# Weak continuity on the variation of Newton Okounkov bodies

## Location

We start by presenting new tools and results suitable for

the study of valuations of higher rank on function fields of algebraic

varieties. This will be based on a study of higher rank quasi-monomial

valuations taking values in the lexicographically ordered group R^k.

This gives us a space of higher rank valuations that we endow with a

weak "tropical" topology. In this setting, we show that the Newton

Okounkov bodies of a given line bundle vary continuously with respect

to the valuation. We explain how this result fits in the literature

and how it gives us a restriction in the existence of mutations of

Newton Okounkov bodies. Joint work with Omid Amini.