Research
Research
Broadly speaking, I am interested in the interactions between model theory, number theory, differential algebra and algebraic geometry.
Many problems in Diophantine geometry are concerned with the solutions to systems of equations coming from some specific set of interest. Classically, the set of interest is the integers, and the problem is to determining whether the system has infinitely many solutions in this set (and if so, give formulas parameterizing the solutions), or only finitely many (and if so, find all of them). This is a very difficult problem in general, but an overarching philosophy is that the presence of an abundance of integer solutions should be explained by the geometry of the system of equations. An example of this is the celebrated theorem of Faltings.
In addition to the integers, many other sets have been studied in a similar way. For example, the torsion points of some algebraic groups like abelian varieties are considered an interesting set and are the focus of the Manin—Mumford conjecture (proved by M. Raynaud). What makes a set interesting depends on context, but usually its interest stems from some arithmetic properties it exhibits.
Many "special" functions are used to understand the sets of interest: complex exponentiation is used to describe the roots of unity, the modular j-function is used to characterize elliptic curves with complex multiplication, the Gamma function is used to study factorials, and Riemann's zeta function is used to prove the prime number theorem on the distribution of prime numbers.
When working with such functions in the context of systems of algebraic equations, facing questions regarding their algebraic properties becomes unavoidable. All of these functions are transcendental, so they only exhibit very specific algebraic properties (if any). One way of studying the properties of these functions is to think of the graph of the function as a new set of interest, which has already led to significant progress in functional transcendence, a major ingredient for solving certain questions in Diophantine geometry such as the André—Oort conjecture.
These properties can be classified into three types:
Functional properties: equations satisfied by the whole function.
Generic properties: equations that are not necessarily satisfied by the whole function but are satisfied by "an abundance" of points in the graph of the function. Examples of this can be found in transcendental dynamics; for instance, one can show that complex exponentiation (and many other transcendental functions) has infinitely many fixed points (in fact, it has infinitely many periodic points of every period), despite the fact that exponentiation is not the identity function.
Special values: equations that are satisfied only by specific points in the graph of the function. These types of equations may lead to questions in transcendental number theory, and as such they are sometimes considered to be out of reach.
A major source of inspiration for this research program are Boris Zilber's results on pseudo-exponentiation (further studied by Martin Bays and Jonathan Kirby, among others). In particular, Zilber's work conjectures a precise algebraic axiomatization of the complex exponential function which claims that the algebraic properties of exponentiation are governed by three properties:
Schanuel's conjecture from transcendental number theory, is a special case of André's generalization of Grothendieck's period conjecture, and tackles transcendence questions of special values of exponentiation.
The Exponential Algebraic Closedness conjecture, also known as the existential closedness problem, naturally lies in the intersection between analytic number theory and model theory. It speaks about the "randomness" of the intersection between algebraic varieties and the graph of exponentiation, giving a precise notion of "abundance" and randomness.
The Zilber—Pink conjecture, also known as the conjecture on the intersection with tori when dealing the exponential function, comes from arithmetic geometry and governs the behavior of "unlikely intersections". This conjectures generalizes many important results in arithmetic geometry, such as the conjectures of Manin—Mumford, Mordell—Lang, and André—Oort (all of which are now theorems).
Although these problems have different origin stories, they interact with each other in fascinating ways. While Zilber's work focused originally on exponentiation, his ideas have been expanded to many other settings in arithmetic geometry and Hodge theory.
The following is a summary of the work I've done together 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 varietiesm. 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 differentially transcendental 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.
We have proven that the modular Zilber—Pink conjecture holds for generic varieties.
We have classified the bialgebraic varieties of the Gamma function.
Here are some recorded talks I have given.
Workshop on Model Theory, Algebraic Dynamics, and Differential-Algebraic Geometry,
Modular Zilber—Pink for generic varieties, June 2025, video.
XXIV Colloquio Latinoamericano de Álgebra,
Likely Intersections, July 2024, video.
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).