Francesco Gallinaro's personal webpage


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.



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.