Abstract: In this talk, we will discuss a preprint of
Fernández-Nickel-Roé 'Newton-Okounkov bodies and Picard numbers on surfaces'
Newton–Okounkov bodies were introduced by A. Okounkov as a tool in
representation theory; later Kaveh-Khovanskii and Lazarsfeld-Mustata
developed a general theory with applications to both convex and algebraic
geometry. In this preprint, the authors study the shapes of all
Newton-Okounkov bodies of a given big divisor on a surface S with respect
to all rank 2 valuations of K(S). They obtain upper bounds for, and in
many cases determine exactly, the possible numbers of vertices of these
bodies. The upper bounds are expressed in terms of Picard numbers. They
also conjecture that the set of all Newton-Okounkov bodies of a single
ample divisor determines the Picard number of S, and proves that this is
the case for Picard number 1, by an explicit characterization of surfaces
of Picard number 1 in terms of Newton-Okounkov bodies.