Gabor Toth (Rutgers Univ., USA)
Lecture 1: Spherical minimal immersion and their moduli
Lecture 2: Isotropic and helical minimal immersions between spheres
Lecture 3: Application: Orthogonal multiplications, and Cayley’s nodal cubic
David Glickenstein (Univ. Arizona, USA)
Lecture 1: Circle packings and discrete conformal structures on surfaces and domains
It is well known that any simply connected domain in the plane can be conformally mapped to the disk (Riemann Mapping Theorem). In 1985, Thurston proposed approximating the Riemann mapping by considering a circle packing that approximates the domain, repacking into a disk while maintaining the tangency pattern, and then considering the mappings between the circle packings as an approximate (or discrete) conformal map. A theory of discrete conformal structures was born, and successively much research has been done to try to understand what constitutes a discrete conformal structure and what it means to approximate with circle packings or other discrete conformal structures. In particular, it was shown that circle packings exhibit uniformization theorems similar to that in conformal geometry of surfaces, and also that the Riemann mapping of a domain to the disk can be approximated by sequences of discrete conformal mappings between circle packings. We will give an overview of these ideas.
Lecture 2: Special metrics and geometric flows
A key element of geometric analysis is the question of which manifolds can admit special metrics, and how many such special metrics there can be on a given manifold. For instance, does a manifold admit a metric of constant sectional curvature? The notion of Ricci flat and Einstein metrics is a natural one, as is the notion of constant scalar curvature. There are often topological obstructions to the existence of certain kinds of special metrics; for instance, the Gauss-Bonnet Theorem implies that any constant curvature metric on the torus must have curvature zero. A related question is how to find special metrics. The philosophy behind parabolic geometric flows is that since any manifold admits a Riemannian metric, a flow of metrics that behaves like a heat equation (by doing some nonlinear averaging) will push the metric toward a special metric that has more symmetry. In particular, the Ricci flow was successfully used to show that every three-dimensional manifold admits a decomposition into manifolds that admit Thurston geometries. We will discuss some notions of special metrics and how geometric flows can be used to find and/or classify them.
Lecture 3: Discrete Riemannian geometry in three dimensions: progress and open problems
While the two dimensional work in discrete conformal structures is well-developed (as seen in the first lecture), the theory of higher dimensions is less so. There is, however, a natural notion of certain curvature measures (Lipschitz-Killing curvatures, including scalar curvature measure), and these can lead to a theory of Einstein metrics and constant scalar curvature metrics. Since the smooth theory of geometric flows on to find these metrics is also well developed in dimension three (as seen in the second lecture), one might hope to be able to understand these in dimension three as well. We will survey some of the known results about discrete Riemannian geometry based on piecewise flat (or polyhedral) geometry in three dimensions as well as some interesting directions for future research.
Jinjin Liang (Univ. Arizona, USA)
Asymptotic behavior of solutions to a polygon flow
We investigate a nonlinear dynamical system of $N$ particles in the plane. We consider the $N$ particles as the vertices of a polygon, and therefore geometrically, this system determines an evolution of the planar polygons. We obtain some analogous result as the smooth curve shortening flow. In particular, we have shown that it shrinks any planar polygon to a point; besides, the regular shape is asymptotic stable to any small perturbations except at the dimension four; while in dimension four, we shall see that the shape of square is locally stable at most at a hypersurface; Furthermore, we have a global picture of the solutions with the additional assumption that if the angle at each particle satisfies some lower bound during the evolution, then there exists a rescaled sequence extracted from the evolution converges to a limiting polygon, which is exactly the self-similar solution of this system.
Alexandre Ramos Peón (Universitetet i Oslo, Norway)
Oka theory, or the h-principle in complex analysis
Heuristically, an h-principle (homotopy principle) is said to hold for a certain geometric problem if a solution exists provided there are no topological obstructions. In real differential topology, examples include the Smale-Hirsch theory of smooth immersions, as well as Nash's isometric immersions.
In complex analysis the h-principle is known as the Oka principle. One of the principal examples is Oka's solution to the second Cousin problem and Grauert's equivalence of classification of complex vector bundles over Stein spaces: continuous sections can be deformed to holomorphic sections.
In this talk, I will not present new results but rather briefly discuss the above Oka principle of Grauert (recalling the relevant concepts from complex analysis) and introduce Gromov's linearizing concept of "dominating sprays": "elliptic manifolds" are those admitting such sprays. I will sketch how these are the key to generalize the Oka principles to sections of more general bundles, and hence how elliptic manifolds are naturally "dual" to Stein manifolds: this is the basis of a recently emerging area of complex geometry called Oka theory. If time permits I will indicate applications to the important problem of holomorphic embeddings of Stein manifolds into affine space of minimal dimension and to that of the Gromov-Vaserstein problem.
Grardo Arizmendi Echegaray (IMATE-UNAM, Mexico)
Twistor space for Riemannian manifolds with even Clifford structure
The notion of an even Clifford structure (ECS) on Riemannian manifolds was introduced by Uwe Semmelmann and Andrei Moroianu, which encloses almost Hermitian and quaternion-Hermitian structures as particular cases. In this talk I will explain the contruction of a twistor space for Riemannian manifolds with ECS which is a natural generalization of the twistor space of a Quaternion-Kahler manifold, and present some results analogous to the Quaternion-Kahler case.