Galois cohomology of real algebraic groups

I shall speak about joint results with Mikhail Borovoi on computation of Galois cohomology of linear algebraic groups over real numbers. The real case is crucial in computation of Galois cohomology over number fields. The Levi decomposition reduces the problem to the case of reductive groups and a theorem of Borovoi (1988) reduces computation to maximal anisotropic tori. Based on this, we obtain an explicit combinatorial description of Galois cohomology for a real reductive group in terms of some special integer labelings of its affine Dynkin diagram and some subquotients of the cocharacter lattice of the central torus. As a by-product, we obtain a transparent description for the component group of the real locus of a connected reductive group in terms of the cocharacter lattice of a maximal split torus, which reinforces a classical result of Matsumoto (1964).