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Description

Quantum groups are a fertile area for explicit computations of cohomology and support varieties because of the availability of geometric methods involving complex algebraic geometry. Ginzburg and Kumar have shown that for l>h (l is order of the root of unity and h is the Coxeter number), the cohomology ring identifies with the coordinate algebra of the nilpotent cone of the underlying Lie algebra g=Lie(G). Bendel, Pillen, Parshall and the speaker have determined the cohomology ring when l is less than or equal to h and have shown that in most cases this identifies with the coordinate algebra of a G-invariant irreducible subvariety of the nilpotent cone. The latter computation employs vanishing results on partial flag variety G/P via the Grauert-Riemenschneider theorem. Support varieties have been determined for tilting modules (by Bezrukavinov), induced/Weyl modules (by Ostrik and Bendel-Nakano-Pillen-Parshall), and simple modules (by Drupieski-Nakano-Parshall). The calculations for tilting modules and simple modules employed the deep fact that the Lusztig Character Formula holds for quantum groups when l>h. In this talk, I will survey several of the main results of the topic and indicate the combinatorial and geometric techniques necessary to make such calculations. Open problems will also be presented.