### Article

## Twisted homology of configuration spaces and homology of spaces of equivariant maps

We calculate homology groups with certain twisted coefficients of configuration spaces of projective spaces. This completes a calculation of rational homology groups of spaces of odd maps of spheres S^m \to S^M, m<M, and of the stable homology of spaces of non-resultant polynomial maps R^{m+1} -> R^{M+1}. Also, we calculate the homology of spaces of Z_r-equivariant maps of odd-dimensional spheres, and discuss further generalizations.

I shall describe the recent progress in the study of cohomology rings of spaces of knots in R^n, H^∗({knots in R^n}), with arbitrary n>23. ‘Any dimensions’ in the title can be read as dimensions n of spaces R^n, as dimensions i of the cohomology groups H^i, and also as a parameter for different generalizations of the notion of a knot. An important subproblem is the study of knot invariants. In our context, they appear as zero-dimensional cohomology classes of the space of knots in R^3. It turns out that our more general problem is never less beautiful. In particular, nice algebraic structures arising in the related homological calculations have equally (or maybe even more) compact description, of which the classical ‘zero-dimensional’ part can be obtained by easy factorization. There are many good expositions of the theory of related knot invariants. There- fore, I shall deal almost completely with results in higher (or arbitrary) dimensions.

The stabilization of cohomology rings of spaces of non-resultant homogeneous polynomial systems of growing degree in $\R^3$ is studied. The rational stable cohomology rings are explicitly calculated, and the instant of stabilization is estimated

Rational homology groups of spaces of non-resultant (that is, having only trivial common zeros) systems of homogeneous quadratic polynomial systems in R^3 are calculated

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.