A version of the Donaldson-Thomas invariants of a Calabi-Yau threefold is proposed as a conjectural mathematical definition of the Gopakumar-Vafa invariants. These invariants have a local version, which is verified to satisfy the required properties for contractible curves. This provides a new viewpoint on the computation of the local Gromov-Witten invariants of contractible curves by Bryan, Leung, and the author.

Let (Mn, g) be a complete non-compact K¨ahler manifold with non-negative and bounded holomorphic bisectional curvature. We prove that M is holomorphically covered by a pseudoconvex domain in Cn which is homeomorphic to R2n, provided (Mn, g) has uniform linear average quadratic curvature decay.

We prove a general maximum principle at infinity for properly immersed minimal surfaces with boundary in R^3. An important corollary of this maximum principle at infinity is the existence of a fixed sized regular neighborhood for any properly embedded minimal surface of bounded curvature.

We investigate the minimal and isoperimetric surface problems in a large class of sub-Riemannian manifolds, the so-called Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of Bryant, Griffiths and Grossman, derive succinct forms of the Euler-Lagrange equations for critical points for the associated variational problems. Using the Euler-Lagrange equations, we show that minimal and isoperimetric surfaces satisfy a constant horizontal mean curvature conditions away from characteristic points. Moreover, we use the formalism to construct a horizontal second fundamental form, II0, for vertically rigid spaces and, as a first application, use II0 to show that minimal surfaces cannot have points of horizontal positive curvature and that minimal surfaces in Carnot groups cannot be locally strictly horizontally geometrically convex. We note that the convexity condition is distinct from others currently in the literature.

We describe a general geometrical construction of representations of fundamental groups of 3-manifolds into PU(2, 1) and eventually of spherical CR structures defined on those 3-manifolds. We construct branched spherical CR structures on the complement of the figure eight knot and the Whitehead link. They have discrete holonomies contained in PU(2, 1,Z[ω]) and PU(2, 1,Z[i]) respectively.