扩大他们的调查，以考虑模型理论的非代数闭值领域的分析结构，并考虑问题的一致性在不同领域的理论刚性subanalytic集。 所用的方法来自模型论、交换代数和代数几何，由解析函数间的方程组和不等式所定义的真实的数域上的欧氏空间中的点集称为半解析集。这类集合是解析几何的基础。半解析集在低维子空间上的投影（即阴影）称为次解析集。亚解析集比半解析集多，其行为也更复杂。有一个自然的兴趣在subanalytic集。在形式逻辑的意义上，这些正是可以从半解析集合中数学定义的集合。此外，真实的子解析集出现在数学的几个分支，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。这种域称为非阿基米德域。然而，相应的子解析集与它们的真实的表亲有许多共同的好性质。Lipshitz和罗宾逊将继续他们的调查这些非阿基米德次解析集的性质，使用的方法从数理逻辑，交换代数和代数几何。在非阿基米德的情况下发展了理论的关键要素，他们提出应用他们的想法来扩展这些结果适用的领域。 特别是，他们计划将在非阿基米德设置开发的想法应用到真实的情况下，从而扩大类的真实的集，其良好的几何性质可以建立这些measure.As的许多程序用于提取几何信息aboutthese套不因领域而异，他们还计划仔细研究这种均匀性的性质。

Lipshitz and Robinson propose to continue their collaborative investigationinto the model theory of valued fields with analytic structure. Classical rigid analytic geometry is based on rings of strictly convergent power series,i.e., 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. In addition they are particularly well suited for model theoretic applications. Lipshitz and Robinson propose to continue to develop the commutative algebra of rings of separated power series and the corresponding rigid geometry in analogy to the classical case. On the model theory side they propose to broaden their investigation to consider the model theory of non-algebraically-closed valued fields with analytic structure and also to consider questions of uniformity over different fields in the theory of rigid subanalytic sets. The methods to be employed come from model theory, commutative algebra and algebraic geometry.The sets of points in Euclidean space over the field of real numbers definedby systems of equations and inequalities among analytic functions are calledsemi-analytic sets. This class of sets is basic to analytic geometry. Theprojection (i.e. the shadow) of a semi-analytic set on a lower dimensionalsubspace is called subanalytic. There are more subanalytic sets than semi-analytic sets, and their behavior is more complicated. There is a natural interest in subanalytic sets. These are exactly the sets that can bemathematically defined from the semi-analytic sets, in the sense of formallogic. Furthermore, real subanalytic sets arise in several branches ofmathematics 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. Such fields are called non-Archimedean. The corresponding subanalytic sets, however, share many of the nice properties of their real cousins. Lipshitz and Robinson will continue their investigation of the properties of these non-Archimedean subanalytic sets, using methods from mathematical logic, commutative algebra andalgebraic geometry. Having developed key elements of theory in the Non-Archimedean case, they propose to apply their ideas to extend the classes of fields to whichthese results apply. In particular, they plan to apply ideas developed inthe non-Archimedean setting to the real case, thereby enlarging the class ofreal sets whose nice geometric properties can be established by these means.Since many of the procedures used to extract geometric information aboutthese sets do not vary from field to field, the also plan a careful study of thenature of this uniformity.