Model Theory, Diophantine Geometry and Combinatorics

逻辑是一个传统上在数学、哲学和计算机科学学科中实践的科学领域。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，我们倾向于称他们的研究为“驯服几何”。这个术语最早是由法国几何学家格罗滕迪克（Grothendieck）使用的，他在他的Esquisse d'un Programme（1984）中设想了一个“拓扑莫氏<s:2> <s:2> <s:2>”。他问是否存在一种严格的数学方法来隔离具有更好的几何和拓扑性质的几何物体类别。模型理论，通过极小性，或者更一般地说，驯服的几何，为格罗滕迪克的问题提供了一个答案：我们可以把重点放在那些用数学逻辑的特定语言“可定义”的几何对象上。这种有意的限制从数学逻辑中产生了新的工具，然后用于获得引人注目的应用。事实上，实几何、复杂几何和代数几何以及其他数学领域中存在已久的问题，已经用驯服几何的技术解决了。本奖学金介绍了一套新的工具和思想，在驯服几何，以统一的方式解决重要的问题，从模型理论，丢番图几何和组合。模型理论的中心设置是具有NIP（非独立性）的结构，这在统计学习和极值组合学中强大的Vapnik-Chervonenkis理论中也很熟悉。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，这在纯模型理论及其应用中都被证明是基本的。跨学科研究将在三个紧密交织的线索的关系中进行：1。NIP理论和可定义群：可定义群已经成为模型理论的核心至少三十年了，主要是因为它们在重要应用中的突出作用。例子包括实李群（在实域中可定义）和代数群（在复域中可定义）。实域和复杂域都是NIP结构，其他更一般的拓扑或代数性质的结构也是NIP结构。该领域最诱人的开放问题之一是从更简单的拓扑和代数“部分”的角度理解NIP结构，这可以为一般的NIP设置提供新技术和应用。这条线索的目的是在可定义群体的水平上实质性地推进这一问题的最新进展。在组合学中的应用：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设置中，它们仍然是开放的。他们在NIP环境中的解决方案将大大扩展这些猜想的适用性范围，但也标志着以下潜在的变革原则：非常抽象和纯逻辑的假设可以对组合问题产生影响。这条线索推进了这一原则，利用普通几何中的工具，处理图组合学和加性组合学中的重要猜想。丢芬图和代数几何的应用：丢芬图几何中一些著名猜想的解，如赫鲁晓夫斯基的莫德尔-朗猜想和皮拉的安德列-奥尔特猜想的某些解，都对模型论中的重要工具进行了重要的应用；分别是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，人们可以恢复无限代数子集。这个线程将这些定理扩展到更丰富的几何设置，为进一步的丢番图应用提供新的强大工具。

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.