Periodic Reporting for period 1 - CaLIGOLA (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)
Reporting period: 2023-01-01 to 2024-12-31
CaLIGOLA aims to cross-fertilize with ideas coming from Lie Theory, Integrable Systems and Quantum Groups, multidisciplinary fields, like quantum computing, vision algorithms and machine learning, which need a rigorous mathematical approach to boost their advancement. This is essential to provide mathematical models for far reaching applications, strategic to put Europe among the leading actors in the fields which constitute a priority in Horizon Europe.
Objectives and overview of the research and innovation programme
1) Develop new methods in Lie theory, more specifically in Cartan Geometry and infinite dimensional representations Harish-Chandra theory of real Lie (super)groups, (super)algebras, to boost the application in physics (e.g. Quantum Geometry, Supersymmetry) and provide the appropriate language for applications in (4) and (5).
2) Investigate the classification of integrable dynamical systems, in particular in its combinatorial aspects, linked to the theory of Harish-Chandra representations in (1). Exploit this study to reach a better understanding of supersymmetric Gauge theories, SUSY (supersymmetric) curves and Dubrovin manifolds (see (1)).
3) Bridge the gap between Connes approach to noncommutative geometry and the Hopf algebraic one, via Spectral triples of quantum symmetric spaces. Develop new tools in quantum representation theory (quantum BGG). Build quantum invariant differential calculus (quantum Dolbeault-Dirac operators); (1) and (2) will be paramount in reaching this objective. Once achieved, it will have a direct impact first on (4) and then on (5)..
4) Define the Quantum Fisher information (Quantum Geometric Tensor) for modeling of optimization tasks in devising quantum algorithms and circuits; . Develop models from quantum representation theory and quantum differential calculus for a topological approach to quantum fault tolerant computation, making essential use of the machinery developed in (3), guided by the insight obtained in (1) and (2)..
5) Develop new vision models and algorithms with the use of persistent homology, Lie Group statistics to boost the machine learning techniques. Understand (Geometric) Deep Learning with the new invariance mathematical tools developed. Provide a new perspective on quantum computing and artificial intelligence algorithms
Objectives and overview of the research and innovation programme
1) Develop new methods in Lie theory, more specifically in Cartan Geometry and infinite dimensional representations Harish-Chandra theory of real Lie (super)groups, (super)algebras, to boost the application in physics (e.g. Quantum Geometry, Supersymmetry) and provide the appropriate language for applications in (4) and (5).
2) Investigate the classification of integrable dynamical systems, in particular in its combinatorial aspects, linked to the theory of Harish-Chandra representations in (1). Exploit this study to reach a better understanding of supersymmetric Gauge theories, SUSY (supersymmetric) curves and Dubrovin manifolds (see (1)).
3) Bridge the gap between Connes approach to noncommutative geometry and the Hopf algebraic one, via Spectral triples of quantum symmetric spaces. Develop new tools in quantum representation theory (quantum BGG). Build quantum invariant differential calculus (quantum Dolbeault-Dirac operators); (1) and (2) will be paramount in reaching this objective. Once achieved, it will have a direct impact first on (4) and then on (5)..
4) Define the Quantum Fisher information (Quantum Geometric Tensor) for modeling of optimization tasks in devising quantum algorithms and circuits; . Develop models from quantum representation theory and quantum differential calculus for a topological approach to quantum fault tolerant computation, making essential use of the machinery developed in (3), guided by the insight obtained in (1) and (2)..
5) Develop new vision models and algorithms with the use of persistent homology, Lie Group statistics to boost the machine learning techniques. Understand (Geometric) Deep Learning with the new invariance mathematical tools developed. Provide a new perspective on quantum computing and artificial intelligence algorithms
CaLIGOLA is articulated into theoretical and more practical interdisciplinary workpackages:
– WP1, WP2. These are theoretical workpackages providing methods and models, pertaining to Cartan Geometry, Harish-Chandra representation theory, integrable systems and Lie Theory on symmetric spaces for the applications developed in WP3, WP4. These are the cornerstones on which the other WPs are based.
– WP3. In its theoretical tasks, WP3 develops the theory of quantum groups and quantum symmetric spaces with the input of differential geometry, Lie theory and integrable systems of WP1, WP2. WP3 is essential for effective models for quantum computing in WP4 and to understand a noncommutative approach to discrete differential geometry in WP5.
– WP4. In its more applied tasks regarding the quantum computing objectives, we shall put into action the theory developed in the previous WPs. Our aim is twofold: give a theoretical vest to the theory of fault resistant quantum computing in its topological realization and apply the theory of quantum representations to understand the categorical approach via anyons to quantum computing.
– WP5. This WP is devoted to the application of the theory developed in the previous WPs. We shall implement ideas from Cartan geometry and SUSY (WP1, WP2) to understand (topological data analysis) TDA, (stochastic gradient descent) SGD and to model the Deep Learning algorithm via the new discrete quantum calculus approach (WP3).
– WP1, WP2. These are theoretical workpackages providing methods and models, pertaining to Cartan Geometry, Harish-Chandra representation theory, integrable systems and Lie Theory on symmetric spaces for the applications developed in WP3, WP4. These are the cornerstones on which the other WPs are based.
– WP3. In its theoretical tasks, WP3 develops the theory of quantum groups and quantum symmetric spaces with the input of differential geometry, Lie theory and integrable systems of WP1, WP2. WP3 is essential for effective models for quantum computing in WP4 and to understand a noncommutative approach to discrete differential geometry in WP5.
– WP4. In its more applied tasks regarding the quantum computing objectives, we shall put into action the theory developed in the previous WPs. Our aim is twofold: give a theoretical vest to the theory of fault resistant quantum computing in its topological realization and apply the theory of quantum representations to understand the categorical approach via anyons to quantum computing.
– WP5. This WP is devoted to the application of the theory developed in the previous WPs. We shall implement ideas from Cartan geometry and SUSY (WP1, WP2) to understand (topological data analysis) TDA, (stochastic gradient descent) SGD and to model the Deep Learning algorithm via the new discrete quantum calculus approach (WP3).
CaLIGOLA will foster and consolidate existing research collaborations and will create and establish new ones, not only within our Consortium beneficiaries, but also outside, with our excellent partners in TC and AC. WP4 and WP5 with their vocational, application oriented, purpose are a paradigm of the new interdisciplinary and intersectorial interactions that will be created. New algorithms in WP4 and WP5 will emerge from the new theory (Cartan Geometry, representations, integrable systems and quantum groups) developed in WP1, WP2, WP3. The solid academic standing of our beneficiaries together with the excellencies of our AC, TC partners, including institutions (ETH, UC) routinely in the top 10 ranking (QS) of every discipline, will be the ground of the innovation brought along from the Cartan geometric approach (WP1) to quantum groups (WP3) and quantum technologies (WP4), one of the priorities of Horizon Europe, reaching a new perspective on machine learning algorithm, via Lie group thermodynamics and Geneos (WP5). New perspectives on algorithms of Geometric Deep Learning, with group equivariant properties, will be our cutting-edge research questions leading to concrete application: we will test our new understanding with new intersectorial challenges on AI brought along by our outstanding industrial partner Thales, leader in Lie sensors and statistical approach to sensing theory.