Geometric Model Theory

摘要奖：DMS-0301771主要研究者：托马斯W.这些项目研究模型理论和几何之间的联系。 在这方面，模型论包括对混凝土结构的公理化和量词消除结果的研究，以及一般稳定性和维数理论的发展。 要研究的几何问题出现在代数，复杂的解析，刚性解析，微分代数几何。 具体项目包括将紧致复流形的模型理论扩展到非标准解析空间和刚性解析空间。 这项工作的一些调查在何种程度上凯勒流形是免费的frommodel理论的病理。 在丢番图几何中，新的一致性和有限性原理被认为是丰富域的类似结果的反映。 特别地，一个基于可分离闭域的模型理论的程序旨在解决Drinfeld模的Mordell-Lang猜想的Denis类似物的函数域变体，将研究D-域上可定义集的精细结构，旨在将相对完备性和量词消去定理推广到具有更丰富解析结构的值D-域，并在这种设置下产生有效的量词消去定理。 微分喷射空间和相关的构造将被用来揭示在偏微分域中正则型的组合几何。 我们将研究一阶结构的欧拉特征和格罗滕迪克环。逻辑学的分支称为模型论，其思想大致是，如果我们知道关于一个对象的所有简单陈述的真理，那么我们就应该知道如何唯一地识别这个对象，或者任何其他共享同一组一阶性质的东西应该像原来的一样有启示性，而且有时可能更容易研究。 更准确地说，模型理论研究数学结构时考虑的是这些结构中的一阶句子为真，以及满足所有这些一阶句子的替代结构族。（逻辑中的句子是由一小部分元素和结构组成的。 “一阶”指的是句子中量词的数量，是衡量句子复杂性的一个标准。） 在这些项目中寻找的算法和边界的模型是长除法：如果你有两个整数要用手除法，那么你可以通过比较被除数和除数的小数展开式中的位数来估计长除法所需的步骤数。

AbstractAward: DMS-0301771Principal Investigator: Thomas W. ScanlonThese projects study the connections between model theory andgeometry. In this context, model theory includes the search foraxiomatizations and quantifier elimination results for concretestructures and the development of general stability and dimensiontheories. The geometric problems to be studied arise inalgebraic, complex analytic, rigid analytic, and differentialalgebraic geometry. Specific projects include the extension ofthe model theory of compact complex manifolds to nonstandardanalytic spaces and rigid analytic spaces. Some of this workinvestigates the extent to which Kaehler manifolds are free frommodel theoretic pathologies. In diophantine geometry newuniformity and finiteness principles are being sought asreflections of similar results for enriched fields. Inparticular, a program based on the model theory of separablyclosed fields aims to resolve a function field variant of Denis'analogue of the Mordell-Lang conjecture for Drinfeld modules.The fine structure of definable sets over D-fields will beinvestigated, aiming to extend relative completeness andquantifier elimination theorems to valued D-fields with richeranalytic structure and to produce effective versions ofquantifier elimination theorems in this setting. Differentialjet spaces and related constructions will be used to uncover thecombinatorial geometry of regular types in partial differentialfields. Abstract Euler characteristics and Grothendieck rings offirst-order structure will be investigated.The idea of the branch of logic called model theory is, roughly,that if we know all of the simply-stated truths about an objectthen either we should know how to recognize that object uniquely,or anything else sharing the same collection of first-orderproperties should be revealing like the original and mightsometimes be easier to study. To be more precise, model theorystudies mathematical structures by considering the first-ordersentences true in those structures, and the family of alternatestructures that also satisfy all of those first-order sentences.(Sentences in logic are built out of a small repertoire ofelements and constructions. "First-order" refers to the numberof quantifiers in a sentence, a measure of complexity.) A modelfor the algorithms and bounds sought in some of these projects islong division: if you are given two whole numbers to divide byhand then you can estimate the number of steps required by longdivision by comparing the number of digits in the decimalexpansions of the dividend and divisor.