I will discuss recent work with David Wiygul to determine the index and nullity of the Lawson surfaces \(\xi_{g,1}\) (in Lawson's original notation) of any genus \(g\ge2\) (arXiv:1904.05812).

I will discuss the current status of doubling and desingularization constructions for minimal surfaces with emphasis to (current and/or possible) applications to minimal surfaces in the round three-sphere and the Yau question on the existence of infinitely many minimal surfaces in Riemannian manifolds.

I will discuss doubling constructions for the Clifford torus and the equatorial two-sphere in the round three-sphere. In doubling constructions minimal surfaces are constructed resembling two copies of the given minimal surface joined by many small catenoidal bridges. I will then discuss desingularization constructions where minimal surfaces are constructed by replacing the intersection curves of minimal two-surfaces in a Riemannian three-manifold with handles modeled after the singly periodic Scherk surfaces and then perturbing to minimality. Finally I will discuss some applications and open questions for closed embedded minimal surfaces in the three-sphere.

I will survey the recent work of Haskins and myself constructing new special Lagrangian cones in $C^n$ for all $nge3$ by gluing methods. The link (intersection with the unit sphere S^{2n-1}) of a special Lagrangian cone is a special Legendrian (n-1)-submanifold. I will start by reviewing the geometry of the building blocks used. They are rotationally invariant under the action of SO(p) imes SO(q) (p+q=n) special Legendrian (n-1) submanifolds of S^{2n-1}. These we fuse (when p=1, p=q) to obtain more complicated topologies. The submanifolds obtained are perturbed to satisfy the special Legendrian condition (and their cones therefore the special Lagrangian condition) by solving the relevant PDE. This involves understanding the linearized operator and its small eigenvalues, and also ensuring appropriate decay for the solutions.. Nicos Kapuleas visitará granada desde el 10 al 17 de mayo.