The Model Theory of Valued Fields with Analytic Structure

Lipshitz和Robinson建议继续合作研究具有解析结构的值场模型理论。经典刚性解析几何是基于严格收敛幂级数的环，即。，幂级数收敛于“闭”盘的积。提出了收敛于“闭”盘和“开”盘积上的幂级数环。这些分离幂级数的环具有严格收敛幂级数的小环所具有的许多代数性质。此外，它们特别适合于模型理论应用。Lipshitz和Robinson建议继续发展分离幂级数环的交换代数和相应的刚性几何类比于经典情况。在模型理论方面，他们提出扩大研究范围，考虑具有解析结构的非代数闭值域的模型理论，并考虑刚性亚解析集理论中不同域上的一致性问题。所采用的方法来自于模型论、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实数域上欧几里德空间中的点的集合称为半解析集。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（即影子）称为子解析集。子解析集比半解析集多，其行为也更为复杂。人们对子分析集很感兴趣。从形式化的意义上讲，这些集合可以在数学上由半解析集合定义。此外，实次解析集出现在数学的几个分支中，如微分方程和几何。类似的集合自然出现，例如在数论中，在不同于实数的域上，距离的概念具有相当不同的性质。这样的领域被称为非阿基米德。然而，相应的子分析集却拥有它们真正的同类的许多优良性质。Lipshitz和Robinson将使用数理逻辑、交换代数和代数几何的方法，继续研究这些非阿基米德子解析集的性质。在发展了非阿基米德理论的关键要素之后，他们打算运用他们的思想来扩展这些结果所适用的领域类别。特别是，他们计划将在非阿基米德背景下发展的思想应用到实际情况中，从而扩大实数集合的类别，这些实数集合的良好几何性质可以通过这些方法建立起来。由于用于提取这些集合的几何信息的许多程序在不同的场之间并没有变化，因此我们还计划仔细研究这种一致性的性质。

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.