Applications of operads to homotopy theory and homological algebra

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Project objective

Research objectives and content This project is concerned with the following applications of operads: The unstable Adams spectral sequence is the main tool for calculating the homotopy type of a topological space. One version of this spectral sequence is based on a model of the loop space construction. Another spectral sequence in homotopy theory based on the loop space construction is Barratt's desuspension spectral sequence. I propose myself to improve these calculational tools by generalizing the construction of these spectral sequences to iterated loop spaces. The little $r$-cubes operad encodes the structure of an $r$-fold loop space and provides an efficient model for this generalization. Operads theory furnishes a nice homology theory for Poisson algebras. I expect to show that this homology theory determines the cyclic homology of some filtered algebras, generalizing to the non smooth case a method due to Brylinsky. Training content (objective, benefit and expected impact) I need knowledge in combinatorial homotopy theory which I may learn from Professor Baues. Moreover, fruitful applications of operads in low dimensional topology, and geometrical physics have been found recently. At the Max Planck Institut, I could establish fruitful international contacts with mathematicians working on these aspects of operads theory. Links with industry / industrial relevance (22)

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Participants

• MAX-PLANCK-GESELLSCHAFT ZUR FOERDERUNG DER WISSENSCHAFTEN E.V. · MUENCHENCoordinatorGermany

Links

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