The aim of the MSCA-DN project CaLiForNIA is to provide a novel interdisciplinary training to a new generation of researchers in the key theoretical areas of Cartan geometry, Lie theory, and representation
theory, as well as their quantum counterparts. Theoretical training will be complemented by strategic applications to Horizon Europe priority areas, in particular, quantum computing and machine learning,
going far beyond the reach of ordinary PhD programs in Mathematics, Physics or Computer Science.
Objectives:
1) Develop new methods in Cartan, control, contact and sub Riemannian geometry, (WP1) towards a better understanding of infinite dimensional representations (Harish-Chandra) theory of real Lie
(super)groups, (super)algebras and provide our key applications with the necessary language: quantum (SUSY) symmetric spaces (WP2), (quantum) information geometry and machine learning algorithms.
2) Understand quantum symmetric spaces from both C* and quantum algebras point of view, using (1) as key input of, in a graph theoretical language (WP2). Develop new methods in quantum
representation theory (quantum BGG), quantum invariant differential calculus (quantum Dolbeault- Dirac operators), towards a new understanding of Connes spectral triples (WP2). Develop new SUSY
techniques towards physical applications.
3) Master a geometric approach to (quantum) information geometry using the input from (1) for modelling optimization tasks (WP3) in a variety of applications including quantum algorithms and
circuits, towards machine learning tasks.
4) Develop new theoretical approaches, via quantum geometry (2), sheaf theory, to invariant versions of geometric deep learning models (WP4), where quantum differential calculus (WP2) becomes an
essential tool to describe discrete differential operators and sheaf theory the language to implement them, in the key message passing mechanism. Provide a new machine learning perspective on quantum
computing and artificial intelligence algorithms.
5) Train a new generation of researchers with skills on Lie, Cartan and Quantum Theories, but also fully capable of applying them to strategic topics as quantum computing and geometric deep learning.
Build a new scientific community by disseminating results within network both in the academic and the industrial sector.
