I will discuss a basic problem in analytic number theory which has appeared recently in my work. This will be a gentle introduction to the Gauss circle problem, hopefully with a discussion of some extensions and applications to understanding L-functions.

I will introduce the Hodge-de-Rham spectral sequence and formulate an algebraic Hodge decomposition theorem. Time permitting, I will sketch Deligne and Illusie’s proof of the Hodge decomposition using positive characteristic methods.

I will introduce various heights on number fields and outline how solving polynomial equations with heights conditions is related to Arakelov geometry and a continuous logic theory called GVF.

In 2019 Masser and Zannier proved that "most" abelian varieties over the algebraic numbers are not isogenous to the jacobian of any curve (where "most" refers to an ordering by some suitable height function). We will see how this result fits in the general Zilber-Pink Conjecture picture and we discuss some (rather concrete) analogous problems in a power of the modular curve Y(1).