### Article

## Witt vectors as a polynomial functor

For every commutative ring *A*, one has a functorial commutative ring *W*(*A*) of *p*-typical Witt vectors of *A*, an iterated extension of *A* by itself. If *A* is not commutative, it has been known since the pioneering work of L. Hesselholt that *W*(*A*) is only an abelian group, not a ring, and it is an iterated extension of the Hochschild homology group HH_0(A) by itself. It is natural to expect that this construction generalizes to higher degrees and arbitrary coefficients, so that one can define “Hochschild–Witt homology” WHH∗(A,M) for any bimodule *M* over an associative algebra *A* over a field *k*. Moreover, if one want the resulting theory to be a trace theory, then it suffices to define it for A=k. This is what we do in this paper, for a perfect field *k* of positive characteristic *p*. Namely, we construct a sequence of polynomial functors W_m, m≥1 from *k*-vector spaces to abelian groups, related by restriction maps, we prove their basic properties such as the existence of Frobenius and Verschiebung maps, and we show that W_m are trace functors. The construction is very simple, and it only depends on elementary properties of finite cyclic groups.

We give a direct interpretation of the Witt vector product in terms of tame residue in algebraic K-theory.

A review of the classical construction of Witt vectors is presented, and some recent generalizations of it to the non-commutative case are described.

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.

In a previous paper, we have defined polynomial Witt vectors functor from vector spaces over a perfect field k of positive characteristic p to abelian groups. In this paper, we use polynomial Witt vectors to construct a functorial Hochschild- Witt complex WCH_∗(A) for any associative unital k-algebra A, with homology groups WHH∗(A). We prove that the group WHH_0(A) coincides with the group of non-commutative Witt vectors defined by Hesselholt, while if A is commutative, finitely generated, and smooth, the groups WHH_i(A) are naturally identified with the terms WΩ^i_A of the de Rham- Witt complex of the spectrum of A.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

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.