Representation theory of quantum groups at roots of unity is connected with various problems in vertex algebras, TQFT, quantum topology and modular representation theory. We have evidence of strong connection between the structure of the center of the small quantum group at a root of unity associated to the Lie algebra $g$ and the geometry of the group G. I will review the results on the center and higher Hochschild cohomologies of the small quantum group. This is a joint work with Qi <a href="http://You. 

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