Model Theory of Nonabelian Free Groups

本研究项目围绕模型理论及其与双曲几何的相互作用。模型论关注的是用形式化的数学语言来研究结构。人们可以追溯其起源在哥德尔的基本工作的完整性的一阶逻辑。哥德尔在他著名的完备性定理中一劳永逸地解决了真理和可证明性是如何联系在一起的问题。粗略地说，一个数学陈述是可证明的，当且仅当它在所有可能的世界中为真。虽然模型论似乎起源于哲学和数学基础的问题，但近年来，人们发现它与核心数学学科和计算机科学有着深刻的联系。双曲几何源于对欧几里得第五公设的反驳，也被称为平行公设。经过二十多个世纪的努力，以证明第五公设的后果，其余的，罗巴切夫斯基和独立波尔约开始发展几何综合利用其否定。这导致了今天所谓的双曲几何。虽然双曲几何是违反直觉的，但在现实中，空间可以在某些地方弯曲，看起来像一个马鞍，双曲几何是正确的模型。本项目通过研究非交换自由群的模型论中的主要开放问题来研究模型论与双曲几何的联系。非交换自由群生成的自由群是双曲群的典型例子，即Cayley图是双曲度量空间的非交换生成群。自从Sela，Kharlampovich-Myasnikov等给出了它们具有相同的一阶理论的深刻结论后，它们就引起了模型理论界的广泛关注。后一个结果回答了塔斯基的一个问题，这个问题持续了50多年。也许令人惊讶的是，这个常见的一阶理论是稳定的。稳定性是Shelah分类程序中的一个驯服条件，也可能是其中最突出的分界线。该项目的一个主要目的是研究自然结构，如场和群，可以用非交换自由群的一阶理论解释，这是PI与Ayala Byron和Chloé Perin，Anand Pillay和Katrin Tent的先前工作的延续。非交换自由群的一阶理论似乎在分叉几何方面表现出一些意想不到的行为，这将由PI进行研究。项目的第二个组成部分涉及模型完整性和模型伴侣的概念。非交换自由群的一阶理论不是模型完备的，但它可能承认一个模型伴侣。在同一思路的性质存在闭子群的欧米茄剩余自由群将研究。PI将使用几何稳定性理论和几何群论的技术。特别是，如PI和其他人以前的结果所示，有用的工具和概念包括，分叉演算，一个basedeness和一般的充足层次结构，欧米茄剩余自由塔，测试序列，并通过RIP机的群体行动的理解。该项目的一个更普遍的目标是开发方法来解决更广泛类别的组中的类似问题，如双曲线组，组的自由产品或直角Artin组。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This research project centers around model theory and its interactions with hyperbolic geometry. Model theory is concerned with studying structures in a formal mathematical language. One can trace its origins in Gödel’s fundamental work on the completeness of first-order logic. Gödel, in his famous completeness theorem, settled once and for all the question of how truth and provability are connected. Roughly speaking a mathematical statement is provable if and only if it is true in all possible worlds. Although model theory seems to have originated in questions around the philosophy and foundations of mathematics, in recent years profound connections with core mathematical disciplines and computer science have been found. Hyperbolic geometry emerged from the refutation of Euclid’s fifth postulate, also known as the parallel postulate. After more than twenty centuries of efforts to prove the fifth postulate as a consequence of the rest, Lobachevsky and independently Bolyai started developing geometry synthetically employing its negation. This resulted to what is nowadays known as hyperbolic geometry. Although hyperbolic geometry is counterintuitive, in reality space can be curved at places, looking like a horse saddle, and hyperbolic geometry is the right model to apply. This project investigates the connections of model theory with hyperbolic geometry by studying major open questions about the model theory of nonabelian free groups.Nonabelian finitely generated free groups are prototypical examples of hyperbolic groups, i.e. finitely generated groups whose Cayley graph is a hyperbolic metric space. They have attracted much model theoretic attention after the profound result that they share the same first-order theory (Sela, Kharlampovich-Myasnikov). The latter result answered a question of Tarski that remained open for more than fifty years. Maybe surprisingly this common first-order theory is stable. Stability is a tameness condition in Shelah’s classification program and maybe the most prominent dividing line in it. A major aim of this project is the study of natural structures, like fields and groups, interpretable in the first-order theory of nonabelian free groups, which continues prior work of the PI with Ayala Byron and with Chloé Perin, Anand Pillay and Katrin Tent. The first-order theory of nonabelian free groups seems to exhibit some unexpected behavior with respect to the geometry of forking, that will be investigated by the PI. The second component of the project involves the notions of model completeness and model companion. The first-order theory of nonabelian free groups is not model complete, but it might admit a model companion. In the same line of thought properties of existentially closed subgroups of omega residually free groups will be studied. The PI will use techniques from geometric stability theory and geometric group theory. In particular, as demonstrated by previous results of the PI and others, useful tools and notions include, forking calculus, one-basedeness and in general the ample hierarchy, omega residually free towers, test sequences, and the understanding of group actions through Rips machine. A more general goal of the project is to develop methods to tackle similar questions in wider classes of groups, like hyperbolic groups, free products of groups or right-angled Artin groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.