Model Theory, Diophantine Geometry and Combinatorics

逻辑是一个科学领域，传统上在数学，哲学和计算机科学的学科中实践。模型论是数理逻辑的一个分支，它使用逻辑工具来探索已知的和新的数学结构（模型）。当这些结构具有几何性质时，我们倾向于称它们的研究为“驯服几何”。这个术语最早是由法国几何学家Grothendieck使用的，他在他的Esquisse d 'un Programme（1984）中设想了一个“拓扑模式”。他询问是否有一种严格的数学方法来隔离具有更好几何和拓扑属性的几何对象类别。模型论，通过o-极小性，或者更一般地说，驯服几何，为格罗滕迪克的问题提供了一个答案：我们可以专注于那些几何对象，这些对象是“可定义的”在一些特定的语言从数学逻辑。这种有意识的限制从数理逻辑中产生了新的工具，这些工具随后被用来获得引人注目的应用。事实上，长期存在的问题，从真实的，复杂的和代数几何，以及其他领域的数学已经解决了使用技术从驯服geometrics.This奖学金介绍了一套新的工具和思想驯服几何，以统一的方式来解决重要的问题，从模型论，丢番图几何和组合数学。中心模型理论的设置是具有NIP（非独立性）的结构，这也是统计学习和极值组合学中强大的Vapnik-Chervonenkis理论所熟悉的。独立性属性允许数学结构对集合的子集进行统一编码。禁止这种编码（NIP）提供了一条分界线，这条分界线在纯模型理论及其应用中都被证明是基本的。学科内的研究将在三个紧密交织的线程的连接处进行：1。NIP理论和可定义群：可定义群在模型论的核心地位至少有三十年，主要是因为它们在重要应用中的突出作用。例子包括真实的李群（可以在真实的域中定义）和代数群（可以在复数域中定义）。真实的和复数域都是NIP结构，其他更一般的拓扑或代数性质的结构也是如此。在这一领域最诱人的开放问题之一是了解NIP结构的更简单的拓扑和代数的“部分”，然后可以产生新的技术和应用程序的一般NIP设置。这条线索的目的是在可定义的群体的水平上实质性地推进这个问题的最新发展。2.组合数学的应用：重要的图组合问题，如Erdös-Hajnal猜想，已经解决了许多代数和拓扑结构，但在一般的NIP设置，他们仍然开放。他们在NIP环境中的解决方案将大大扩展这些假设的适用范围，但也标志着以下潜在的变革原则：非常抽象和纯粹的逻辑假设可以对组合问题产生影响。这条线索推进了这一原则，解决了图组合学和加法组合学的重要问题，使用了驯服几何学的工具。丢番图和代数几何的应用：著名的丢番图几何的解决方案，如Hrushovski的Mordell-Lang和Pila的André-Oort的某些情况，分别使用了模型论的重要工具，即Zilber二分法和Pila-Wilkie定理。这些定理将逻辑与数学的其他领域联系起来，例如数论：在可定义集合的某些数论假设下，人们可以恢复无限的代数子集。这个线程将这些定理扩展到更丰富的几何设置，产生新的强大的工具，进一步丢番图应用。

Logic is a scientific field traditionally practiced within the disciplines of mathematics, philosophy and computer science. Model theory is a branch of mathematical logic which uses logical tools to explore known and new mathematical structures (models). When those structures are of a geometric nature, we tend to call their research "tame geometry". This terminology was first used by the French geometer Grothendieck, who envisioned in his Esquisse d'un Programme (1984) a "topologie modérée". He asked whether there is a strict mathematical way to isolate classes of geometric objects which enjoy better geometrical and topological properties. Model theory, via o-minimality, or more generally, tame geometry, offers one answer to Grothendieck's question: we can focus on those geometric objects that are "definable" in some specific language from mathematical logic. This intentional restriction yields new tools from mathematical logic which are then used to obtain striking applications. Indeed, long-standing problems from real, complex and algebraic geometry, and other areas of mathematics have been solved using techniques from tame geometry.This Fellowship introduces a novel set of tools and ideas in tame geometry in order to tackle in a uniform way important problems from model theory, Diophantine geometry and combinatorics. The central model-theoretic setting is that of structures with NIP (Not the Independence Property) which are also familiar in the powerful Vapnik-Chervonenkis theory in statistical learning and extremal combinatorics. The Independence Property allows a mathematical structure to code uniformly the subsets of a set. Forbidding this coding (NIP) provides a dividing line which has proven fundamental in both pure model theory and its applications. Intradisciplinary research will be pursued at the nexus of three closely interwoven threads:1. NIP theories and definable groups: Definable groups have been at the core of model theory for at least three decades, largely because of their prominent role in important applications. Examples include real Lie groups (which are definable in the real field) and algebraic groups (which are definable in the complex field). Both the real and the complex field are NIP structures, and so are other structures of more general topological or algebraic nature. One of the most tantalizing open questions in this area is to understand NIP structures in terms of their simpler topological and algebraic 'parts', which can then yield new techniques and applications to the general NIP setting. This thread aims to advance substantially the state-of-the-art of this question at the level of definable groups.2. Applications to combinatorics: Important graph-combinatorial questions, such as the Erdös-Hajnal conjecture, have been solved for many algebraic and topological structures, but in the general NIP setting they remain open. Their solution in the NIP setting would both significantly expand the range of applicability of those conjectures, but also mark the following potentially transformative principle: very abstract and purely logical assumptions can have an impact on combinatorial questions. This thread advances this principle, tackling important conjectures from graph combinatorics and additive combinatorics, using tools from tame geometry.3. Applications to Diophantine and algebraic geometry: The solutions of famous conjectures from Diophantine geometry, such as Mordell-Lang by Hrushovski and certain cases of André-Oort by Pila, made crucial use of important tools from model theory; namely, the Zilber Dichotomy and the Pila-Wilkie theorem, respectively. These theorems relate logic with other areas of mathematics, such as number theory: under certain number-theoretic assumptions on definable sets, one can recover infinite algebraic subsets. This thread will extend these theorems to richer geometric settings, yielding new strong tools for further Diophantine applications.