Model Theory, Diophantine Geometry and Combinatorics

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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.