# Model theory, Diophantine geometry and combinatorics

Model theory, Diophantine geometry and combinatorics

Logic is a scientific field traditionally practised within the disciplines of mathematics, philosophy and computer science. Model theory is a branch of mathematical logic that 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 strictly 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 that 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 fields 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.

### Impact

The impact of this research will mostly be academic and of a public engagement in science nature. The former is detailed in "Academic Impacts" and "Academic Beneficiaries". The short term economic and societal impacts will be focused on public engagement in science. Outreach activities will be planned to increase accessibility, visibility, and awareness on topics related to the Fellowship. They will target a broad audience, including high-school pupils and A-level students, who might get excited by research and encouraged to pursue a Maths degree in a strong UK university; undergraduate students, who might get inspired to follow postgraduate studies in mathematics; maths teachers, who might enrich their general mathematical background, often adopting new perspectives in explaining mathematical concepts; academic non-mathematicians, who might get inspired by ideas from another discipline and engage in interdisciplinary research; and members of the wider public who normally do not engage in science, but they appreciate its role in our society and might benefit from the associated enhancement to their quality of life.

The research topics of this Fellowship are ideal for promotion to the targeted groups. The Objectives concern structural theorems in tame geometry, drawing a sharp dividing line between what is considered tame and what non-tame. This dividing line can be conveyed clearly to the above audiences, by visualizing, at the one extreme, beautiful algebraic surfaces (that can be defined via polynomial equations), and at the other extreme, exotic fractals (whose definition demands infinite repetition). The dividing line dates back to the origins of mathematical logic, with Gödel's famous Incompleteness Theorem, which drew a line between what is provable/non-provable, and Turing machines which drew a line between decidable/undecidable. These elements of the foundations of mathematics are particularly potent for catching the public imagination in all targeted groups, since they draw a deep connection between mathematics and computers, which are used in our everyday life, and tackle fundamental philosophical queries about the nature of mathematics, and more extendedly, of our everyday reasoning.

The PDRA and I will promote these ideas to the targeted groups via popular science press, Café Scientific talks, festivals, school presentations, a summer school week, interdisciplinary seminars across the University, and monitoring Wikipedia pages. It is of course very hard to predict the longer-term impact of pure mathematics research. However, as noted on the EPSRC "Logic and combinatorics" portfolio webpage, "Both Logic and Combinatorics are underpinning fundamental research areas and so play a key role in supporting ongoing research in other areas of the mathematical sciences and other disciplines such as Information and Communication Technologies (ICT)."

The project as a whole may thus also be viewed as a "Pathway to Impact" for model theory, promoting possible future economic and societal impacts of the field by way of its influence on combinatorics. For example, the NIP combinatorial condition, localized to a formula, asserts that a family of sets (defined by that formula) has finite VC-dimension. This brings connections to the powerful VC-theory in statistics, and industry of applications in statistical learning theory, neural networks, probably-approximately-correct learning in AI, and compression of information in computer science.

There are potential long-term applications of NIP theories to machine learning, and a need to identify problems from both areas that can be solved using tools from each other. I will facilitate this interaction and build links between model theorists and computer scientists by inviting them to the Leeds workshops. The Fellowship will also have a considerable impact through the training and professional development of the PDRA for both an academic and non-academic career.

### Project website

https://gow.epsrc.ukri.org/NGBOViewGrant.aspx?GrantRef=EP/V003291/1