The goal of this school is to present the main research directions of combinatorial commutative algebra with a strong focus on its applicability in various fields, like combinatorics, statistics, and biology.
For this purpose we plan to offer integrated courses where the theoretical developments are directly motivated by topics from the above mentioned fields. This activity will be complemented by informal tutorial sessions, computer demos, and problem sessions lead by senior participants, where the audience has the chance to ask questions and discuss aspects of the material.
The EMS Summer School on Multigraded Algebra and Applications will take place in the frame of the National School of Algebra, an event with a long tradition in Romania which is organized by the Ovidius University of Constanta, the Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, and the University of Bucharest.
There will be the following two main courses:
Monomial and binomial ideals, by Juergen Herzog and Apostolos Thoma.
Contents: Toric rings and toric ideals; Groeobner bases; Convex polytopes; Edge rings of finite graphs; Toric rings and ideals arising from finite distributive lattices; Lattice and lattice basis ideals; The cotangent functors T^i and deformation theory; The cotangent functor T^1 for Stanley-Reisner rings and separability; Bi-Cohen-Macaulay graphs; T^1 for toric rings and separation.
Applicable combinatorial commutative algebra, by Ezra Miller and Thomas Kahle.
Contents: Multigradings, multigraded Hilbert series, affine semigroups; Multigraded free and injective resolutions; Irreducible resolutions and Alexander duality; Review of homology and persistent homology; Computing with Macaulay2; Solving polynomial equations with Groebner bases; Primary decomposition in applications; Random walks on discrete objects: Markov bases; Binomials, monoid algebras, congruences, and decompositions.