Mathematical Sciences: Rigid Analytic Geometry and Logic

9401451 Lipshitz L.Lipshitz，Z.Robinson和他们的学生将继续他们对刚性次解析和半解析集(即代数闭域上关于非阿基米德绝对值完备的次解析和半解析集)的研究。这一研究将进一步加强对可分幂级数环的代数和几何的研究。在其他主题中，他们将研究这些环上的唯一因式分解、一致化、Stein因式分解、奇异轨迹定理和层理论。使用的方法将是非标准模型和数理逻辑中的量词消除，以及交换代数和代数几何的技术。由解析函数之间的方程组和不等式定义的实空间中的点集称为解析集。这样的集合在低维子空间上的投影(即阴影)称为次解析。实次解析集出现在数学的几个分支中，例如微分方程式和几何。例如，在数论中，类似的集合类自然地出现在与实数不同的域上，其中距离的概念具有相当不同的性质。然而，这些集合分享了它们真正的表亲的许多美好的特性。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。***

9401451 Lipshitz L. Lipshitz, Z. Robinson and their students will continue their investigation of rigid subanalytic and semianalytic sets (i.e. subanalytic and semianalytic sets over an algebraically closed field which is complete with respect to a non-Archimedean absolute value). This investigation will entail further work on the algebra and geometry of the rings of separated power series. Among other topics, they will investigate unique factorization, uniformization, Stein factorization, the singular locus theorem, and sheaf theory over these rings. Methods to be used will be nonstandard models and quantifier elimination from mathematical logic, together with techniques from 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. ***