P.R.A.G.MAT.I.C. (Promotion of Research in Algebraic Geometry for MAThematicians in Isolated Centres) is a project for stimulating researches in Algebraic Geometry and Commutative Algebra among young people.
Pragmatic aims to give the opportunity of new collaborations and scientific horizons to young talented mathematicians, trying to create a center for training young mathematicians and to help young scientists to find their own area of research.
In pursuit of these goals Pragmatic will organize a period of lectures and seminars on very concrete problems and on techniques to solve them related to fundamental topics in Algebraic Geometry and in Commutative Algebra.

Abstract:

The two courses aim at presenting two apparently different but intimately related theories about sheaves on manifolds: the one about atomic sheaves and the one about twisted ones.

The first course will present some recent advances in the theory of stable sheaves on higher dimensional hyper-Kähler manifolds. First we will review Mukai's theory on K3 surfaces. Then we will consider hyper-Kähler fourfolds and this will naturally lead us to introduce
the recent theory of atomic sheaves. In this direction, we will discuss general results due to O'Grady, Markman, Beckmann, and Bottini.

The second course will be about twisted sheaves and their many appearances in algebraic geometry. After surveying some foundational material about Brauer groups and twisted sheaves, we will focus on the construction of moduli spaces of twisted sheaves and their relation with Hodge theory. In this direction we will review the proof of Shafarevich conjecture and the role of twisted sheaves in the recent proofs of some cases of the Hodge conjecture. On the more arithmetic side we will discuss some recent striking results proving the so-called period-index conjecture for Brauer groups which encodes a deep relationship between two natural invariants associated to elements of the Brauer group.

One of the objectives of the two courses is to show the fruitful interplay between the two subjects.