Model Theory, Algebra and Geometry

摘要奖：DMS-0303618主要研究者：托马斯W.该奖项支持的研究将由Matthias Aschenbrenner执行，并由WisconasW赞助。斯坎伦 这些项目在模型论及其应用代数和几何是有关渐近微分代数，o-极小几何，边界和算法在代数。 渐近微分代数的项目将追求哈代域和transseries的模型理论和代数性质，以及它们之间的关系。哈代域是定义在真实的直线的正无穷邻域上的实值、一次可微函数芽的有序微分域;它们在微分方程的渐近理论中很重要，并且自然地出现在与实域的o-极小展开的联系中。 一个跨级数领域的例子是关于真实的数的幂指数级数领域，它已经被分析家和模型理论家所探索。 在代数的算法问题中，将研究整数上多项式环和幂级数环的性能界算法;寻找好算法的问题包括理想成员的测试。逻辑学的分支称为模型论的思想大致是，如果我们知道关于一个对象的所有简单陈述的真理，那么我们要么应该知道如何唯一地识别这个对象，或者任何其他具有相同一阶性质的集合的东西都应该像原始的一样有启发性，而且有时可能更容易研究。更精确地说，模型论通过考虑在这些结构中一阶句子为真以及满足所有这些一阶句子的替代结构家族来研究数学结构。 （逻辑中的句子是由少量元素和结构组成的。 “一阶”指的是句子中量词的数量，是衡量复杂性的一个标准。）一个模型的算法和边界寻求在一些这些项目是长除法：如果你给两个整数除以手然后你可以估计的步骤数所需的长除法通过比较的数字在十进制扩展的除数和除数。

AbstractAward: DMS-0303618Principal Investigator: Thomas W. ScanlonThe research supported by this award is to be performed byMatthias Aschenbrenner, under the sponsorship of ThomasW. Scanlon. These projects in model theory and its applicationsto algebra and geometry are concerned with asymptoticdifferential algebra, o-minimal geometry, and bounds andalgorithms in algebra. The project on asymptotic differentialalgebra will pursue model-theoretic and algebraic properties ofHardy fields and transseries, and the relationship between them.A Hardy field is an ordered differential field of germs ofreal-valued, once-differentiable functions defined onneighborhoods of positive infinity in the real line; they areimportant in the asymptotic theory of differential equations andappear naturally in connection with o-minimal expansions of thereal field. An example of a field of transseries is the field oflogarithmic-exponential series over the real numbers, which hasbeen explored by analysts as well as model-theorists. Among thealgorithmic issues in algebra that will be investigated arealgorithms with performance bounds for polynomial rings over theintegers and for rings of power series; problems for which goodalgorithms are sought include tests for ideal membership.The idea of the branch of logic called model theory is, roughly, thatif we know all of the simply-stated truths about an object then eitherwe should know how to recognize that object uniquely, or anything elsesharing the same collection of first-order properties should berevealing like the original and might sometimes be easier to study.To be more precise, model theory studies mathematical structures byconsidering the first-order sentences true in those structures, andthe family of alternate structures that also satisfy all of thosefirst-order sentences. (Sentences in logic are built out of a smallrepertoire of elements and constructions. "First-order" refers to thenumber of quantifiers in a sentence, a measure of complexity.)A model for the algorithms and bounds sought in some of these projectsis long division: if you are given two whole numbers to divide by handthen you can estimate the number of steps required by long divisionby comparing the number of digits in the decimal expansionsof the dividend and divisor.