Combinatorics and Algebraic Geometry have enjoyed a fruitful interplay since the nineteenth century. Classical interactions include invariant theory, theta functions, and enumerative geometry. The aim of this volume is to introduce recent developments in combinatorial algebraic geometry and to approach algebraic geometry with a view towards applications, such as tensor calculus and algebraic statistics. A common theme is the study of algebraic varieties endowed with a rich combinatorial structure. Relevant techniques include polyhedral geometry, free resolutions, multilinear algebra, projective duality and compactifications.

Koszul algebras, Koszul homology and syzygies.- Infinite-dimensional systems of polynomial equations with symmetry.- Maximum Likelihood Geometry.- Linear Toric fibrations and Cayley polytopes.- Toroidal compactifications and tropicalizations of moduli spaces.