Francesco Gallinaro's personal webpage

Research

What is it about?

The focus of my research is Boris Zilber's Exponential-Algebraic Closedness Conjecture.

When people who do math ask me what it is, I say that it is a conjecture predicting that certain systems of equations involving polynomials and exponentials have solutions in the complex numbers, that the interest of the conjecture comes from from model theory because, if confirmed, it would imply that all subsets of the complex numbers that are definable using polynomials and exponentials are countable or cocountable (another conjecture due to Zilber, usually referred to as the Quasiminimality Conjecture) and that similar questions can be asked concerning analytic functions other than the complex exponential, for example the exponential maps of complex abelian varieties and the modular invariant j .

When people who do not do math ask me what it is, I ask them if they remember learning in high school that solving a system of two linear equations in two variables is the same thing as intersecting two lines in the plane; regardless of whether they do, I then mumble something about my research being on doing a similar thing with more complicated sets: tackling a question of the form "does this system of equations have a solution" by trying to intersect some geometric sets.

Publications

• "On some systems of equations in abelian varieties", International Mathematics Research Notices, June 2023. We remove an assumption needed in the first part of "Solving Systems of Equations of Raising-to-Powers Type": the main result is that algebraic varieties which split as the product of a linear subspace of the tangent space at identity of an abelian variety and an algebraic subvariety of the abelian variety intersect the graph of the exponential. The ideas in the proof are mainly homological: the intersection product on homology groups of abelian varieties plays an important role.
• "Exponential sums equations and tropical geometry", Selecta Mathematica (New Series), 29 (2023). The main result is that algebraic varieties which split as the product of a linear subspace of a power of the complex number and an algebraic subvariety of a power of the complex multiplicative group intersect the graph of the exponential function. The proof uses methods from the theory of amoebas, tropical geometry, and toric varieties.
• "Solving systems of equations of raising-to-powers type", Israel Journal of Mathematics (to appear). The first main result deals with abelian varieties. A sufficient condition is given for the image of a linear subspace of the tangent space at identity under the exponential map to be dense in the Euclidean topology, and that is used to show that if the Cartesian product of the linear space and an algebraic subvariety of the abelian variety satisfies the assumptions of the Exponential-Algebraic Closedness conjecture and this additional assumption, then it intersects the graph of the exponential function. The second main result, which deals with the j -function, has a similar flavour: first it is shown that the images of certain Möbius subvarieties of Cartesian powers of the upper half plane have dense image under the j -function, and then that if the Cartesian product of a Möbius variety and an algebraic variety satifies the assumptions of the j -Algebraic Closedness Conjecture then it intersects the graph of j .
• "Dividing lines between positive theories" (with Anna Dmitrieva and Mark Kamsma), Journal of Symbolic Logic, published online 2023:1-25. We generalise to positive model theory properties from classification theory such as the Order Property, the Independence Property, the Tree Property and so on, and we prove various implications between them.

Preprints

• "Quasiminimality of complex powers" (with Jonathan Kirby, submitted). We use the main theorem from "Exponential Sums Equations and Tropical Geometry" and a strategy due to Bays and Kirby to show that the complex field equipped with all operators of raising to a complex power is quasiminimal.

• "The Fundamental theorem of tropical differential algebra over nontrivially valued fields and the radius of convergence of nonarchimedean differential equations" (with Stefano Mereta, submitted). We prove a version of the fundamental theorem of tropical geometry for tropical differential equations, showing that tropicalizing the set of solutions to a partial differential equations over a non-trivially valued field is the same as computing the solutions of the topicalized equation. Moreover, we develop parts of the theory of initial forms and of Berkovich analytification in this setting. This generalizes work of various authors, who had considered this problem for ODEs and/or over a field with trivial valuation.

PhD Thesis

I wrote my PhD thesis, "Around Exponential-Algebraic Closedness", between 2018 and 2022 at the University of Leeds, under the supervision of Vincenzo Mantova. It contains roughly the same things as the first three papers listed above (although I would recommend looking at the papers), together with a general introduction to the problem and some of the further directions of research that I cared about as of April 2022.