Interactions between Computability Theory and Model Theory

可计算理论是从计算的角度研究数学对象的复杂性，而模型理论是从结构的角度研究数学对象的复杂性。这个项目旨在加深对这两种不同的复杂性概念之间相互作用的理解。一般来说，人们认为对象越结构化，就越容易计算有关它的信息。本研究项目旨在阐明具有可理解理论的数学对象如何与有关该对象的事实的可计算性相关。特别是，该项目致力于理解可计算模型理论中的某些问题：可计算模型的谱问题询问哪些维集可以作为某些不可数范畴理论的可计算模型的维集。该项目将根据理论的几何性质来研究区分这个问题，目的是在理论满足Zilber三分法的情况下推进理解。本项目还旨在促进对理论度谱的理解。一个理论的度谱衡量了计算该理论的任何模型的难度。研究者的目标是发展一个新的概念，将强调理论的性质和光谱的性质之间的联系，提供可计算性和模型理论之间的进一步联系。研究了模型理论概念的指标集；这是一种测量属性复杂性的可计算性理论方法。了解属性的确切复杂性有助于以最直接和有效的方式处理这些属性，从而促进进一步的研究进展。

While computability theory studies complexity of mathematical objects from a computational point of view, model theory studies complexity of mathematical objects from a structural point of view. This project aims to deepen understanding of the interactions between these two different notions of complexity. In general, one expects that the more structured an object is, the easier it is to compute information about it. This research project aims to elucidate how a mathematical object having an understandable theory relates to computability of facts about the object.In particular, this project works towards understanding certain problems in computable model theory: the spectrum of computable models question asks which sets of dimensions can arise as the dimensions of computable models of some uncountably categorical theory. The project will investigate distinguishing this problem based on the geometric nature of the theory, with the goal of advancing understanding in the cases where the theory satisfies the Zilber trichotomy. This project also aims to advance understanding of the degree spectra of theories. The degree spectrum of a theory measures how hard it is to compute any model of that theory. The investigator aims to develop a newer notion that will highlight connections between properties of the theory and the properties of the spectrum, providing a further link between computability and model theory. The project also investigates index sets of model theoretic notions; this is a computability-theoretic way to measure the complexity of a property. Knowing the exact complexity of a property aids in working with these properties in the most direct and efficient manner possible, thus spurring further research progress.