Model Theory, Diophantine Geometry and Combinatorics

模型理论、丢番图几何和组合数学

基本信息

批准号：
EP/V003291/1
负责人：
Panteleimon Eleftheriou
金额：
$ 107.99万
依托单位：
University of Leeds
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV003291%2F1
关键词：
Model Theory Diophantine Geometry Combinatorics

项目摘要

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.

逻辑是传统上在数学，哲学和计算机科学学科中实践的科学领域。模型理论是数学逻辑的一个分支，它使用逻辑工具探索已知和新的数学结构（模型）。当这些结构具有几何特性时，我们倾向于将其研究称为“驯服几何”。该术语最初是由法国地理表Grothendieck使用的，他在他的Esquisse d'Un计划（1984年）中设想了“拓扑古Modérée”。他问是否有一种严格的数学方法来隔离享有更好几何和拓扑特性的几何对象类别。模型理论是通过O-Wimimation或更一般的TAME几何形状来解决Grothendieck的问题的一个答案：我们可以专注于从数学逻辑中某种特定语言中“定义”的几何对象。这种有意的限制产生了数学逻辑中的新工具，然后将其用于获取引人注目的应用程序。的确，使用驯服几何学的技术解决了实际，复杂和代数几何形状以及其他数学领域的长期问题。该奖学金在驯服的几何学中引入了一系列新颖的工具和思想，以便以统一的方式解决模型理论，二磷甘氨酸的质量和组合师的重要问题。中心模型的理论环境是具有NIP（不是独立性）的结构，这些结构在统计学习和极端组合学中强大的Vapnik-Chervonenkis理论中也很熟悉。独立属性允许数学结构均匀地编码集合的子集。禁止此编码（NIP）提供了一条分界线，该线路在纯模型理论及其应用中都证明了基础。学科的研究将在三个紧密交织的线程的联系中进行：1。 NIP理论和定义组：可确定的群体至少是模型理论的核心，这主要是因为它们在重要应用中的重要作用。示例包括真实的谎言组（在实际字段中定义）和代数组（在复杂场中定义）。真实和复杂领域都是nip结构，其他更通用拓扑或代数性质的结构也是如此。在该领域，最诱人的开放问题之一是从简单的拓扑和代数“零件”来了解NIP结构，然后可以为一般的NIP设置产生新的技术和应用。该线程旨在在可定义的组级别上大大推进该问题的最新问题。2。组合学的应用：为许多代数和拓扑结构解决了重要的图形组合问题，例如Erdös-Hajnal概念，但在一般的NIP设置中，它们仍然保持开放。他们在NIP设置中的解决方案都可以显着扩大这些猜想的适用性范围，但也标志着以下潜在的变换原理：非常抽象和纯粹的逻辑假设可能会对组合问题产生影响。该线程通过使用驯服几何形状的工具3。应用于Diophantine和代数的几何形状：来自Diophantine几何形状的著名猜想的解决方案，例如Hrushovski的Mordell-Lang，以及Pila的某些André-Oort案例，使模型理论中重要的工具对重要工具进行了至关重要的使用；也就是说，Zilber二分法和Pila-Wilkie定理。这些定理将逻辑与其他数学领域（例如数字理论）相关联：在可定义集的某些数字理论假设下，人们可以恢复无限的代数子集。该线程将将这些定理扩展到更丰富的几何环境，从而为进一步的二磷剂应用提供新的强大工具。