Pragmatic 2011

Scientific activities

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

  1. Hilbert functions of graded modules I (Mats Boij)

  2. Complex and algebroid curves, the Puiseux theorem. (Ralf Fröberg)

June 21 2011

  1. Hilbert functions of graded modules II (Mats Boij)

  2. Semigroups of curves and blowups. (Ralf Fröberg)

June 22 2011

  1. Introductions to Macaulay2 and computations of betti numbers of general points in projective spaces. (Mats Boij)

  2. Resolutions of Du Val singularities as GIT quotients (Ralf Fröberg)

June 23 2011

  1. Cone of Betti diagrams ((Mats Boij)

  2. Variation of GIT quotients: flips and flops (Ralf Fröberg)

June 24 2011

  1. Introduction to toric geometry (Alexander Engström)

June 27 2011

  1. Cone of Hilbert functions and Betti diagrams (Mats Boij)

  2. Cox rings: an introduction (Ralf Fröberg)

Individual meetings

June 28 2011

  1. Open problems for Pragmatic 2011 (Mats Boij)

  2. Mori Dream Spaces and Small Q-factorial models (Ralf Fröberg)

Individual meetings

June 29 2011

  1. Schur modules (Mats Boij)

  2. Cones in Mori Theory: effective, movable and nef cones (Ralf Fröberg)

Individual meetings

June 30 2011

  1. Hilbert functions of points (Mats Boij)

  2. Essential curves and division of movable cones by flopping classes (Ralf Fröberg)

Individual meetings

July 1 2011

  1. Introduction to geometric invariant theory (Alexander Engström)

Individual meetings

July 4 2011

  1. Parameter spaces of points in the plane (Mats Boij)

  2. Combinatorial approach to contractions: marked polytopes and polynomials (Ralf Fröberg)

Individual meetings

July 5 2011

  1. Cone of Hilbert functions of non-standard graded rings (Mats Boij)

  2. Mory Theory as a Grand Unified Theory (Ralf Fröberg)

Individual meetings

July 6 2011

  1. 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