Mathematical Sciences: Topics in Logic and Set Theory

小行星9505118 安德烈亚斯·布拉斯的研究项目在四个方面：（1）红衣主教 连续统的特征，包括一般理论， 这些特征（涉及Galois-Tukey连接）和详细的 研究特殊的特征，特别是最近推出的 规避和预测的数字。(2)集合论的应用，特别是 在阿贝尔群理论中。(3)线性逻辑的语义，特别是 游戏语义及其变体。(4)拓扑的逻辑，特别是 几何态射的逻辑性质 Glen的研究项目 Whitney关注于解释一般的递归定义，尤其是在 非确定性和并发上下文。 具体问题包括： (1)用递归运算找到一个通用的结构类， 每个结构都包含在这个类的一个结构中。（二） 并发编程语言语义的比较建议。 (3)寻求与有限模型理论中使用的不动点逻辑的联系。 (4)扩展当前的理论，以允许更详细的建模， 参数和子程序之间的消息传递。 Whitney将要研究的递归FLR的形式语言和Blass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模型 递归，即，使用与程序相同的子程序的程序 自己 惠特尼的项目旨在提高对 递归在并行计算的上下文中，即计算完成 多台机器同时操作。 LL对数据类型进行建模;它的游戏语义对访问数据的情况进行建模， 涉及数据的提供者和用户之间的交互（访问协议），而不仅仅是数据的传输。 这两种理论目前都处于数学探索阶段，但希望所获得的信息将用于未来编程语言的设计和程序正确性的形式化验证。 连续体的基本特征是一种展示和澄清数学各个领域困难问题的组合内容的方法-最初是拓扑学和分析，最近还有代数。 它们在这些不同领域之间提供联系，从而使一个领域的问题的进展能够应用于另一个领域。 此外，他们使这些问题适合治疗的方法，现代集理论。 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，主要是在数理逻辑。 拓扑的几何态射这一特定主题结合了集合论、信息系统（作为计算模型引入）、层理论（拓扑学的一部分）和构造性逻辑的思想。 ***

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. ***