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）。我们倾向于将他们的研究称为“驯服几何”。这个术语首先由法国几何学家格罗腾迪克使用，他在他的《Esquisse d'un Program》（1984）中设想了一种“拓扑学”。是一种严格的数学方法，通过 o 极小性，或者更一般地说，驯服几何，来隔离具有更好几何和拓扑属性的几何对象类别，为格洛滕迪克的问题提供了一个答案：我们可以专注于那些具有更好几何和拓扑属性的几何对象。这种有意的限制从数理逻辑中产生了新的工具，然后这些工具被用来获得引人注目的应用。事实上，来自实数、复杂和代数几何以及其他数学领域的长期存在的问题已经得到了解决。解决了使用驯服几何学的技术。该奖学金介绍了驯服几何学中的一套新颖的工具和思想，以便以统一的方式解决模型理论、丢番图几何和组合学中的重要问题。中心模型理论设置是 NIP 结构的设置。（不是独立性），这在统计学习和极值组合学中强大的 Vapnik-Chervonenkis 理论中也很常见。独立性允许数学结构统一编码 a 的子集。禁止这种编码（NIP）提供了一条分界线，该分界线在纯模型理论及其应用中将在三个紧密交织的线索之间进行：1。至少三十年来一直是模型理论的核心，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（可在实数域中定义）和代数群（可在实数域中定义）。实数域和复数域都是 NIP 结构，其他更一般的拓扑或代数结构也是如此，该领域最诱人的开放问题之一是从更简单的拓扑和结构的角度来理解 NIP 结构。代数“部分”，然后可以在一般 NIP 设置中产生新的技术和应用。该线程旨在在可定义组的级别上大幅推进该问题的最新技术。2。组合学：重要的图组合问题，例如 Erdös-Hajnal 猜想，已经针对许多代数和拓扑结构得到了解决，但在一般 NIP 设置中它们仍然是开放的，它们在 NIP 设置中的解决方案都将显着扩展适用范围。这些猜想，但也标志着以下潜在的变革性原则：非常抽象和纯粹的逻辑假设可以对组合问题产生影响，该线程推进了这一原则，从图形中协商重要的猜想。组合学和加性组合学，使用简单几何中的工具。3.在丢番图和代数几何中的应用：丢番图几何中著名猜想的解决方案，例如赫鲁索夫斯基的莫德尔-朗和皮拉的安德烈-奥尔特的某些案例，都充分利用了模型理论的重要工具；即 Zilber 二分法和 Pila-Wilkie 定理，分别将逻辑与其他领域联系起来。数学，例如数论：在可定义集合的某些数论假设下，可以恢复无限代数子集，该线程将这些定理扩展到更丰富的几何设置，为丢番图的进一步应用提供新的强大工具。