K-stability for Fano varieties is an algebraic stability condition to detect the existence of K/"aher-Einstein metrics. It is widely expected and partially proven that K-stability gives nicely behaved moduli spaces for Fano varieties. In this talk, we will show that the K-moduli space of cubic threefolds is isomorphic to the corresponding GIT moduli space. The main ingredient of our proof is an estimate in dimension 3 of the local volume of klt singularities introduced by Chi Li. This is joint work with Chenyang Xu.

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