Cartan and differential geometry, Lie theory, quantum groups and non commutative geometry For novel and Innovative Applications to quantum algorithms and geometric deep learning

Project Information

CaLiForNIA

Grant agreement ID: 101119552

DOI

10.3030/101119552

EC signature date 5 July 2023

Start date 1 January 2024

End date 31 December 2027

Funded under

• Marie Skłodowska-Curie Actions (MSCA)

Total cost

No data

EU contribution

€ 2 504 779,20

Coordinated by

ALMA MATER STUDIORUM - UNIVERSITA DI BOLOGNA
Italy

Periodic Reporting for period 1 - CaLiForNIA (Cartan and differential geometry, Lie theory, quantum groups and non commutative geometry For novel and Innovative Applications to quantum algorithms and geometric deep learning)

Reporting period: 2024-01-01 to 2025-12-31

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

CaLiForNIA scientific part is articulated in Workpackages (WP) and Tasks, each corresponding to a DC project.

WP1: Cartan Geometry and Lie Theory.
Task 1.1 Foundations of Cartan connections and representation theory.
Task 1.2 Geometric Control Theory.
Task 1.3 Contact and SubRiemannian Geometry.
Task 1.4 Palatini-Cartan Formalism.

WP2: Non commutative geometry of symmetric spaces.
Task 2.1 Quantum Flags and C*-Algebras.
Task 2.2 Baum-Connes conjecture for quantum symmetric spaces.
Task 2.3. Quantum Harmonic Superspace and Unitary Representations.

WP3: Quantum Computing and Quantum Information Geometry
Task 3.1 Quantum Information Geometry.
Task 3.2 Quantum Algorithms.

WP4: Geometric Deep Learning.
Task 4.1 Foundations of Sheaf Neural Networks.
Task 4.2 Geometric Deep Learning and Symmetric Spaces.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The aim of CaLiForNIA is to create a critical mass of researchers with expertise in strategic areas of Mathematics, Physics, Quantum Computing, AI and Machine Learning applications (Cartan Geometry,
Non commutative Geometry for Quantum Computing and Geometric Deep Learning) which constitute a priority in Horizon Europe.
Our diverse intersectorial excellent network will allow us to set up an interdisciplinary training program for the young generations to acquire the strategic expertise (quantum computing, geometric deep
learning) and transferable skills to excel in research, leading Europe towards the new challenges coming from these new technologies.
We envision the impact in research and training according to the following outline:
– Outstanding research: publication in top tier mathematical and physical journals, according to IF;
– Network: connections across the network will allow young researchers to secure a position in the most delicate years of their career, that is immediately after their PhD;
– Network and local Training: PhD courses and secondments in our excellent partners across sectors will furnish the DCs with the skills and operate the transfer of knowledge necessary to be leading
worldwide research in the future;
– Dissemination and public engagement: involving DCs, as essential part of their training, in the organization of all dissemination and public engagement events will increase exponentially their
visibility, enhancing their possibility of future development of their academic careers.

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Poster of the project