What are globally valued fields?

Given a projective tuple of rational numbers (q_1:\dots:q_n) a basic measure of complexity is its height, defined as \log \max_i |q_i|, when (q_1,\dots,q_n) are integers generating the unit ideal. Heights are fundamental in the study of diophantine equations, paralleling the other important notions in number theory, e.g. valuations or derivations on fields. A globally valued field is a field with an abstract notion of complexity of projective tuples, satisfying suitable axioms. These were defined by Ben Yaacov and Hrushovski and are closely related to M-fields of Gubler and adelic curves of Chen and Moriwaki. I will talk about various examples of these structures and we will relate them to product formulas. Many features of global fields generalise to globally valued fields – there are variants of geometry of numbers, heights of cycles (say Faltings heights), equidistribution. If time permits I will talk about a joint work with Nuno Hultberg and Antoine Sedillot, where we use these features to study arithmetic Siu inequality for globally valued fields over globally valued fields extending function fields.