In this paper twistor methods are used to construct a family of multivalued harmonic functions on R^3 which were obtained by Dashen Yan using different methods. The branching sets for the solutions are ellipses and the functions have quadratic growth at infinity.

We develop the deformation theory of Calabi-Yau threefolds, by which we mean 3-dimensional complex manifolds with a nowhere-vanishing holomorphic 3-form, on manifolds with boundary. The boundary data is a closed, real 3-form on the 5-dimensional boundary. In the case of strongly pseudoconvex boundary, we obtain an analogue of Hitchin's local Torelli Theorem for compact manifolds, modulo a finite dimensional obstruction space, which we show is zero in many cases of interest.

This is an expository article discussing some of the work of Uhlenbeck, focusing mainly on work concerning harmonic maps and Yang-Mills fields.

We study an intrinsic volume form defined on a pseudoconvex hypersurface in a complex Calabi-Yau manifold. We compute first and second variation formulae and discuss possible analogues of the affine isoperimetric inequality. In the last section of the paper we explore infinite dimensional aspects, including moment maps for diffeomorphism group actions.

We consider the question if a five dimensional manifold can be embedded into a Calabi-Yau manifold of complex dimension three such that the real part of the holomorphic volume form induces a given closed 3-form on the 5-manifold. We define an open set of 3-forms in dimension five which we call strongly pseudoconvex, and show that for closed strongly pseudoconvex 3-forms the perturbative version of this embedding problem can be solved if a finite dimensional vector space of obstructions vanishes.

The first part of this article is a short and selective survey of developments in differential and algebraic geometry from the 1980's involving enumerative questions and nonlinear elliptic partial differential equations. In the second part we discuss possible extensions of these ideas to structures on 4-manifolds, in particular to complex structures on surfaces of general type

This is an expository article on the 1978 Atiyah-Drinfeld-Hitchin-Manin construction of Yang-Mills instanton connections over the 4-sphere.

The first part of the article surveys Atiyah’s work in algebraic geometry during the 1950s, mainly on holomorphic vector bundles over curves. In the second part we discuss his work from the late 1970s on mathematical aspects of gauge theories, involving differential geometry, algebraic geometry, and topology.

We consider harmonic sections of a bundle over the complement of a codimension 2 submanifold in a Riemannian manifold, which can be thought of as multivalued harmonic functions. We prove a result to the effect that these are stable under small deformations of the data. The proof is an application of a version of the Nash-Moser implicit function theorem.

In this article we discuss the work of Karen Uhlenbeck, mainly from the 1980s, focused on variational problems in differential geometry.The calculus of variations goes back to the 18th century.In the simplest setting we have a functional () = ∫ Φ(, ′ ),

We discuss a model for associative submanifolds in $G_{2}$-manifolds with K3 fibrations, in the adiabatic limit. The model involves graphs in a 3-manifold whose edges are locally gradient flow lines. We show that this model produces analogues of known singularity formation phenomena for associative submanifolds.

This is a survey article, to appear in the Proceedings of the 2018 International Congress of Mathematicians. (Revised, with added and updated references.)

lection of more personal recollections-what it was like to be his student, to work alongside him, to have avenues of exploration pointed out, or to be inspired and energized by his unique personality.The

This is a survey article, based on the author's lectures in the 2015 AMS Summer Research Institute in Algebraic Geometry, and to appear in the Proceedings.

The variational point of view on exceptional structures in dimensions 6, 7 and 8 is one of Nigel Hitchin’s seminal contributions. One feature of this point of view is that it motivates the study of boundary value problems, for structures with prescribed data on a boundary. This chapter considers the case of 7 dimensions and G2 structures. It briefly reviews a general framework and then goes on to examine in more detail symmetry reductions to dimensions 4 and 3. In the latter case, the chapter presents an interesting variational problem related to the real Monge–Ampère equation and describes a generalization of this.

We show that the $G_{2}$ holonomy equation on a manifold with boundary, with prescribed 3-form on the boundary, is elliptic. The main point is to set up a suitable linear elliptic boundary value problem. This result leads to a deformation theory. In particular we establish the existence of certain $G_{2}$ cobordisms between two small deformations of a Calabi-Yau 3-fold.