Mathematical Sciences: Topics in Logic and Set Theory

Andreas Blass的研究项目包括四个方面：(1)连续统的基本特征，包括这些特征的一般理论（涉及Galois-Tukey连接）和特定特征的详细研究，特别是最近引入的逃避和预测数。(2)集合论的应用，特别是在阿贝尔群理论中的应用。(3)线性逻辑的语义，特别是博弈语义及其变体。拓扑逻辑，特别是几何态射的逻辑性质。Glen Whitney的研究项目涉及解释一般递归定义，特别是在非确定性和并发上下文中。具体问题包括：(1)用递归操作找到一个结构的通用类，使每个结构都包含在这个类的一个类中。(2)比较并发编程语言的语义建议。(3)寻找与有限模型理论中不动点逻辑的联系。(4)扩展当前的理论，以便对子程序之间的参数和消息传递进行更详细的建模。惠特尼将要研究的递归形式语言FLR和格拉斯的研究课题之一吉拉德的线性逻辑LL都可以作为计算的某些方面的数学模型。FLR模拟递归，即使用与程序本身相同的子程序的程序。惠特尼的项目旨在提高对并行计算背景下递归的理解，即由许多机器同时操作协同完成的计算。LL模型数据类型；它的游戏语义模拟了这样一种情况：访问数据更多地涉及到数据提供者和用户之间的交互（访问协议），而不仅仅是数据传输。这两种理论目前都处于数学探索阶段，但希望从中获得的信息可以用于未来编程语言的设计和程序正确性的形式化验证。连续统的基本特征是一种展示和澄清来自数学各个领域的困难问题的组合内容的方法-最初是拓扑和分析，最近也是代数。它们提供了这些不同领域之间的联系，从而使一个领域的问题进展能够应用于另一个领域。此外，它们使这些问题易于用现代集合论的方法来处理。布拉斯提出的研究的最后一个主题，拓扑理论，也用于连接明显不同的领域，主要是在数学逻辑。拓扑的几何态射这个特殊主题结合了集合论、信息系统（作为计算模型引入）、层理论（拓扑的一部分）和构造逻辑的思想。＊＊＊

9505118 Blass The research project of Andreas Blass is in four areas: (1) Cardinal characteristics of the continuum, including both the general theory of such characteristics (involving Galois-Tukey connections) and the detailed study of particular characteristics, especially the recently introduced evasion and prediction numbers. (2) Applications of set theory, especially in the theory of abelian groups. (3) Semantics of linear logic, especially game semantics and its variants. (4) Logic of topoi, particularly the logical properties of geometric morphisms. The research project of Glen Whitney concerns interpreting general recursive definitions, especially in non-determinisitic and concurrent contexts. Specific questions include: (1) Finding a universal class of structures with a recursion operation, so that every structure will be contained in one from this class. (2) Comparing proposals for the semantics of concurrent programming languages. (3) Seeking connections with fixpoint logics used in finite model theory. (4) Extending the current theory to allow more detailed modelings of parameter and message passing between subprograms. Both the formal language of recursion FLR, which Whitney will study, and Girard's linear logic LL, one of Blass' research topics, can serve as mathematical models of certain aspects of computing. FLR models recursion, i.e., programs using subroutines identical to the programs themselves. Whitney's project is directed toward improved understanding of recursion in the context of parallel computing, i.e. computations done collaboratively by many machines operating simultaneously. LL models data types; its game semantics models the situation where accessing data involves more of an interaction (access protocol) between the provider and the user of the data than mere transmission of data. Both theories are currently at the stage of mathematical exploration, but it is hoped that the information obtained w ill be of use in the design of future programming languages and in the formal verification of correctness of programs. Cardinal characteristics of the continuum are a way to exhibit and clarify the combinatorial content of difficult problems from various areas of mathematics -- originally topology and analysis, more recently algebra as well. They provide connections between these different areas and thus allow progress on a problem in one area to be applied in another. In addition, they make these problems amenable to treatment by the methods of modern set theory. The final topic in Blass' proposed research, topos theory, also serves to connect apparently disparate areas, mostly within mathematical logic. The particular topic of geometric morphisms of topoi combines ideas from set theory, from information systems (introduced as a model of computation), from sheaf theory (a part of topology), and from constructive logic. ***