Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is motivated by the problem of constructing strict deformation quantizations of symplectic manifolds. We show that this type of approximation exists for any real number and also investigate what happens if the number is rational or a quadratic irrational.

We consider a Hamiltonian action of $n$-dimensional torus, $T^n$, on a compact symplectic manifold $(M,\omega)$ with $d$ isolated fixed points. For every fixed point $p$, there exists (although not unique) a class $a_p \in H^*_{T}(M; \mathbb{Q})$ such that the collection $\{a_p\}$, over all fixed points, forms a basis for $H^*_{T}(M; \mathbb{Q})$ as an $H^*(BT; \mathbb{Q})$ module. The map induced by the inclusion, $\iota^*:H^*_{T}(M; \mathbb{Q}) \rightarrow H^*_{T}(M^{T}; \mathbb{Q})= \oplus_{j=1}^{d}\mathbb{Q}[x_1, \ldots, x_n] $ is injective. We use such classes $\{a_p\}$ to give necessary and sufficient conditions for $f=(f_1, \ldots ,f_d)$ in $\oplus_{j=1}^{d}\mathbb{Q}[x_1, \ldots, x_n]$ to be in the image of $\iota^*$, i.e., to represent an equivariant cohomology class on $M$. In the case when $T$ is a circle and present these conditions explicitly. We explain how to combine this $1$-dimensional solution with Chang-Skjelbred Lemma in order to obtain the result for a torus $T$ of any dimension. Moreover, for a Goresky-Kottwitz-MacPherson (GKM) $T$-manifold $M$ our techniques give combinatorial description of $H^*_{K}(M; \mathbb{Q})$, for a generic subgroup $K \hookrightarrow T$, even if $M$ is not a GKM $K$-manifold.

For a compact contact manifold $M^{2n + 1}$, it is shown that the anisotropic Folland-Stein function spaces $\Gamma^{s} (M), s \geq (2n + 4)$ form an algebra. The notion of anisotropic regularity is extended to define the space of $\Gamma^{s}$-contact diffeomorphisms, which is shown to be a topological group under composition and a smooth Hilbert manifold. These results are used in a subsequent paper to analyse the action of the group of contact diffeomorphisms on the space of CR structures on a compact, three-dimensional manifold.

Let $\mathcal{L} \to \mathcal{M}$ be a Hermitian line bundle over a compact manifold. Write $\mathcal{S}$ for the space of all unitary connections in $\mathcal{L}$ whose curvatures define symplectic forms on $\mathcal{M}$ and $\mathcal{G}$ for the identity component of the group of unitary bundle isometries of $\mathcal{L}$, which acts on $\mathcal{S}$ by pullback. The main observation of this note is that $\mathcal{S}$ carries a $\mathcal{G}$-invariant symplectic structure, there is a moment map for the $\mathcal{G}$-action and that this embeds the components of $\mathcal{S}$ as $\mathcal{G}$-coadjoint orbits. Restricting to the subgroup of $\mathcal{G}$ which covers the identity on $\mathcal{M}$, we see that prescribing the volume form of a symplectic structure can be seen as finding a zero of a moment map. When $\mathcal{M}$ is a Kähler manifold, this gives a moment-map interpretation of the Calabi conjecture. We also describe some directions for future research based upon the picture outlined here.

Starting with the data of a curve of singularity types, we use the Legendre transform to construct weak geodesic rays in the space of locally bounded metrics on an ample line bundle $L$ over a compact manifold. Using this we associate weak geodesics to suitable filtrations of the algebra of sections of $L$. In particular this works for the natural filtration coming from an algebraic test configuration, and we show how this recovers the weak geodesic ray of Phong-Sturm.

We give a presentation of the cohomology ring of spatial polygon spaces $M(r)$ with fixed side lengths $r \in \mathbb{R}^n_+$. These spaces can be described as the symplectic reduction of the Grassmaniann of 2-planes in $\mathbb{C}^n$ by the $U(1)^n$-action by multiplication, where $U(1)^n$ is the torus of diagonal matrices in the unitary group $U(n)$. We prove that the first Chern classes of the $n$ line bundles associated with the fibration ($r$-level set) $\to M(r)$ generate the cohomology ring $H*(M(r),\mathbb{C})$. By applying the Duistermaat-Heckman Theorem, we then deduce the relations on these generators from the piece-wise polynomial function that describes the volume of $M(r)$. We also give an explicit description of the birational map between $M(r)$ and $M(r ')$ when the lengths vectors $r$ and $r '$ are in different chambers of the moment polytope. This wallcrossing analysis is the key step to prove that the Chern classes above are generators of $H*(M(r))$, (This is well-known when $M(r)$ is toric, and by wall-crossing we prove that it holds also when $M(r)$ is not toric).