Mixed Hodge structures in homotopy theory

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This research project focuses on the study of homotopical structures of algebraic varieties, with a special emphasis on their motivic formulation, using techniques of algebraic geometry and homotopy theory. In particular, we plan to study the existence and properties of mixed Hodge structures on homotopy theoretic invariants arising from rational homotopy, intersection cohomology and deformation theory. We aim to develop suitable algebraic frameworks for motivic rational homotopy theory.The main algebraic objects under consideration in the present research project are mixed Hodge diagrams of differential graded algebras. These are a multiplicative analogue of the mixed Hodge complexes of Deligne involving filtered and bifiltered differential algebras over the rational and complex fields that encode the weight and Hodge filtrations up to filtered quasi-isomorphisms. The development of the present project requires to extend their scope of application to different homotopical frameworks, and to develop concrete applications to the homotopy theory of algebraic varieties.