Cartan geometry, Lie and representation theory, Integrable Systems, quantum Groups and quantum computing towards the understanding of the geometry of deep Learning and its Applications

CaLIGOLA aims at advancing the research in Cartan Geometry, Lie Theory, Integrable Systems and Quantum Groups to provide insight into a variety of multidisciplinary fields oriented towards the applications with a special interest in machine learning and quantum computing. Sound mathematical models for quantum computing, vision and more generally machine learning are a priority for Horizon Europe and strategic to include Europe among the leading actors in such fields. Through the theory of symmetric spaces from the Cartan Geometric and Lie theoretic point of view, we shall implement the Erlangen philosophy for mathematical and physical questions (integrable systems and SUSY gauge field theory), but also for more applied themes including Quantum Computing and (geometric) Deep Learning. Quantum symmetric spaces and quantum representations will be the key to approach the questions of fault tolerant quantum algorithms in topological quantum computing and quantum information geometry on homogeneous spaces. With the language of Cartan geometry and Quantum Groups, we shall reformulate group invariant neural network models. Persistent homology and topological data analysis will take a step forward towards a metric theory on the space of observers. With the help of Lie group thermodynamic, we shall push the understanding of symmetries at a deeper level. Overall, the new algorithms of Deep Learning and Geometric Deep Learning will find a better modeling and understanding towards a comprehensive theory of dimensionality reduction of parameter space via group equivariance.