Mathematical Sciences: Computability in Mathematics

Peter A. Cholak的主要兴趣是各种数学结构和不同可计算性度量之间的相互作用。其中一个重要的数学结构是可计算枚举集的结构——那些可以被图灵机枚举的集合，具有集合包含的关系。Cholak的项目旨在理解可定义集合之间的关系，这些集合的各种动态特性，以及这种结构的自同构。例如，Cholak希望确定两个可计算枚举集合是否在同一轨道上这个问题的复杂性（根据上述度量）。这可以帮助证明某些跳转类在该结构中是不变的。项目的另一部分是探索是否每对划分都有一个非高齐次集。这一结果将提高我们对拉姆齐定理和弱柯尼格引理的证明理论强度的认识。Cholak研究可计算性理论，这是数理逻辑的一个领域。Cholak的主要兴趣是各种数学结构和不同可计算性度量之间的相互作用。例如，最常用的可计算性模型是图灵机的模型——或多或少是一台没有任何内存或时间限制的理想计算机。Cholak的研究涉及可以用图灵机列出的集合，希望进一步理解可以用公式定义的集合之间的关系。此外，他的研究还涉及这些集合的各种动态特性——枚举的快慢以及它们的信息内容。

DMS-9634565 Peter A. Cholak Cholak's main interest is in the interaction between various mathematical structures and different measures of computability. One such important mathematical structure is that of the computably enumerable sets -- those sets which can be enumerated by a Turing machine, with the relation of set inclusion. Cholak's project aims at understanding the relationships between the definable sets, the various dynamic properties of these sets, and the automorphisms of this structure. For example, Cholak wishes to determine the complexity (in terms of the above measures) of the question whether two computably enumerable sets are in the same orbit. This could help one show that certain jump classes are invariant within this structure. Another part of the project is to explore whether every partition of pairs has a non-high homogeneous set. Such a result would improve our understanding of the proof-theoretic strength of Ramsey's Theorem for pairs and of Weak Konig's Lemma. Cholak works in computability theory, an area of mathematical logic. Cholak's main interest is in the interaction between various mathematical structures and different measures of computability. For example, the most generally used model of computability is that of the Turing machine -- more or less an ideal computer running without any memory or time bounds. Cholak's research concerns the sets which can be listed using a Turing machine, with the hope of further understanding the relationships between sets which can be defined by formulas. Additionally, his research concerns various dynamic properties of these sets -- how fast or slow their enumeration is, and their information content.