Commutative Algebra and Combinatorics

1. Finite free resolutions of modules and related invariants: Betti numbers, Castelnuovo-Mumford regularity, depth. 2. Exterior, symmetric and Rees algebras of finitely generated modules and their invariants. 3. Combinatorial objects from a Commutative Algebra point of view and their invariants: graphs, lattices, simplicial complexes, polyominoes. 4. Hilbert series of graded algebras. 5. Cohen-Macaulay and Gorenstein rings, and their generalizations such as almost Gorenstein rings, nearly Gorenstein rings and sequentially Cohen-Macaulay modules. 6. Numerical semigroups, generalized numerical semigroups, and affine semigroups. 7. Graded Betti numbers, projective dimension, and regularity of the quotient ring associated with k-partite planar graphs 8. Implementation (of the number) of spanning trees and determination of graceful labeling of connected cycle-related graphs. 9. Existence and convergence of solutions of nonlinear equations on locally finite weighted graphs. 10. Topological and variational methods for non-compact orientable differentiable manifolds.