Research on non-abelian Hodge structures of algebraic varieties

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There were the following three purposes for this project.1) Study of the non-abelian Hodge decomposition (or filtration) in the case of positive characteristic.2) Study of the relation between defomation theory of non-commutative schemes and non-abelian Hodge theory.3) Study of non-abelian mixed Hodge structures of algebraic varieties.As for the first purpose, I realized that it would be important to construct the crystellin homotopy theory using n-stacks. If such a theory is constructed, the original problem would be clear and there would be many applications. It would be an extention of the work of N. Katz on Frobenius maps and Hodge filtrations in the abelian cases. I am now trying this next big project. On the second one, after the study of non-abelian schemes I understood that it is more natural to describe the deformation parameter space as an n-stacks. Such description would be useful to study the relationship with non-abelian Hodge structures. On the third one, there was a great development by C. Simpson. He describe the non-abelian mixed Hodge structures as kinds of filtrations on non-abelian cohomology n-stacks. I studied his theory and make some explicit calculations of such filtrations on n-stacks. Moreover I worked with Y. Takeda on the pre-Tango structures on curves. The work is related on the first one. With K. Ohba, I studied on the relation between the study of gerbs (using n-stacks) and the study of cycles of moduli spaces.