Model Theory, Algebra and Geometry

模型理论、代数和几何

• 批准号: 0513494
• 负责人: Matthias Aschenbrenner
• 金额: --
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-20 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0513494&HistoricalAwards=false
• 关键词: Model; Theory; Algebra; Geometry

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.

奖项：dms -0303618首席研究员：ThomasW. scanlon本奖项支持的研究将由matthias Aschenbrenner在ThomasW赞助下进行。斯坎伦。这些项目在模型理论及其在代数和几何中的应用涉及渐近微分代数，零最小几何，代数中的界和算法。渐近微分代数项目将研究hardy域和横列的模型论和代数性质，以及它们之间的关系。Hardy域是定义在实线正无穷邻域上的实值、一次可微函数的芽的有序微分域；它们在微分方程的渐近理论中是重要的，并且自然地与实场的0极小展开联系在一起。异列领域的一个例子是实数上的对数-指数序列领域，它已经被分析学家和模型理论家所探索。将研究的代数算法问题包括整数上的多项式环和幂级数环的性能界算法；寻找好的算法的问题包括理想隶属度的检验。逻辑的一个分支被称为模型论，粗略地说，如果我们知道关于一个物体的所有简单陈述的真理，那么我们应该知道如何唯一地识别这个物体，或者任何其他具有相同一阶属性集合的东西应该像原始的一样被揭示出来，有时可能更容易研究。更精确地说，模型理论通过考虑在这些结构中真实的一阶句子以及满足所有这些一阶句子的替代结构族来研究数学结构。逻辑中的句子是由少量的元素和结构构成的。“一阶”指的是句子中量词的数量，衡量句子的复杂程度。)在这些项目中，算法和边界的一个模型是长除法：如果给你两个整数要用手除，那么你可以通过比较被除数和被除数的十进制展开式中的位数来估计长除法所需的步数。