Newton-Okounkov Bodies are convex sets in Euclidean space that encode (asymptotic) information about sections of line bundles on algebraic varieties. This construction depends on a non-canonical choice of valuation on the function field of the variety. For example, Gelfand-Tsetlin polytopes and Feigin–Fourier–Littelmann–Vinberg polytopes can both be realized as Newton-Okounkov bodies of the same line bundles on flag varieties for different choices of valuations. In this talk, I will explain the construction of Newton-Okounkov bodies, their basic properties, and what kinds of problems they are useful for.
