Mathematical Sciences: Model Theory and Rigid Analytic Geometry

Lipshitz和Robinson建议在两个截然不同但相关的方向上继续他们的合作研究：非阿基米德（超度量）绝对值场的解析几何，以及具有解析结构的超度量场的模型理论（即，亚解析集理论）。经典仿射刚性解析几何是基于严格收敛幂级数的环-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。Lipshitz和Robinson建议在这些环上发展局部和全局刚性几何，类似于经典情况。这将代表经典理论在“开放”多盘的相对刚性几何情况下的扩展。在模型理论方面，Lipshitz和Robinson提出通过证明统一化定理和奇异轨迹定理来完成基于分离幂级数环的刚性亚解析集理论的发展。他们还提出在严格收敛幂级数环的基础上继续发展刚性次解析集理论。所采用的方法来自模型论、交换代数和代数几何。由解析函数之间的方程组和不等式所定义的实空间中的点的集合称为解析集。这种集合在低维子空间上的投影（即影子）称为亚解析。实子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。然而，这些集合共享了它们真正的表兄弟的许多漂亮属性。Lipshitz， Robinson和他们的学生将继续研究这些集合的性质，使用数理逻辑，交换代数和代数几何的方法。

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.