Cel This project focuses on questions related to the deformation theory of algebraic structures in the very general setting of algebras over a prop (which includes various kinds of algebras and bialgebras). In particular, it aims to understand realizations of an algebraic structure at the (co)homology level in an algebraic structure up to homotopy at the (co)chain level. When this complex is of topological or geometric nature (e.g. the de Rham complex of a manifold), one expects to extract from these realizations new topological and geometric invariants. These realizations are not understood at present in various cases where bialgebra structures play a crucial role, especially with Poincaré duality, String topology and Deformation quantization. We use methods relying in particular on Quillen’s work on homotopical algebra, Lurie's higher category theory and Toen-Vezzosi’s derived algebraic geometry. We define and study the homotopy type of realization spaces of algebraic structures and develop a derived geometry approach to deformation theory of such structures. This machinery is set up to address three kinds of problems related to topology, geometry and mathematical physics. The first one is to use realization spaces of Poincaré duality structures and equivariant string topology to build new topological and geometric invariants of manifolds, solve an open problem of Sullivan about the realisation of known geometric invariants and understand the action of the Grothendieck-Teichmüller group on such structures. The second one is to use a derived geometry approach to these moduli spaces to compare deformation theories of bialgebras and algebras over the little disks operad and solve a longstanding formality conjecture of Kontsevich. The third one is to use deformation quantization of moduli stacks of algebraic structures up to homotopy to perform a far reaching generalization of Turaev's work to the construction of link invariants in higher dimensional manifolds. Dziedzina nauki natural sciencesmathematicsapplied mathematicsmathematical physicsnatural sciencesmathematicspure mathematicstopologynatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsalgebraalgebraic geometry Słowa kluczowe operads props bialgebras moduli spaces deformation theory homotopical algebra higher category theory derived algebraic geometry Poincaré duality string topology deformation quantization Program(-y) H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions Main Programme H2020-EU.1.3.2. - Nurturing excellence by means of cross-border and cross-sector mobility Temat(-y) MSCA-IF-2015-EF - Marie Skłodowska-Curie Individual Fellowships (IF-EF) Zaproszenie do składania wniosków H2020-MSCA-IF-2015 Zobacz inne projekty w ramach tego zaproszenia System finansowania MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF) Koordynator KOBENHAVNS UNIVERSITET Wkład UE netto € 166 829,00 Adres NORREGADE 10 1165 Kobenhavn Dania Zobacz na mapie Region Danmark Hovedstaden Byen København Rodzaj działalności Higher or Secondary Education Establishments Linki Kontakt z organizacją Opens in new window Strona internetowa Opens in new window Uczestnictwo w unijnych programach w zakresie badań i innowacji Opens in new window sieć współpracy HORIZON Opens in new window Koszt całkowity € 166 829,00
