Toric geometry provides a useful dictionary between combinatorics and algebraic geometry. The theory of Newton-Okounkov bodies allows the combinatorial techniques of toric geometry to be applied to more general projective varieties. In a recent paper with Harada, we described a phenomenon of wall-crossing for Newton-Okounkov bodies, which involves piecewise-linear mutation maps between different Newton-Okounkov bodies associated to the same variety. Similar phenomena appear in the work of Rietsch-Williams, as well as Bossinger-Cheung-Magee-Nájera Chávez on Newton-Okounkov bodies associated to compactifications of cluster varieties. In addition, Kaveh and Manon have analyzed the theory of valuations into semifields of piecewise linear functions, and explored their connections to families of toric degenerations. Inspired by these ideas, in joint work with Harada and Manon, we wrap the data of a collection of lattices related by piecewise-linear bijections together into a single semi-algebraic object, equipped with its own notions of convexity and polyhedra. In certain situations, such a (generalized) polytope encodes a compactification of an affine variety, and aspects of the geometry of the compactification (including some of its toric degenerations) can be understood combinatorially.
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