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Skein theory offers several plausible strategies for extending link homology theories, such as Khovanov homology, to topological field theories in 4 or 5 dimensions. In this talk, I will focus on a categorified analog of a TFT of Turaev-Viro type. Joint work with Matthew Hogancamp and David Rose. 

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Alpha-induction is a tensor functor arising from a Frobenius algebra on a braided fusion category to a new fusion category using braiding.  A bi-unitary connection consists of partial data of generalized quantum 6j-symbols and describes a commuting square in subfactor theory.  A finite family of bi-unitary connections gives operator-algebraic description of a fusion category.  Last year, I showed that if we have a commutative Frobenius algebra, then the resulting bi-unitary connection from alpha-induction is flat, which means that quantum 6j-symbols are in a certain canonical form.  I now show that the converse of this statement also holds.

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In this talk I will describe some relatively new symmetric tensor categories in positive characteristic. We will discuss their construction and known and conjectural properties.

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We explain how braided fusion categories can be  canonically  reconstructed as gaugings of categories of a particular  type, which we call  reductive. This leads to a parameterization  of braided fusion categories that could be helpful for classification purposes.   The key role is played by a new notion of  the Tannakian radical.  This is a joint work with Jason Green.

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