Mathematical Sciences: Logic and Analysis

9503398 Jockusch这个项目涉及数理逻辑及其与其他数学领域的联系，特别是代数和分析。卡尔·约克施将致力于可计算性理论(递归理论)及其与其他领域的联系。特别是，他将研究不可解度，既有它们本身的权利，也有与有效的组合学、群论和代数有关的。L.van den Dries将通过分析中熟悉的准解析函数类来研究实场的展开。我们的目的是证明这种展开式的o-极小和多项式有界性。这是创建一个框架来解决希尔伯特关于极限环和类似问题的第16个问题的努力的一部分。数理逻辑为数学断言和演绎系统提供了一个精确的框架。它还提供了抽象计算机可计算性的精确概念，即没有运行时间或存储空间限制的计算机。Jockusch将研究可计算性理论及其与数学其他分支的联系。这项研究的中心兴趣在于，理论上限制了可以计算的内容，以及在获得其他信息的情况下计算信息的可能性。特别是，他将分析与代数和数论中的可计算性有关的问题。卢·范登德里斯将使用数理逻辑的工具，特别是模型理论，研究解析几何。这些研究有望为有关解析矢量场的问题提供新的线索。目前在实数研究中获得的信息目前正被应用于许多领域，包括机器人和神经网络。***

9503398 Jockusch This project concerns mathematical logic and its connections with other areas of mathematics, especially algebra and analysis. Carl Jockusch will work in computability theory (recursion theory) and its connections with other areas. In particular, he will study degrees of unsolvability, both in their own right and in connection with effective combinatorics, group theory, and algebra. L. van den Dries will investigate expansions of the real field by quasi-analytic classes of functions that are familiar from analysis. The goal is to prove o-minimality and polynomial boundedness of such expansions. This is part of an effort to create a framework for solving Hilbert's 16th problem on limit cycles and similar problems. Mathematical logic provides a precise framework for mathematical assertions and deductive systems. It also provides a precise concept of computability by abstract computers, i.e. ones without limitations of running time or memory space. Jockusch will study the theory of computability and its connections with other branches of mathematics. Theoretical limitations on what can be computed and the possibility of computing information given access to other information are of central interest for this research. In particular, he will analyze questions related to computability in algebra and number theory. Lou van den Dries will work in analytic geometry, using tools from mathematical logic, especially model theory. These investigations are expected to shed new light on questions about analytic vector fields. Information gained in current research on real numbers is presently being applied to many areas, including robotics and neural networks. ***