Model Theory, Algebra and Geometry

摘要奖：DMS-0303618首席研究员：Thomas W.Scanron本奖项支持的研究将由Matthias Aschenbrenner进行，由ThomasW赞助。斯坎隆。模型理论的这些项目及其在代数和几何中的应用涉及渐近微分代数、o-极小几何以及代数中的界限和算法。渐近微分代数的项目将探索Hardy场和跨序列的模型论和代数性质，以及它们之间的关系。Hardy场是定义在实线上的正无穷邻域上的实值一次可微函数芽的有序微分场；它们在微分方程组的渐近理论中很重要，与实场的O-极小展开一起自然地出现。跨系列领域的一个例子是实数上的对数-指数级数领域，这一领域已经被分析家和模型理论家探索过。在代数的算法问题中，将被研究的是整数上多项式环和幂级数环的具有性能界限的算法；寻找好的算法的问题包括对理想成员的测试。被称为模型理论的逻辑分支的想法是，粗略地说，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该对象，要么我们应该知道任何具有相同一阶性质集合的东西应该像原始的一样，有时可能更容易研究。更准确地说，模型理论研究数学结构是通过考虑那些结构中为真的一阶语句，以及也满足所有这些一阶语句的可选结构家族。(逻辑中的句子是由少量的元素和结构组成的。“一阶”指的是句子中数量词的数量，这是一种复杂性的衡量标准。)其中一些项目中寻找的算法和界限的模型是长除法：如果给你两个整数用手除，那么你可以通过比较被除数和除数的小数展开中的位数来估计长除法所需的步骤数。

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.