Computability on Cones

可计算性理论（Computability Theory）是数学逻辑中的一个领域，研究可数数学对象的复杂性。一些数学对象、构造和证明比其他的更复杂。逻辑学家已经开发了各种方法来衡量这种复杂性。当一个人对可数对象感兴趣时--数学的很大一部分都与之有关--测量这些复杂性的工具来自可计算性理论。这个项目的目的是绘制复杂性问题和结构性问题之间的联系，以提高对复杂性的形状和形式的理解。该项目是正在进行的结构研究的一部分，该结构是在考虑圆锥上的可计算性理论属性时出现的，即相对于马丁度量而言几乎所有预言都成立的属性。特别是，该项目将探索Vaught猜想和可计算性理论之间的联系，以及均匀马丁猜想的推广。几年前，可计算结构理论和沃特猜想之间的联系出人意料地出现了。虽然可能需要很多年才能看到Vaught的猜想得到解决，但这个项目追求一个特定的方面：它与可计算性理论的联系。

Computability theory is an area within mathematical logic that studies the complexity of countable mathematical objects. Some mathematical objects, constructions, and proofs are more complicated than others. Logicians have developed various ways of measuring this complexity. When one is in interested in countable objects -- with which a large portion of mathematics is concerned -- the tools to measure these complexities come from computability theory. The objective of this project is to draw connections between complexity issues and structural issues to improve understanding of what shapes and forms complexity can take. This project is part of an ongoing study of the structure that emerges when considering computability theoretic properties on a cone, that is, properties that hold relative to almost every oracle with respect to Martin's measure. In particular, the project will explore the connections between Vaught's conjecture and computability theory, and the generalizations of the uniform Martin's conjecture. A few years ago, connections between computable structure theory and Vaught's conjecture emerged unexpectedly. While it may take many years to see Vaught's conjecture settled, this project pursues a particular aspect: its connections to computability theory.