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In the last 10-15 years the theory of linear algebraic groups has witnessed an intrusion of the cohomological methods of modern algebraic geometry and algebraic topology. These new methods have led to breakthroughs on a number of classical problems in algebra, which are beyond the reach of earlier purely algebraic techniques. The workshop will focus at the following new emerging techniques and applications: 

- The proof of the Grothendieck-Serre conjecture based on the theory of affine Grassmannians.
The Grothendieck-Serre conjecture stated in 60's says that any rationally trivial G-bundle is locally trivial (in Zariski topology), where G is a smooth reductive group scheme over a local regular ring. In a recent preprint (Nov. 2012) Fedorov and Panin proved the conjecture in the geometric case using new connections with the theory of affine Grassmannians.

- Computation of generalized equivariant cohomology of projective homogeneous and toric varieties.
The notion of an algebraic oriented cohomology theory was introduced and extensively studied during the last decade by Levine, Morel, Panin, Smirnov and others. The universal such theory called algebraic cobordism has many applications in the theory of quadratic forms and homogeneous spaces. Its equivariant version is believed to contain all the information concerning cohomological behaviour of homogeneous spaces. In particular, its computation can lead to a better understanding of cohomology and related combinatorics of toric varieties.