Toric geometry (discovered in 1970s-1980s) and its modern generalization in a theory of Newton-Okounkov bodies serve as bridge between two mathematical "worlds"

— convex geometry, and in toric case —combinatorics and piecewise-linear geometry

— "mature" geometries (projective, algebraic, Kähler, symplectic, ...)

The notions of algebraic geometry (projective varieties, affine varieties, schemes, non-normal varieties; ampleness, discrepancies and types of singularities (smooth, terminal, canonical, log-terminal, (Q-)factorial, (Q-)Gorenstein..., Betti numbers) have counterparts in the world of convex and piecewise-linear lattice geometry (rational convex polytopes, rational convex cones, fans, sets in a lattice, convexity, reflexivity, simpliciality or simpleness, Delzant, ..., f-polynomials).

The bridge has proved to be useful in both directions:

— Stanley's proof that Dehn--Sommerville relations completely determine possible numbers of faces of a convex simplicial polytope: by associating to each such polytope P a family of toric varieties X and showing that the respective numbers correspond to Betti numbers of (intersection) homology of X. In this light Dehn-Sommervile relations are seen as Poincaré duality and Hard Lefschetz theorem.

— Bernstein—Kushnirenko theorem (1975) for number of roots of a system of polynomial equations as mixed volume of their Newton polytopes

— explicit toric MMP, and later the use of birational cobordism and torification techniques to reproof resolution of singularities and weak facotorization theorem (AKMW, 2002)

— Batyrev's construction (and its generalization by Borisov) of trillions of mirror-dual pairs of Calabi-Yau threefolds as resolutions of singularities of hypersurfaces in toric fourfolds associated with reflexive polytopes in 4 dimensions (resp. complete intersections associated with with higher-dimensional polytopes and nef partitions)

I will give basic definitions and sketch a dictionary, proving (or outlining/explaining) the results above.