Pragmatic 2023
Cohomology and Frobenius
Catania, Italy
 
June 11th - July 1st, 2023
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Research school in
Algebraic Geometry and Commutative Algebra
Sponsored by UNICT, PRIN 2020 "Squarefree Gr÷bner degenerations, special varieties and related topics", GNSAGA, DMI UNICT
Frobenius everywhere
Lecturer:
Luis Nu˝ez-Betancourt (CIMAT, Mexico)
Collaborator:
Eamon Quinlan-Gallego (University of Utah, USA)

Cohomology of line bundles on flag varieties
Lecturer:

Claudiu Raicu (Notre Dame University, USA)
Collaborator:
Alessio Sammartano (Politecnico di Milano, Italy)

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Pragmatic 2023
 
(First announcement)

Catania, Italy
June 12th - June 30th, 2023
(arrival date June 11th; departure date July 1st)

Organizers:

Marco D'Anna  - UniversitÓ di Catania
Elena Guardo  - UniversitÓ di Catania
Francesco Russo  - UniversitÓ di Catania
Giuseppe ZappalÓ - UniversitÓ di Catania
Alfio Ragusa  - UniversitÓ di Catania


"Frobenius everywhere"
Lecturer:
Luis Nu˝ez-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 the line 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.


People who wish to be considered either for participation and/or for  financial support should fill out the enclosed application form, which  should be sent to the following address:

E-mail: pragmatic@dmi.unict.it

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 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.

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APPLICATION FORM TO PRAGMATIC 2023


Surname:

Name:

Male or Female:

Date of birth:

Nationality:

Institution:

Academic qualification (Ph.D., Master,...):

Name of advisor:

Professional Status and Current Position:

E-mail address:

Field of interest:

Can you be supported by your institution?: (Possible answers: totally,  partly, no)

If you have a Ph.D, when did you get it and where?

Reasons for wanting to participate in the session:

It is suggested that the applications include both a recommendation  letter and a brief Curriculum vitae