Guest-Ohnita and Crawford have shown the path-connectedness of the space of harmonic maps from $S^2$ to $\mathbf{C} P^n$ of a fixed degree and energy.It is well-known that the $\partial$ transform is defined on this space. In this paper,we will show that the space is decomposed into mutually disjoint connected subspaces on which $\partial$ is homeomorphic.

An abstract polytope of rank $n$ is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. This paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks $3$, $4$, and $5$.

For every polytope $\mathcal{P}$ there is the universal regular polytope of the same rank as $\mathcal{P}$ corresponding to the Coxeter group $\mathcal{C} =[\infty, \dots, \infty]$. For a given automorphism $d$ of $\mathcal{C}$, using monodromy groups, we construct a combinatorial structure $\mathcal{P}^d$. When $\mathcal{P}^d$ is a polytope isomorphic to $\mathcal{P}$ we say that $\mathcal{P}$ is self-invariant with respect to $d$, or $d$-invariant. We develop algebraic tools for investigating these operations on polytopes, and in particular give a criterion on the existence of a $d$\nobreakdash-auto\-morphism of a given order. As an application, we analyze properties of self-dual edge-transitive polyhedra and polyhedra with two flag-orbits. We investigate properties of medials of such polyhedra. Furthermore, we give an example of a self-dual equivelar polyhedron which contains no polarity (duality of order 2). We also extend the concept of Petrie dual to higher dimensions, and we show how it can be dealt with using self-invariance.

In this paper, we characterize unitary representations of $\pi:=\piS$ whose generators $u_1, \dots, u_l$ (lying in conjugacy classes fixed initially) can be decomposed as products of two Lagrangian involutions $u_j=\s_j\s_{j+1}$ with $\s_{l+1}=\s_1$. Our main result is that such representations are exactly the elements of the fixed-point set of an anti-symplectic involution defined on the moduli space $\Mod:=\Hom_{\mathcal C}(\pi,U(n))/U(n)$. Consequently, as this fixed-point set is non-empty, it is a Lagrangian submanifold of $\Mod$. To prove this, we use the quasi-Hamiltonian description of the symplectic structure of $\Mod$ and give conditions on an involution defined on a quasi-Hamiltonian $U$-space $(M, \w, \mu\from M \to U)$ for it to induce an anti-symplectic involution on the reduced space $M/\!/U := \mu^{-1}(\{1\})/U$.

We introduce and investigate using Hilbert modules the properties of the {\em Fourier algebra} $A(G)$ for a locally compact groupoid $G$. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.

We determine and completely describe all pure realizations of the finite regular toroidal polyhedra of types $\{3,6\}$ and $\{6,3\}$.