HIDRA · Homological Invariants of Deformations of Groups and Algebras
Horizon Europe — Marie Skłodowska-Curie Actions
- Duration
- 2023-09-01 → 2025-08-31
- EU contribution
- €221,669
- Participants
- 1
- Scheme
- HORIZON-TMA-MSCA-PF-EF
Lines connect the coordinator with its partners.
Results in brief
Homological Invariants of Deformations of Groups and Algebras
Symmetry is a powerful concept in both nature and mathematics. Traditionally, mathematicians study symmetry using objects called groups. This project explores new and broader ways to understand symmetry, using more flexible mathematical structures called groupoids. These help us study complex systems, such as certain types of chaotic behavior known as hyperbolic dynamical systems. Another path we're exploring involves quantum groups—mathematical structures that behave like groups but are designed to work in noncommutative spaces (spaces where the usual rules of arithmetic don’t always apply). These are part of a modern mathematical field called noncommutative geometry. Our focus is on identifying and studying invariants - mathematical properties that stay the same even when the system changes. By developing new ways to compute and compare these invariants, we hope to better understand the underlying structure of these complex systems. Goals of the Project: Investigate how different mathematical tools (like homology and K-theory) help us understand chaotic systems. Study specific phenomena like torsion and explore advanced mathematical ideas, such as the Baum-Connes conjecture, in the context of quantum groups. Although highly theoretical, this kind of research provides the foundation for many advanced technologies. By deepening our understanding of symmetry and structure, we may uncover insights relevant to physics, computer science, and beyond.
Data: CORDIS, © European Union
Project objective
The pervasive role of algebraic topology in mathematics is proof of the powerful effects that homological invariants produce in the development of the discipline. Extending these techniques beyond the category of topological spaces, in order to include ""quantized"" systems arising from dynamical systems and (quantum) groups, is going to be extremely useful to make fast progress in these fields. The framework of operator algebras and noncommutative geometry is extremely well-suited for these developments and has already been applied with some success. The goal of this proposal is to further develop these homological techniques by supporting them with novel methods based on triangulated categories, homotopy theory, and index theory. The research problems tackled in this Action are deeply related to important topics which attracted a great deal of interest in the mathematical community. For example, we study the celebrated Baum-Connes conjecture (for both groupoids and quantum groups) through a relatively unexplored perspective and relate it to the computation of K-theoretic and homological invariants for notable dynamical systems (e.g., Smale's Axiom A diffeomorphisms). This research will provide mathematicians with both conceptually new approaches and powerful computational tools. Some of these results are relevant not only for pure mathematics, but also for solid-state physics and quantum information theory. This Action will take us one step closer to the solution of significant problems or the formulation of more and more refined research questions. This fellowship will allow V. Proietti to work under the supervision of M. Yamashita (a world-class expert on quantum groups) at the University of Oslo (a leading institution in operator algebras). It will expand the fellow's technical expertise and integrate it with essential management, administrative, and dissemination skills which will help V. Proietti reach a position of professional maturity.""
Original text from CORDIS.
Participants
- UNIVERSITETET I OSLO · OsloCoordinatorNorway
Links
- View on CORDIS
- DOI: 10.3030/101063362
- https://ec.europa.eu/research/participants/documents/downloadPublic?documentIds=080166e502b1dd1f&appId=PPGMS
- https://ec.europa.eu/research/participants/documents/downloadPublic?documentIds=080166e5206f10f2&appId=PPGMS
Data: CORDIS, © European Union