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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.","funder_award_id":"101086123","funder":{"id":"https://openalex.org/F4320338438","display_name":"HORIZON EUROPE Marie Sklodowska-Curie Actions","doi":"10.13039/100018694"},"funded_outputs":["https://openalex.org/W4388637407","https://openalex.org/W4391013562","https://openalex.org/W4402772627","https://openalex.org/W4403788857","https://openalex.org/W4404826732","https://openalex.org/W4405285589","https://openalex.org/W4405286288","https://openalex.org/W4405627392","https://openalex.org/W4406033466","https://openalex.org/W4415014266","https://openalex.org/W4416454626","https://openalex.org/W4416677975","https://openalex.org/W4416715876","https://openalex.org/W6979252028","https://openalex.org/W7136629128"],"funded_outputs_count":15,"parent_awards":[],"sub_awards":[],"sub_awards_count":0,"amount":874000.0,"currency":"EUR","amount_usd":932208.4,"amount_usd_basis":"daily_rate","recipients":[{"id":null,"display_name":"BUNDESANSTALT FUER MATERIALFORSCHUNG UND -PRUEFUNG","ror":null,"country_code":"DE","roles":["partner"],"share":null,"basis":"funder_reported","asserted_by":"funder"},{"id":null,"display_name":"SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI DI TRIESTE","ror":null,"country_code":"IT","roles":["partner"],"share":null,"basis":"funder_reported","asserted_by":"funder"},{"id":null,"display_name":"THALES LAS FRANCE SAS","ror":null,"country_code":"FR","roles":["partner"],"share":null,"basis":"funder_reported","asserted_by":"funder"},{"id":null,"display_name":"UNIVERSITA DEGLI STUDI DEL PIEMONTE ORIENTALE AMEDEO AVOGADRO","ror":null,"country_code":"IT","roles":["partner"],"share":null,"basis":"funder_reported","asserted_by":"funder"},{"id":null,"display_name":"UNIVERSITA DEGLI STUDI DI FERRARA","ror":null,"country_code":"IT","roles":["partner"],"share":null,"basis":"funder_reported","asserted_by":"funder"},{"id":null,"display_name":"UNIVERZITA KARLOVA","ror":null,"country_code":"CZ","roles":["partner"],"share":null,"basis":"funder_reported","asserted_by":"funder"},{"id":"https://openalex.org/I16097986","display_name":"Universitat de Val\u00e8ncia","ror":"https://ror.org/043nxc105","country_code":"ES","roles":["partner"],"share":null,"basis":"affiliation_matcher","asserted_by":"funder"},{"id":"https://openalex.org/I21449261","display_name":"Masaryk University","ror":"https://ror.org/02j46qs45","country_code":"CZ","roles":["partner"],"share":null,"basis":"affiliation_matcher","asserted_by":"funder"},{"id":"https://openalex.org/I50357001","display_name":"Universidad Carlos III de Madrid","ror":"https://ror.org/03ths8210","country_code":"ES","roles":["partner"],"share":null,"basis":"affiliation_matcher","asserted_by":"funder"},{"id":"https://openalex.org/I9360294","display_name":"University of Bologna","ror":"https://ror.org/01111rn36","country_code":"IT","roles":["lead"],"share":null,"basis":"affiliation_matcher","asserted_by":"funder"}],"recipient_countries":[{"country_code":"IT","share":0.4211,"amount_usd":392552.96,"basis":"organisation_location"},{"country_code":"CZ","share":0.3895,"amount_usd":363095.17,"basis":"organisation_location"},{"country_code":"ES","share":0.1684,"amount_usd":156983.89,"basis":"organisation_location"},{"country_code":"DE","share":0.0158,"amount_usd":14728.89,"basis":"organisation_location"},{"country_code":"FR","share":0.0053,"amount_usd":4940.7,"basis":"organisation_location"}],"recipient_regions":[],"research_scope":{"is_research":true,"basis":"funder_default","tier":1},"funding_type":"fellowship","funder_scheme":"HORIZON - 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