This paper gives a generalization of some results on Hilbert schemes of points on surfaces. Let M^G(r,n) (resp. M^U(r,n)) be the Gieseker (resp. Uhlenbeck) compactification of the moduli spaces of stable bundles on a smooth projective surface. We show that, for surfaces satisfying some technical condition:

(a) The natural map M^G(r,n) → M^U(r,n) generalizing the Hilbert-Chow morphism from the Hilbert scheme of n points on S to the n-th symmetric power, is strictly semi-small in the sense of Goresky-MacPherson with respect to some stratification.

(b) Let P_t(X) be the Intersection Homology Poincare polynomial of X. Generalizing the computation due to Gottsche and Sorgel we prove that the ratio ∑_n qⁿP_t(M^G(r,n))/∑_n qⁿP_t(M^U(r,n)) is a character of a certain Heisenberg-type algebra.

(c) Generalizing results of Nakajima we show how to obtain the action of the Heisenberg algebra on the cohomology using correspondences.

We show that a compact self-dual four-manifold with a smooth action of a two-torus and with non-zero Euler characterestic is necessarily diffeomorphic to a connected sum of copies of complex projective planes, and furthermore the self-dual structure is isomorphic to one of those constructed by Joyce in [11]. This settles a conjecture of Joyce [11] affirmatively. Our method of proof is to show, by complex geometric techniques, that the associated twistor space, which is a compact complex threefold with the induced holomorphic action of algebraic two-torus, has a very special structure and is indeed determined by a certain invariant which is eventually identified with the invariant associated with the Joyce's construction of his self-dual manifolds.

This paper studies knots that are transversal to the standard contact structure in $\mathbb{R}^3$ bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type $\mathcal{TK}$ is transversally simple if it is determined by its topological knot type $\mathcal{K}$ and its Bennequin number. The main theorem asserts that any $\mathcal{TK}$ whose associated $\mathcal{K}$ satisfies a condition that we call exchange reducibility is transversally simple.

As a first application, we prove that the unlink is transversally simple, extending the main theorem in [10]. As a second application we use a new theorem of Menasco [17] to extend a result of Etnyre [11] to prove that all iterated torus knots are transversally simple. We also give a formula for their maximum Bennequin number. We show that the concept of exchange reducibility is the simplest of the constraints that one can place on $\mathcal{K}$ in order to prove that any associated $\mathcal{TK}$ is transversally simple. We also give examples of pairs of transversal knots that we conjecture are not transversally simple.

Quaternionic maps (Q-maps) between hyperkähler manifolds are quaternionic analogue of Cauchy-Riemann equations between Kähler manifolds. We provide a necessary and sufficient condition on when a quaternionic map becomes holomorphic with respect to some complex structures in the hyperkähler 2-spheres, and give examples of Q-maps which cannot be holomorphic. When the domain is real 4-dimensional, we analyze the structure of the blow-up set of a sequence of Q-maps, and show that the singular set of a stationary $Q$-map is $\mathcal{H}^1$-rectifiable.