Research

Broadly speaking, I am interested in the interactions between model theory, number theory, differential algebra and algebraic geometry. In particular, I am very interested in understanding the properties of important analytic functions one encounters in maths. 

Research Program 

When studying a specific function there are many topics one can focus on: does the function have nice analytic properties (continuous, differentiable, etc.)?, does it satisfy a differential equation?, does it satisfy any polynomial equations?, does it behave well with respect to some kind of algebraic structure (e.g. is it a group homomorphism)?, does it encode arithmetic information?

I mostly work with functions coming from arithmetic geometry, such as the complex exponential function or the modular j function, and there are three main problems I focus on:

1.  Schanuel's conjecture (and its variants) from transcendental number theory, a special case of the far-reaching Grothendieck-André generalised period conjecture.

2.  The Existential Closedness problem, a natural intersection between analytic number theory and model theory, speaks about the size of the intersection between algebraic varieties and the graph of the function being considered.

3.  The Zilber-Pink conjecture from arithmetic geometry, governing the behaviour of "unlikely intersections". This conjectures generalizes many important results in arithmetic geometry, such as the conjectures (which are now theorems) of Manin-Mumford, Mordell-Lang, and André-Oort. 

Although these problems have different origin stories, they interact with each other in fascinating ways, and they all point to very deep properties of the functions in question. There are, as usual, both weaker and stronger variants of these conjectures, as well as versions that can be stated in other settings (such as in the context of differential fields), and my research also looks at those problems. In some cases, it is possible to show that the stronger versions actually follow from the original statements.

A major motivation of this research program are Boris Zilber's results on pseudo-exponentiation (further studied and completed by Martin Bays and Jonathan Kirby, among others). The three conjectures above arise somewhat naturally when one is trying to understand the algebraic properties of important transcendental functions in arithmetic geometry. In particular, Zilber's work conjectures a precise algebraic axiomatisation of the complex exponential function. Even though Zilber's work focused originally on exponentiation, his ideas have been expanded to many other settings in arithmetic geometry and Hodge theory. 

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Some of our contributions 

The following is a summary of the work I've done along with my collaborators.

• We have solved many cases of the existential closedness problem for the modular j function, and more generally for the uniformisation maps of Shimura varieties. The methods we have used can be extended even further to more general settings, in particular, we have shown that they can be adapted to include the derivatives of j, and they can even handle some non-periodic functions such as the Gamma function.

• We have completely solved differential versions of the existential closedness problem for both the exponential function and the j function. As before, our methods seem to be very general and can be applied to many other settings. 

• We have shown that the three conjectures above (Schanuel, Existential Closedness and Zilber-Pink), all together imply a strong version of existential closedness. Furthermore, combining this with some of the unconditional results already mentioned, we can show many cases in which one obtains the strong version of existential closedness unconditionally. 

• We have proven a strong counterpart of the Zilber-Pink conjecture (in a very general setting), namely that "likely intersections" exist and are in fact as likely as they can be. 

• We have proven a special case of the Zilber-Pink conjecture involving multiplicative relations among differences of singular moduli. 

Some Talks 

Here are some recorded talks I have given. 

• Géométrie et Théorie des Modèles
  Generic Solutions to Systems of Equations Involving Functions from Arithmetic Geometry, March 2023, video.

• Geometry, Arithmetic and Differential Equations of Periods seminar
  Generic Solutions to Equations Involving the Modular j-Function, June 2022, video. 

• Kolchin Seminar in Differential Algebra
  Existential Closedness and Differential Algebra, April 2022, video.

• Diophantine Problems Seminar, part of the MSRI program on Decidability, Definability and Computability in Number Theory 
  The Existential Closedness Problem for the Modular j-Function, October 2020, video.

• Seminario LATEN
  Algunas Consecuencias del Teorema de Ax-Schanuel para la Función j modular,  September 2020, video (in Spanish).