Logic, Sets, Categories, and Applications

提议者继续他的研究应用集理论的方法阿贝尔群理论，特别强调产品的无限循环群和相关群体。 除了应用集理论的思想，这分支代数，他正在寻找新的集理论的概念和结果的动机代数问题。 他正在探索之间的联系普遍代数（在Lawvere的范畴理论制定），理论分类拓扑，和理论的统一（其中发挥了突出作用，在某些领域的计算机科学）。 在绝对自由代数的情况下，这种联系已经被理解了，但是在代数服从非平凡恒等式的更一般的情况下，需要进一步的探索。 在恒等式存在下的统一通常是一个棘手的话题，希望与其他更好理解的领域的联系可以澄清它。提出者还在研究一种拓扑理论方法，将Borel函数的概念扩展到更高的类型。 这样的扩展对于连续统的基数特征理论是有用的。在基型（真实的数）上的波莱尔函数已经在这个理论中发挥了重要作用，但是顺序组合的操作也必然导致更高的类型。 提出者还研究了构造性逻辑、范畴论和博弈语义学之间的几个问题。 这项研究包括试图扩大游戏和自由（或co-free）bicomplete类别之间的联系，公理化的游戏语义有效性，和游戏理论的原则，可能会卓有成效地添加到直觉逻辑。 最后，作者还研究了集合论中的几个问题，这些问题大多与超滤子有关，特别是与超滤子上的近相干等价关系有关。 在无限组合学中也有一些问题是由阿贝尔群论中的考虑引起的。 该项目建立在提议者过去在几个数学领域的经验-集合论，范畴论（特别是拓扑理论），阿贝尔群论，博弈语义学，构造数学和计算机科学-为了解决这些领域的问题。 提案中的几乎所有主题都涉及两个或更多领域。 在每一种情况下，我们都有理由期待，来自一个领域的思想和推理风格可以使其他领域受益。 这项研究的潜在成果包括：（1）更好地理解统一，这是逻辑编程的一个重要组成部分;（2）更清楚地了解基于某些类型辩论中的策略的构造性数学推理;（3）在许多方面类似于传统集合论的背景下的应用，但避免了有时被视为病态的现象。

The proposer continues his research on applications of set-theoretic methods to abelian group theory, with particular emphasis on products of infinite cyclic groups and related groups. In addition to applying set-theoretic ideas to this branch of algebra, he is looking for new set-theoretic concepts and results motivated by algebraic problems. He is exploring the connection between universal algebra (in Lawvere's category-theoretic formulation), the theory of classifying topoi, and the theory of unification (which plays a prominent role in some areas of computer science). The connection is already understood in the case of absolutely free algebras, but further exploration is needed in the more general case of algebras subject to nontrivial identities. Unification in the presence of identities is in general a troublesome topic, and it is hoped that the connection with other, better understood areas can clarify it. The proposer is also studying a topos-theoretic approach to extending the notion of Borel functions to higher types. Such an extension is useful for the theory of cardinal characteristics of the continuum.Borel functions at the base type (real numbers) already play a major role in this theory, but so does an operation of sequential composition that necessarily leads to higher types. The proposer is also studying several questions at the border between constructive logic, category theory, and game semantics. This research includes an attempt to amplify the connection between games and free (or co-free) bicomplete categories, axiomatization of game-semantical validity, and a game-theoretic principle that may be fruitfully added to intuitionistic logic. Finally, the proposer is studying several questions in set theory, most of which are connected with ultrafilters and in particular with the equivalence relation of near coherence on ultrafilters. There are also some questions in infinite combinatorics growing out of considerations in abelian group theory. The project builds on the proposer's past experience in several areas of mathematics --- set theory, category theory (especially topos theory), abelian group theory, game semantics, constructive mathematics, and computer science --- in order to address problems bridging these areas. Almost all of the topics in the proposal connect two or more of these areas. In each case, it is reasonable to expect that ideas and styles of reasoning from one area can benefit other areas. Among the potential outcomes of this research are: (1) a better understanding of unification, which is an essential ingredient of logic programming, (2) a clearer view of constructive mathematical reasoning as based on strategies in certain sorts of debates, and (3) applications of a context that resembles traditional set theory in many ways yet avoids the phenomena sometimes regarded as pathological.