New Directions in Reverse Mathematics and Applied Computability Theory

今天的数学得益于“坚实的基础”，我们通常指的是一个公理系统，足以证明我们关心的各种定理。但是，给定一个特定的定理，我们能精确地指定导出它所需要的公理吗？这是一个自然的问题，也是一个古老的问题：2000多年前，希腊数学家们就在问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。这提供了对证明给定定理所需的基本思想和方法的更深入的了解。更确切地说，逆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各个分支的基础，在这一领域反复证明的一个惊人的事实是，绝大多数数学命题都可以被归类为少数数学命题之一。类别但对于一些非常重要的基本定理，情况并非如此。Dzhafarov的研究集中在定理这一点？不规则”类型，包括拉姆齐定理，选择公理的各种等价物，以及认知科学中某些问题所产生的原理。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，从以前未探索的数学领域中找到这些定理的新例子，并将反向数学分析应用于数学之外的问题。这将有助于从可计算性理论和证明理论的方法的应用，并从各种合作的想法在纯数学和应用数学的一些领域，以及与康涅狄格州逻辑组的多学科大学的成员的互动。

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.