Research School PRAGMATIC - Announcement

it is our pleasure to announce the new edition of PRAGMATIC Research School.

Pragmatic 2023 “Cohomology and Frobenius"
Catania, Italy, June 12th - June 30th, 2023
(arrival date June 11th; departure date July 1st)

People who wish to be considered either for participation and/or for financial support should fill out the application form available at:
and then send it to the following address:
E-mail: <>

The deadline for applications is March 31th, 2023. The Committee of Pragmatic will decide about financial supports and admissions within April 15th, 2023.
Interested people who need more information about Pragmatic can write to the above e-mail address or contact any member of the local committee.

The Organization of Pragmatic 2023 will cover the boarding and lodging expenses of the participants which are not supported by their institutions but not the travel expenses.

At the following addresses you will find all the necessary information about the application form, the abstracts, the venue and the past editions of Pragmatic:


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/expand 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.

The first edition was in 1997 and this will be the 19th edition (for many reasons the last one was in 2017).


"Frobenius everywhere"
Lecturer: Luis Nuñez-Betancourt (CIMAT, Mexico)
Collaborator: Eamon Quinlan-Gallego (University of Utah, USA)
Abstract: The Frobenius morphism has proven to be a powerful tool in commutative algebra. Peskine and Szpiro proved Auslander's zero
divisor conjecture and answered Bass' question regarding Cohen-Macaulay rings using this map. They also provided a series of
important results regarding the structure of local cohomology. Hochster and Roberts proved that rings of invariants are
Cohen-Macaulay. Later, Hochster and Huneke developed tight closure theory and used it to solve several homological conjectures in prime
characteristic. Since then, the use of the Frobenius map has grown to touch other areas such as differential algebra, combinatorics,
representation theory and birational geometry. The first lectures will focus on the basics on the Frobenius maps, its splittings, and local
cohomology. Then, we focus on the use of Frobenius in singularity theory and combinatorics. We will also discuss connections with
D-modules. There will also be an introduction to Macaulay2 packages in prime characteristic, and related topics. The list of open problems to
discuss will try to reflect the diversity of uses of the Frobenius map.

"Cohomology of line bundles on flag varieties"
Lecturer: Claudiu Raicu (Notre Dame University, USA)
Collaborator: Alessio Sammartano (Politecnico di Milano, Italy)
Abstract: The goal of these lectures is to discuss the problem of computing cohomology of line bundles on flag varieties, and to explore
various applications to the study of homological invariants in commutative algebra and algebraic geometry. Over fields of
characteristic zero, the cohomology calculation is well-understood, and is the subject of the celebrated Borel-Weil-Bott theorem. In
positive characteristic however, the description of cohomology is largely unknown, even when it comes to deciding its vanishing and
non-vanishing behavior. The question of computing cohomology turns out to be quite versatile – any reasonable restrictions, either on theline bundles considered, or on the flag varieties themselves, lead to
special cases that are of interest on their own, and the participants will have the chance to examine a number of particular scenarios. The
applications we envision include, but are not limited to: Castelnuovo-Mumford regularity for powers and symbolic powers of
determinantal varieties, Hilbert functions of Koszul modules, or the homology of certain arithmetic Koszul-type complexes.

Organizing committee:

Marco D'Anna  - Università di Catania
Elena Guardo  - Università di Catania
Alfio Ragusa  - Università di Catania
Francesco Russo  - Università di Catania
Giuseppe Zappalà - Università di Catania

Data di pubblicazione: 24/01/2023