in this talk we discuss a classification of ribbon categories with the tensor product rules of the finite-dimensional complex representations of $so(n)$, for $n \geq 5$ and $n=3$. the equivalence class of a category with $so(n)$ fusion rules depends only on the eigenvalues of the braid operator on $x \otimes x$, where $x$ corresponds to the defining representation. the classification applies both to generic $so(n)$ tensor product rules, and to certain fusion rings having only finitely many simple objects.

in recent years, k-stability of fano varieties has been proved to be a rich topic for higher dimensional geometers. the transition of knowledge is mutual. in one direction, we use the powerful machinery from higher dimensional geometry, especially the minimal model program, to have a better understanding of various concepts in k-stability. on the other direction, k-stability provides the right subclass to construct moduli spaces of fano varieties, which had been once considered beyond reach by algebraic geometers. in the first half hour, i will explain how people change their viewpoint on the definition of k-stability. then in the main talk, i will focus on the moduli of fano varieties.