On a smooth $n$ -dimensional complete variety $X$ over $\mathbb{C}$ we show that the trace map ${{\bar{\theta }}_{X}}\,:\,{{H}^{n}}(X,\,\Omega _{X}^{n})\,\to \,\mathbb{C}$ arising from Lipman's version of Grothendieck duality in $\left[ \text{L} \right]$ agrees with

$${{(2\pi i)}^{-n}}{{(-1)}^{n(n-1)/2}}\,\int_{X}{:\,H_{DR}^{2n}\,(X,\,\mathbb{C})\,\to \,\mathbb{C}}$$

under the Dolbeault isomorphism.

Recently [8], a harmonic theory was developed for a compact contact manifold from the viewpoint of the transversal geometry of contact flow. A contact flow is a typical example of geodesible flow. As a natural generalization of the contact flow, the present paper develops a harmonic theory for various flows on compact manifolds. We introduce the notions of $H$ -harmonic and ${{H}^{*}}$ -harmonic spaces associated to a Hörmander flow. We also introduce the notions of basic harmonic spaces associated to a weak basic flow. One of our main results is to show that in the special case of isometric flow these harmonic spaces are isomorphic to the cohomology spaces of certain complexes. Moreover, we find an obstruction for a geodesible flow to be isometric.

We define the $\mathcal{M}$ -harmonic conjugate operator $K$ and prove that it is bounded on the non-isotropic Lipschitz space and on $\text{BMO}$ . Then we show $K$ maps Dini functions into the space of continuous functions on the unit sphere. We also prove the boundedness and compactness properties of $\mathcal{M}$ -harmonic conjugate operator with ${{L}^{p}}$ symbol.

Let $G$ be a finitely generated, infinite group, let $p\,>\,1$ , and let ${{L}^{p}}\left( G \right)$ denote the Banach space $\left\{ \sum{_{x\in G}{{a}_{x}}x}|\sum{_{x\in G}|{{a}_{x}}{{|}^{p}}<\infty } \right\}$ . In this paper we will study the first cohomology group of $G$ with coefficients in ${{L}^{p}}\left( G \right)$ , and the first reduced ${{L}^{p}}$ -cohomology space of $G$ . Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.

We prove a quantized version of a theorem by M. V. Sheĭnberg: A uniform algebra equipped with its canonical, i.e., minimal, operator space structure is operator amenable if and only if it is a commutative ${{C}^{*}}$ -algebra.

Define a group $G$ to be in the class $S$ if for any finitely generated subgroup $K$ of $G$ having the property that there is a positive integer $n$ such that ${{g}^{n\,}}\in \,K$ for all $g\,\in \,G,\,K$ has finite index in $G$ . We show that a free product with amalgamation $A{{*}_{_{C}}}B$ and an $\text{HNN}$ group $A{{*}_{C}}$ belong to $S$ , if $C$ is in $S$ and every subgroup of $C$ is finitely generated.

This paper studies the following question: Given an ${\omega }'$ -symplectic action of a Lie group on a manifold $M$ which coincides, as a smooth action, with a Hamiltonian $\omega$ -action, when is this action a Hamiltonian ${\omega }'$ -action? Using a result of Morse-Bott theory presented in Section 2, we show in Section 3 of this paper that such an action is in fact a Hamiltonian ${\omega }'$ -action, provided that $M$ is compact and that the Lie group is compact and connected. This result was first proved by Lalonde-McDuff-Polterovich in 1999 as a consequence of a more general theory that made use of hard geometric analysis. In this paper, we prove it using classical methods only.

Let ${{A}_{\theta }}$ be the universal ${{C}^{*}}$ -algebra generated by two unitaries $U,\,V$ satisfying $VU\,=\,{{e}^{2\pi i\theta }}UV$ and let $\Phi $ be the antiautomorphism of ${{A}_{\theta }}$ interchanging $U$ and $V$ . The $K$ -theory of ${{R}_{\theta }}\,=\,\{a\,\in \,{{A}_{\theta }}\,:\,\Phi (a)\,=\,{{a}^{*}}\}$ is computed. When $\theta $ is irrational, an inductive limit of algebras of the form ${{M}_{q}}(C(\mathbb{T}))\,\oplus \,{{M}_{{{q}'}}}(\mathbb{R})\,\oplus \,{{M}_{q}}(\mathbb{R})$ is constructed which has complexification ${{A}_{\theta }}$ and the same $K$ -theory as ${{R}_{\theta }}$ .

The purpose of this note is to extend to an arbitrary generalized Hopf and Calabi-Eckmann manifold the following result of Kalyan Mukherjea: Let ${{V}_{n}}={{\mathbb{S}}^{2n+1}}\times {{\mathbb{S}}^{2n+1}}$ denote a Calabi-Eckmann manifold. If $f,g:\,{{V}_{n}}\to {{\mathbb{P}}^{n}}$ are any two holomorphic maps, at least one of them being non-constant, then there exists a coincidence: $f(x)\,=\,g(x)$ for some $x\in {{V}_{n}}$ . Our proof involves a coincidence theorem for holomorphic maps to complex projective varieties of the form $G/P$ where $G$ is complex simple algebraic group and $P\,\subset \,G$ is a maximal parabolic subgroup, where one of the maps is dominant.

In this paper we show that two minimal hypersurfaces in a manifold with positive Ricci curvature must intersect. This is then generalized to show that in manifolds with positive Ricci curvature in the integral sense two minimal hypersurfaces must be close to each other. We also show what happens if a manifold with nonnegative Ricci curvature admits two nonintersecting minimal hypersurfaces.

In the sequel we construct simple, unital, separable, stable, amenable ${{C}^{*}}$ -algebras for which the ordered ${{K}_{0}}$ -group is strongly perforated and group isomorphic to $Z$ . The particular order structures to be constructed will be described in detail below, and all known results of this type will be generalised.

We show here that the commutator subgroup of a free group of finite rank poses a basis of Bachmuth's type.

We determine an explicit cell decomposition of the wonderful compactification of a semisimple algebraic group. To do this we first identify the $B\,\times \,B$ -orbits using the generalized Bruhat decomposition of a reductive monoid. From there we show how each cell is made up from $B\,\times \,B$ -orbits.

A theorem concerning the asymptotic behaviour of partial sums of the coefficients of products of Dirichlet series is proved using properties of regularly varying functions. This theorem is a multiplicative analogue of Schur's Tauberian theorem for power series.

A Newton polygon is introduced for a ramified point of a Galois covering of curves over a finite field. It is shown to be determined by the sequence of higher ramification groups of the point. It gives a blowing up of the wildly ramified part which separates the branches of the curve. There is also a connection with local reciprocity.

The behaviour of the Hasse-Schmidt algebra of higher derivations under localization is studied using André cohomology. Elementary techniques are used to describe the Hasse-Schmidt derivations on certain monomial rings in the nonmodular case. The localization conjecture is then verified for all monomial rings.

Upper bounds of second order Dehn functions of asynchronously automatic groups are obtained.

Fix an elliptic curve ${{y}^{2}}\,=\,{{x}^{3}}\,+\,Ax\,+\,B$ , satisfying $A,\,B\,\in \,\mathbb{Z},\,A\ge \,\left| B \right|\,>\,0$ . We prove that the $\mathbb{Q}$ -torsion subgroup is one of $(0),\,\mathbb{Z}/3\mathbb{Z},\,\mathbb{Z}/9\mathbb{Z}$ . Related numerical calculations are discussed.