Polyhedral Geometry and Linear Optimization Polyhedral sets and cones Relative volumes of lattice polytopes Hilbert bases and TDI systems Rees cones and clutters The integral closure of a semigroup Unimodularity of matrices and normality Normaliz, a computer program Cut-incidence matrices and integrality Elementary vectors and matroids Commutative Algebra Module theory Graded modules and Hilbert polynomials Cohen-Macaulay modules Normal rings Valuation rings Krull rings Koszul homology A vanishing theorem of Grothendieck Affine and Graded Algebras Cohen-Macaulay graded algebras Hilbert Nullstellensatz Gröbner bases Projective closure Minimal resolutions Rees Algebras and Normality Symmetric algebras Rees algebras and syzygetic ideals Complete and normal ideals Multiplicities and a criterion of Herzog Jacobian criterion Hilbert Series Hilbert-Serre Theorem a -invariants and h -vectors Extremal algebras Initial degrees of Gorenstein ideals Koszul homology and Hilbert functions Hilbert functions of some graded ideals Stanley-Reisner Rings and Edge Ideals of Clutters Primary decomposition Simplicial complexes and homology Stanley-Reisner rings Regularity and projective dimension Unmixed and shellable clutters Admissible clutters Hilbert series of face rings Simplicial spheres The upper bound conjectures Edge Ideals of Graphs Graph theory Edge ideals and B-graphs Cohen-Macaulay and chordal graphs Shellable and sequentially C-M graphs Regularity, depth, arithmetic degree Betti numbers of edge ideals Associated primes of powers of ideals Toric Ideals and Affine Varieties Binomial ideals and their radicals Lattice ideals Monomial subrings and toric ideals Toric varieties Affine Hilbert functions Vanishing ideals over finite fields Semigroup rings of numerical semigroups Toric ideals of monomial curves Monomial Subrings Integral closure of monomial subrings Homogeneous monomial subrings Ehrhart rings The degree of lattice and toric ideals Laplacian matrices and ideals Gröbner bases and normal subrings Toric ideals generated by circuits Divisor class groups of semigroup rings Monomial Subrings of Graphs Edge subrings and ring graphs Incidence matrices and circuits The integral closure of an edge subring Ehrhart rings of edge polytopes Integral closure of Rees algebras Edge subrings of complete graphs Edge cones of graphs Monomial birational extensions Edge Subrings and Combinatorial Optimization The canonical module of an edge subring Integrality of the shift polyhedron Generators for the canonical module Computing the a -invariant Algebraic invariants of edge subrings Normality of Rees Algebras of Monomial Ideals Integral closure of monomial ideals Normality criteria Rees cones and polymatroidal ideals Veronese subrings and the a -invariant Normalizations of Rees algebras Rees algebras of Veronese ideals Divisor class group of a Rees algebra Stochastic matrices and Cremona maps Combinatorics of Symbolic Rees Algebras of Edge Ideals of Clutters Vertex covers of clutters Symbolic Rees algebras of edge ideals Blowup algebras in perfect graphs Algebras of vertex covers of graphs Edge subrings in perfect matchings Rees cones and perfect graphs Perfect graphs and algebras of covers Combinatorial Optimization and Blowup Algebras Blowup algebras of edge ideals Rees algebras and polyhedral geometry Packing problems and blowup algebras Uniform ideal clutters Clique clutters of comparability graphs Duality and integer rounding problems Canonical modules and integer rounding Clique clutters of Meyniel graphs Appendix: Graph Diagrams Bibliography Notation Index Injective closure Minimal resolutions Rees Algebras and Normality Symmetric algebras Rees algebras and syzygetic ideals Complete and normal ideals Multiplicities and a criterion of Herzog Jacobian criterion Hilbert Series Hilbert-Serre Theorem a -invariants and h -vectors Extremal algebras Initial degrees of Gorenstein ideals Koszul homology and Hilbert functions Hilbert functions of some graded ideals Stanley-Reisner Rings and Edge Ideals of Clutters Primary decomposition Simplicial complexes and homology Stanley-Reisner rings Regularity and projective dimension Unmixed and shellable clutters Admissible clutters Hilbert series of face rings Simplicial spheres The upper bound conjectures Edge Ideals of Graphs Graph theory Edge ideals and B-graphs Cohen-Macaulay and chordal graphs Shellable and sequentially C-M graphs Regularity, depth, arithmetic degree Betti numbers of edge ideals Associated primes of powers of ideals Toric Ideals and Affine Varieties Binomial ideals and their radicals Lattice ideals Monomial subrings and toric ideals Toric varieties Affine Hilbert functions Vanishing ideals over finite fields Semigroup rings of numerical semigroups Toric ideals of monomial curves Monomial Subrings Integral closure of monomial subrings Homogeneous monomial subrings Ehrhart rings The degree of lattice and toric ideals Laplacian matrices and ideals Gröbner bases and normal subrings Toric ideals generated by circuits Divisor class groups of semigroup rings Monomial Subrings of Graphs Edge subrings and ring graphs Incidence matrices and circuits The integral closure of an edge subring Ehrhart rings of edge polytopes Integral closure of Rees algebras Edge subrings of complete graphs Edge cones of graphs Monomial birational extensions Edge Subrings and Combinatorial Optimization The canonical module of an edge subring Integrality of the shift polyhedron Generators for the canonical module Computing the a -invariant Algebraic invariants of edge subrings Normality of Rees Algebras of Monomial Ideals Integral closure of monomial ideals Normality criteria Rees cones and polymatroidal ideals Veronese subrings and the a -invariant Normalizations of Rees algebras Rees algebras of Veronese ideals Divisor class group of a Rees algebra Stochastic matrices and Cremona maps Combinatorics of Symbolic Rees Algebras of Edge Ideals of Clutters Vertex covers of clutters Symbolic Rees algebras of edge ideals Blowup algebras in perfect graphs Algebras of vertex covers of graphs Edge subrings in perfect matchings Rees cones and perfect graphs Perfect graphs and algebras of covers Combinatorial Optimization and Blowup Algebras Blowup algebras of edge ideals Rees algebras and polyhedral geometry Packing problems and blowup algebras Uniform ideal clutters Clique clutters of comparability graphs Duality and integer rounding problems Canonical modules and integer rounding Clique clutters of Meyniel graphs Appendix: Graph Diagrams Bibliography Notation Index InAdmissible clutters Hilbert series of face rings Simplicial spheres The upper bound conjectures Edge Ideals of Graphs Graph theory Edge ideals and B-graphs Cohen-Macaulay and chordal graphs Shellable and sequentially C-M graphs Regularity, depth, arithmetic degree Betti numbers of edge ideals Associated primes of powers of ideals Toric Ideals and Affine Varieties Binomial ideals and their radicals Lattice ideals Monomial subrings and toric ideals Toric varieties Affine Hilbert functions Vanishing ideals over finite fields Semigroup rings of numerical semigroups Toric ideals of monomial curves Monomial Subrings Integral closure of monomial subrings Homogeneous monomial subrings Ehrhart rings The degree of lattice and toric ideals Laplacian matrices and ideals Gröbner bases and normal subrings Toric ideals generated by circuits Divisor class groups of semigroup rings Monomial Subrings of Graphs Edge subrings and ring graphs Incidence matrices and circuits The integral closure of an edge subring Ehrhart rings of edge polytopes Integral closure of Rees algebras Edge subrings of complete graphs Edge cones of graphs Monomial birational extensions Edge Subrings and Combinatorial Optimization The canonical module of an edge subring Integrality of the shift polyhedron Generators for the canonical module Computing the a -invariant Algebraic invariants of edge subrings Normality of Rees Algebras of Monomial Ideals Integral closure of monomial ideals Normality cri.