I will talk about an application of the differentiability of the arithmetic volume function and an arithmetic Bertini type theorem to classify when one can find a closed point on the generic fiber of an arithmetic variety, whose heights with respect to some finite tuple of arithmetic R-divisors approximate a given tuple of real numbers.This result is used to prove existential closedness of $\mathbb{Q}^{alg}$ as a globally valued field (abbreviated GVF) - it is an arithmetic analogue of the function field case published recently by Ita Ben Yaacov and Ehud Hrushovski.
De Arthur Forey
De Sylvy Anscombe
De Raf Cluckers
De Pablo Cubides Kovacsics
De Lothar Sebastian Krapp
De Michal Szachniewicz
De Ya'acov Peterzil
De Mariana Vicaria
De Matthias Aschenbrenner
De Philip Dittmann
De Adele Padgett
