Abstract
In my PhD I proposed a novel notion of first order deformation for an (enhanced) triangulated category. I showed that, in the restricted first order case, our theory is better behaved than the classical one; specifically, it manages to give a meaningful interpretation to Hochschild classes containing curvature, thus solving the curvature problem. In this project I will continue to develop this theory, with the aim of obtaining a full understanding of the deformation theory of higher categories. Concretely, I will first expand our notion -- called categorical deformation -- to work over more general bases, and later study the induced deformation functor. This will require a careful study of derived deformations in the framework of stable infinity categories. A comparison with the classical case will also be developed, via a procedure that we call categorical blowup. As an application, I will study absolute (that is, non-linear) deformations of higher categories, and understand their relation to topological Hochschild Cohomology. Finally, I will work on a side project aimed at giving a geometric interpretation to the category of matrix factorizations with N steps.
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