Project description
Study probes deeper into singularities in algebra
Singularities in positive-characteristic algebra are irregularities in mathematical structures that occur in systems defined over fields with a prime number characteristic. With the support of the Marie Skłodowska-Curie Actions programme, the SIPOCAG project aims to understand and classify these singularities using two approaches. First, researchers will investigate how the Frobenius morphism (a special map in algebra) affects a ring’s structure by creating modules, focusing on their smallest, non-divisible building blocks (indecomposable summands). Second, the team will explore arc and jet schemes (spaces representing tiny curves on a variety) to connect them with known singularity types like F-purity (a measure of regularity). New computational tools will also be created to simplify the complex mathematics of these two areas of study.
Objective
"SIPOCAG aims to understand and classify singularities occurring in positive-characteristic from two points of view: First, to understand the indecomposable summands arising via the Frobenius morphism of a ring or a variety, and second, to understand the relation between the theory of arc/jet schemes and the ""standard"" singularities of positive-characteristic commutative algebra.
Towards the first objective, given a ring R in characteristic p, one obtains a new module by restricting scalars along the Frobenius self-map on R. The properties of resulting module reflect the singularities of R; for example, Kunz showed that this module is flat if and only if R is regular. I seek to understand what the indecomposable summands of this module tell us about R, especially as one restricts scalars along higher iterates of the Frobenius. In particular, SIPOCAG focuses on the question of how many different isomorphism classes of indecomposables show up in this process: I propose to understand when this number is finite, and what the types and nature of the summands tell us about the ring R.
Towards the second objective, the arc scheme of a variety X and its truncation, the jet schemes, are moduli spaces of infinitesimal curves on X, and have proved useful in studying singularities in birational geometry. However, the relation between the arc/jet schemes and the usual notions of singularities in commutative algebra, such as F-purity, remains unknown. SIPOCAG will characterize such classes of singularities via the study of their arc/jet schemes, and leverage these connections to better understand the behavior of these singularities.
Finally, I propose to develop useful new computational tools for working with concrete examples of the preceding two phenomena, which will be useful not only for SIPOCAG but also to many other researchers.
These goals will be approached by combining my existing geometric approach to these topics with the algebraic expertise of the host."
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
- natural sciences mathematics pure mathematics algebra commutative algebra
- natural sciences mathematics pure mathematics geometry
- natural sciences mathematics pure mathematics algebra algebraic geometry
You need to log in or register to use this function
Keywords
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
-
HORIZON.1.2 - Marie Skłodowska-Curie Actions (MSCA)
MAIN PROGRAMME
See all projects funded under this programme
Topic(s)
Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
HORIZON-TMA-MSCA-PF-EF - HORIZON TMA MSCA Postdoctoral Fellowships - European Fellowships
See all projects funded under this funding scheme
Call for proposal
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) HORIZON-MSCA-2024-PF-01
See all projects funded under this callCoordinator
Net EU financial contribution. The sum of money that the participant receives, deducted by the EU contribution to its linked third party. It considers the distribution of the EU financial contribution between direct beneficiaries of the project and other types of participants, like third-party participants.
48009 Bilbao
Spain
The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.