Large Cardinals, Small Sets and Absoluteness

数学逻辑的技术可以用来衡量数学概念的复杂性。具有物理应用的数学领域往往出现在相应复杂性层次的较低层次上。进入更高层次的复杂性使数学家能够在数学的不同领域之间建立联系，并发展出富有成效的一般理论。相应地，数学的世界被划分为绝对的部分，其中自然的问题倾向于用标准公理来解决，而更抽象的部分则需要扩展标准公理系统来解决许多基本问题。拉尔森研究的主要焦点是这些部分之间的关系。这个项目中涉及的技术机制包括科恩的强迫技术，在无限游戏中断言获胜策略存在的公理，以及断言无限对象存在的公理，这些对象的存在不能从标准数学公理中证明。这些技术起源于集合论，它是最普遍接受的数学基础。Larson的工作将它们应用到其他领域，包括模型理论、拓扑和分析。Larson希望用这种方法在几个众所周知的长期开放问题上取得进展。拉尔森工作的一个方面是强迫确定性模型产生规范模型。Woodin的Pmax强迫，当应用于确定性模型时，产生了一个对可数序数集合子集最大的模型。一个由伍丁发起，并由拉尔森与其他研究人员合作继续进行的项目是，这种最大值可以在多大程度上容纳更大的枢机。另一个产品，主要是与Jindrich Zapletal合作，将一些经典的ZFC结构转换为偏序，当应用于确定性模型时，产生选择公理片段的模型。这种方法已经解决了许多关于选择公理形式之间关系的经典问题，但也可能能够照亮Borel等价关系理论中的新问题。该项目的其他方面包括对模型理论中最古老的问题之一沃特猜想的新方法，以及对普遍可测量性概念的研究，这是一个来自分析的基本概念，但仍未得到很好的理解。最后，拉尔森正在写一本关于W.H.伍丁未出版作品的书，关于确定性公理的扩展。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Techniques from mathematical logic can be used to measure the complexity of mathematical concepts. Areas of mathematics with physical applications tend to appear in the low levels of the corresponding complexity hierarchy. Passing to higher levels of complexity enables mathematicians to make connections between different areas of mathematics, and to develop productive general theories. There is a corresponding division of the universe of mathematics into an absolute part, where natural questions tend to be resolved by the standard axioms, and a more abstract part where extensions of the standard axiom system are needed to resolve many fundamental questions. The main focus of Larson's research is the relationship between these parts. The technical machinery involved in this project includes Cohen's forcing technique, axioms asserting the existence of winning strategies in infinite games, and axioms asserting the existence of infinite objects whose existence cannot be proved from the standard axioms for mathematics. These techniques originate in set theory, which serves as the most commonly accepted foundations for mathematics. Larson's work applies them to other areas, including model theory, topology and analysis. Larson hopes to make progress with this approach on several longstanding well-known open problems.One aspect of Larson's work concerns forcing over models of determinacy to produce canonical models. Woodin's Pmax forcing, when applied to a model of determinacy, produces a model which is maximal for subsets of the set of countable ordinals. One project, initiated by Woodin, and continued by Larson in collaboration with other researchers, is the extent to which this maximality can be made to hold for larger cardinals. Another product, largely in collaboration with Jindrich Zapletal, converts a number of classical ZFC constructions into partial orders, which when applied to determinacy models produce models of fragments of the Axiom of Choice. This approach has resolved a number of classical questions about the relationship between forms of the Axiom of Choice, but may also be able to shine light on newer problems in the theory of Borel equivalence relations. Other aspects of the project include a new approach to Vaught's Conjecture, one of the oldest problems in model theory, and the study of the notion of universally measurability, a fundamental concept from analysis which is still not well understood. Finally, Larson is working on a book on unpublished work of W.H. Woodin, on extensions of the Axiom of Determinacy.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.