New Directions in Reverse Mathematics and Applied Computability Theory

今天的数学得益于“坚实的基础”，我们通常指的是足以证明我们所关心的各种定理的公理系统。但是给定一个特定的定理，我们能精确地指出需要哪些公理来推导它吗？这是一个自然的问题，也是一个古老的问题：2000多年前，希腊数学家就曾问过关于欧几里得几何的问题。逆向数学是数学逻辑的一个领域，它通过根据逻辑强度对数学定理进行分类，为这类问题提供了一种现代方法。这为证明给定定理所需的基本思想和方法提供了更深入的见解。更准确地说，反向数学提供了一个框架，在其中比较和对比来自不同数学领域的结果，这有助于阐明数学科学的各个分支的基础，从而导致更好地理解数学及其应用。在这一领域反复证明的一个惊人事实是，绝大多数数学命题可以被归类为少数类别之一。但对于一些非常重要和基本的定理，情况并非如此。Dzhafarov的研究重点是这个定理？“不规则”类型，包括拉姆齐定理，选择公理的各种等价，以及认知科学中某些问题产生的原理。在这个项目中，Dzhafarov将致力于更好地理解这些“不规则”定理的复杂性，从以前未被探索的数学领域找到这些定理的新例子，并将反向数学分析应用于数学以外的问题。这将通过应用可计算理论和证明理论的方法，通过在多个纯数学和应用数学领域的各种合作的想法，以及与康涅狄格大学多学科逻辑小组成员的互动来促进。

Mathematics today benefits from having "firm foundations", by which we usually mean a system of axioms sufficient to prove the various theorems we care about. But given a particular theorem, can we specify precisely which axioms are needed to derive it? This is a natural question, and also an ancient one: over 2000 years ago, the Greek mathematicians were asking it about Euclid's geometry. Reverse mathematics is an area of mathematical logic that offers a modern approach to this kind of question, by classifying mathematical theorems according to their logical strength. This offers a deeper insight into the fundamental ideas and methods needed to prove a given theorem. More precisely, reverse mathematics provides a framework in which to compare and contrast results from disparate areas of mathematics, which helps elucidate the underpinnings of various branches of the mathematical sciences, and thereby leads to a better understanding of mathematics and its applications.A striking fact repeatedly demonstrated in this area is that the vast majority of mathematical propositions can be classified into one of a small number of categories. But for some very important and fundamental theorems this is not the case. Dzhafarov's research focuses on theorems of this ?irregular" type, including Ramsey's theorem, various equivalents of the axiom of choice, and principles arising from certain problems in cognitive science. In this project, Dzhafarov will work to achieve a greater understanding of the complexities of these "irregular" theorems, to find new examples of such theorems from previously unexplored areas of mathematics, and to apply the reverse mathematics analysis to questions from outside of mathematics. This will be facilitated by the application of methods from computability theory and proof theory, and by the addition of ideas from various collaborations across a number of areas of pure and applied mathematics, as well as interactions with members of the multidisciplinary University of Connecticut logic group.