Schedule of the courses:

- 1st week 12-16 May: Free resolutions. Exactness criteria. Definition and criteria of regularity. Koszul complexes and their generalizations. Koszul algebras. Eagon-Northcott complex.

- 2nd week 19-23 May: Free resolutions of sets of points. Applications of the Hilbert-Burch Theorem. The conjecture on expected resolutions. Gale-Coble duality and its applications.

- 3rd week 26-30 May: Seminars, exercises, trainings on related topics by the young participants. Meetings for a starting organization of projects of researches. Initial draft of the notes of the course (by the participants)

- 4th week 2-6 June: Curves. The Gruson-Lazarsfeld-Peskine Theorem on regularity. Resolution of rational and elliptic curves. Liaison.

- 5th week 9-13 June: Higher dimensions. General facts. Toric examples. Surfaces in P4. Other examples. Meetings for a final organization of projects of researches. Final draft of the notes of the course (by the participants).