Model theory, Diophantine geometry, and automorphic functions

点击翻译按钮获取中文摘要

My research is in model theory, a branch of mathematical logic with striking applications in many fields ranging from number theory to machine learning. The problems I study lie at the intersection of several mathematical disciplines such as model theory, number theory, algebra, and geometry. The model-theoretic perspective reveals new links between those disciplines providing a deeper insight into the problems.The aim of this fellowship is to develop novel model-theoretic tools to answer some fundamental questions about certain classical functions of a complex variable. A renowned example of such a function is the so-called j-function. It arises naturally in number theory and has applications in cryptography and coding theory. The j-function and other similar (more general) functions, known as automorphic functions, are the main objects of study in this proposal.The proposal focuses on two main problems. The first is to understand when systems of equations in several variables involving addition, multiplication, and automorphic functions have solutions in the complex numbers. This is a natural question in complex geometry and is the automorphic variant of the well-known problem of solvability of systems of polynomial equations in several variables, settled by Hilbert's Nullstellensatz (German for "theorem of zeroes"). I have proposed a conjectural analogue of the Nullstellensatz for automorphic functions (the Existential Closedness conjecture) and aim to prove it.The second main problem is to describe the sets of integral, rational or "special" solutions of polynomial equations in several variables (Diophantine equations) related to automorphic functions. Diophantine problems date back to the third century mathematician Diophantus of Alexandria and are among the oldest and hardest problems in mathematics. Modern methods of studying Diophantine equations are based on the idea that polynomial equations can be replaced by geometric objects (curves, surfaces, etc.), namely the sets of points satisfying these equations. Then geometric tools can be used to study rational points on these objects. One of the mainstream problems in Diophantine geometry is related to "unlikely intersections". For example, in a 3-dimensional space two randomly chosen lines are not likely to intersect, and when they do, it is an unlikely intersection. On the other hand, in a plane two such lines are likely to intersect. A famous open problem in the theory of unlikely intersections is the Zilber-Pink conjecture stating roughly that unlikely intersections of a variety with certain "special" varieties are controlled by finitely many special varieties. It is a far-reaching generalisation of some renowned Diophantine statements such as Mordell-Lang and André-Oort. Making progress on the Zilber-Pink conjecture for automorphic functions is a major aim of the proposal.One of the new ideas of the proposal is to consider Existential Closedness and Zilber-Pink for automorphic functions together with their derivatives. This results in more general and harder problems, but it also gives a deeper insight into the conjectures and into the full model-theoretic picture. Moreover, in that generality I have established new links between Existential Closedness and Zilber-Pink which opened the way to a powerful strategy of using the former to attack the latter. I have applied that strategy to obtain the first Zilber-Pink type theorems "with derivatives", with a lot more to explore. I intend to extend and exploit these links to tackle both problems in parallel. Investigating several other related questions, aimed at understanding the geometry of automorphic functions using model-theoretic techniques, are also among the goals of the fellowship. Making progress towards these questions would improve our understanding of the rich mathematical theory around automorphic functions including a notoriously hard open problem known as the Schanuel conjecture.