In this paper the germ of neighborhood of a compact leaf in a Lagrangian foliation is symplectically classified when the compact leaf is ${{\mathbb{T}}^{2}}$ , the affine structure induced by the Lagrangian foliation on the leaf is complete, and the holonomy of ${{\mathbb{T}}^{2}}$ in the foliation linearizes. The germ of neighborhood is classified by a function, depending on one transverse coordinate, this function is related to the affine structure of the nearly compact leaves.

We prove an algebraic “no-go theorem” to the effect that a nontrivial Poisson algebra cannot be realized as an associative algebra with the commutator bracket. Using it, we show that there is an obstruction to quantizing the Poisson algebra of polynomials generated by a nilpotent basic algebra on a symplectic manifold. This result generalizes Groenewold’s famous theorem on the impossibility of quantizing the Poisson algebra of polynomials on ${{\mathbf{R}}^{2n}}$ . Finally, we explicitly construct a polynomial quantization of a symplectic manifold with a solvable basic algebra, thereby showing that the obstruction in the nilpotent case does not extend to the solvable case.

Let ${{B}_{N}}$ be the unit ball in ${{\mathbb{C}}^{N}}$ and let $f$ be a function holomorphic and ${{L}^{2}}$ -integrable in ${{B}_{N}}$ . Denote by $E\left( {{B}_{N}},\,f \right)$ the set of all slices of the form $\Pi \,=\,L\,\cap \,{{B}_{N}}$ , where $L$ is a complex one-dimensional subspace of ${{\mathbb{C}}^{N}}$ , for which $f{{|}_{\Pi }}$ is not ${{L}^{2}}$ -integrable (with respect to the Lebesgue measure on L). Call this set the exceptional set for $f$ . We give a characterization of exceptional sets which are closed in the natural topology of slices.

In May, 1999 James Greig Arthur, University Professor at the University of Toronto was awarded the Canada Gold Medal by the National Science and Engineering Research Council. This is a high honour for a Canadian scientist, instituted in 1991 and awarded annually, but not previously to a mathematician, and the choice of Arthur, although certainly a recognition of his greatmerits, is also a recognition of the vigour of contemporary Canadian mathematics.

We construct unbounded positive ${{C}^{2}}$ -solutions of the equation $\Delta u\,+\,K{{u}^{\left( n+2 \right)/\left( n-2 \right)}}\,=\,0$ in ${{\mathbb{R}}^{n}}$ (equipped with Euclidean metric ${{g}_{0}}$ ) such that $K$ is bounded between two positive numbers in ${{\mathbb{R}}^{n}}$ , the conformal metric $g\,=\,{{u}^{4/\left( n-2 \right)}}{{g}_{0}}$ is complete, and the volume growth of $g$ can be arbitrarily fast or reasonably slow according to the constructions. By imposing natural conditions on $u$ , we obtain growth estimate on the ${{L}^{2n/\left( n-2 \right)}}$ -norm of the solution and show that it has slow decay.

A generalization of Schmüdgen’s Positivstellensatz is given which holds for any basic closed semialgebraic set in ${{\mathbb{R}}^{n}}$ (compact or not). The proof is an extension of Wörmann’s proof.

Let $G$ be an infinite discrete amenable group or a non-discrete amenable group. It is shown how to construct a net $\left( {{f}_{\alpha }} \right)$ of positive, normalized functions in ${{L}_{1}}\left( G \right)$ such that the net converges weak* to invariance but does not converge strongly to invariance. The solution of certain linear equations determined by colorings of the Cayley graphs of the group are central to this construction.

The zeta function of a nonsingular pair of quadratic forms defined over a finite field, $k$ , of arbitrary characteristic is calculated. A. Weil made this computation when char $k\,\ne \,2$ . When the pair has even order, a relationship between the number of zeros of the pair and the number of places of degree one in an appropriate hyperelliptic function field is established.