Mathematical Sciences: Rigid Analytic Geometry and Logic

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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. ***