We discuss computation of the special values of partial zeta functions associated to totally real number fields. The main tool is the Eisenstein cocycle Ψ , a group cocycle for GLn(Z); the special values are computed as periods of Ψ , and are expressed in terms of generalized Dedekind sums. We conclude with some numerical examples for cubic and quartic fields of small discriminant.

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2 Virasoro as a local Lie algebra on C 3 2.1 Dolbeault resolution of holomorphic vector fields . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Lie algebra extensions and the cocycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.1 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.2 A cocycle for LC . . . . . ....

Let G be a finite abelian group. We will consider a skew product extension of a product of two Cantor minimal Z-systems associated with a G-valued cocycle. When G is non-cyclic and the cocycle is non-degenerate, it will be shown that the skew product system has torsion in its coinvariants.

Given a set S of vertices in a graph, the cocycle determined by S is the set of edges joining a vertex in S to a vertex not in S. A bond is a minimal non-empty cocycle. We characterise graphs that admit an orientation under which every bond of even cardinality has a prescribed directed parity.

We give a complete classification up to cocycle conjugacy of uniformly outer actions of Z on UHF algebras. In particular, it is shown that any two uniformly outer actions of Z on a UHF algebra of infinite type are cocycle conjugate. We also classify them up to outer conjugacy.

The main objective of this article is to analyse the dynamics of Burgers equation on the unit interval, driven by affine linear white noise. It is shown that the solution field of the stochastic Burgers equation generates a smooth perfect and locally compacting cocycle on the energy space L2([0, 1],R). Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-...

We consider continuous SL(2,R)-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an SO(2,R)-cocycle. Using this, we show that if a cocycle’s homotopy class does not display a certain obstruction to uniform hype...

We consider 1-cocycles with values in locally compact, second countable abelian groups on discrete, nonsingular, ergodic equivalence relations. If such a cocycle is invariant under certain automorphisms of these relations we show that the skew product extension defined by the cocycle is ergodic. As an application we obtain an extension of many of the results in [9] to higher-dimensional shifts ...

Theorem 1.1 (Popa’s Cocycle Superrigity, Special Case). Let Γ be a discrete group with Kazhdan’s property (T), let Γ0 < Γ be an infinite index subgroup, (X0,μ0) an arbitrary probability space, and let Γ (X,μ) = (X0,μ0)0 be the corresponding generalized Bernoulli action. Then for any discrete countable group Λ and any measurable cocycle α : Γ × X → Λ there exist a homomorphism : Γ → Λ and a meas...