During
the courses the teachers introduced graded
algebras and modules arising in various possible applications,
such as semigroups, simplicial complexes and low-dimensional
varieties. In particular, were introduced various invariants, such
as Hilbert functions and graded Betti numbers, and their relations to
properties of the algebras and modules as well as at classification
problems, including the study of these invariants up to scaling.
Moreover, for these problems they make use of the computational tools
available through computer algebra systems as Macaulay2 and CoCoA. Many
individual meetings were used for training people on using
Macaulay 2and for approaching the proposed open problems.
In
particular, were addressed about 10 open problems on the previous
topics.
Here is the schedule of the
seminars and individual meetings during the period of the school.
Calendars
of seminars
June 20 2011
Hilbert functions of graded modules I (Mats Boij)
Complex and algebroid curves, the Puiseux theorem. (Ralf Fröberg)
June 21 2011
Hilbert functions of graded modules II (Mats Boij)
Semigroups of curves and blowups. (Ralf Fröberg)
June 22 2011
Introductions to Macaulay2 and computations of betti numbers of general points in projective spaces. (Mats Boij)
Resolutions of Du Val singularities as GIT quotients (Ralf Fröberg)
June 23 2011
Cone of Betti diagrams ((Mats Boij)
Variation of GIT quotients: flips and flops (Ralf Fröberg)
June 24 2011
Introduction to toric geometry (Alexander Engström)
June 27 2011
Cone of Hilbert functions and Betti diagrams (Mats Boij)
Cox rings: an introduction (Ralf Fröberg)
Individual meetings
June 28 2011
Open problems for Pragmatic 2011 (Mats Boij)
Mori Dream Spaces and Small Q-factorial models (Ralf Fröberg)
Individual meetings
June 29 2011
Schur modules (Mats Boij)
Cones in Mori Theory: effective, movable and nef cones (Ralf Fröberg)
Individual meetings
June 30 2011
Hilbert functions of points (Mats Boij)
Essential curves and division of movable cones by flopping classes (Ralf Fröberg)
Individual meetings
July 1 2011
Introduction to geometric invariant theory (Alexander Engström)
Individual meetings
July 4 2011
Parameter spaces of points in the plane (Mats Boij)
Combinatorial approach to contractions: marked polytopes and polynomials (Ralf Fröberg)
Individual meetings
July 5 2011
Cone of Hilbert functions of non-standard graded rings (Mats Boij)
Mory Theory as a Grand Unified Theory (Ralf Fröberg)
Individual meetings
July 6 2011
Weyl group action on movable divisors of MDS's (Alexander Engström)
Individual meetings
July 7 2011
5 Presentations of the open problems by the participants
July 8 2011
5 Presentations of the open problems by the participants