Recent years have witnessed a new way to introduce convex geometric methods to areas of mathematics around algebraic geometry: based on earlier works of Newton and Okounkov,
Kaveh-Khovanskii and Lazarsfeld-Mustata defined convex bodies (so-called Newton-Okounkov bodies), which capture the vanishing behaviour of sections of line bundles.
As a first approximation, the theory of Newton-Okounkov bodies is an attempt to create a correspondence between line bundles and convex bodies known from toric geometry, except that in the absence of a large torus action, one has to make do with an infinite collection of bodies for every line bundle.
This point of view has been fairly succesful in that Newton-Okounkov bodies has been shown to encode positivity of line bundles, and they also serve as targets for completely integrable systems analogous to moment maps.
The theory has exciting connections with symplectic geometry, representation theory, and combinatorics for instance, nevertheless, in these lectures we will focus on its applications to projective geometry.
After introducing Newton-Okounkov bodies and their basic theory, we will discuss the case of surfaces, where there is a particularly satisfying theory, and the connection to (local) positivity of line bundles. With this done, we will proceed to applications to higher syzygies of line bundles on abelian surfaces, and an exciting connection with Diophantine approximation.
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