Charles Boyer (New Mexico)
I will begin these lectures by giving a brief general discussion about Sasakian geometry and extremal Sasakian structures in particular. I want to focus on several foundational problems: 1. Given a manifold determine how many contact structures of Sasaki type there are.
2. Given an isotopy class of contact structures determine the space of compatible extremal Sasakian structures.
3. Given extremal Sasakian structures when do they have constant scalar curvature, and when are they Sasaki-Einstein? We concentrate our discussion on examples obtained by combining the join construction for Sasakian structures with the admissible construction for Kaehlerian ruled manifolds.
Alessandra Iozzi (ETH-Zurich)
CAT(0) groups and bounded cohomology (Lecture 1, Lecture 2, Lecture 3)
Bounded cohomology is by now one of the "classical" tools to approach rigidity questions. Without getting involved into technicalities, we will illustrate for example how it can be used to show that the behavior of "large" discrete groups G<Aut(T)xAut(T') acting cofinitely by automorphisms on a product of trees TxT' is very different from the one of their cousins acting on the product of hyperbolic planes. We will construct concrete examples of such groups G<Aut(T)xAut(T') and illustrate some of their stunning properties.
We will moreover show how the median class of an action on a tree, an easily and concretely defined bounded cohomology class, can be used to construct quasimorphisms of the group and identify properties of the given action. We will then move to the median class of a group acting on a CAT(0) cube complex and propose some few open problems.
The minicourse will be completely self-contained and no prior knowledge of bounded cohomology will be assumed.
Sergei Tabachnikov (Penn State)
Linear difference equations, frieze patterns, projective polygons, and the Pentagram Map.
I shall describe three closely interrelated spaces: the space of linear difference equations with periodic coefficients and (anti)periodic solutions, the space of frieze patterns, and the moduli space of polygons in the projective space. Of these three, frieze patterns, beautiful combinatorial objects introduced and studied by Coxeter and Conway in the 1970s, are lesser known. Recently they have attracted much attention due to their relation to the theory of cluster algebras. I shall describe the combinatorial Gale transform, a somewhat surprising duality between difference equations of different orders with (anti)periodic solutions.
Introduced by R. Schwartz about 20 years ago, the Pentagram Map acts on plane n-gons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new n-gon formed by their intersections. The pentagram map is a discrete completely integrable system whose continuous limit is the Boussinesq equation, a completely integrable PDE of soliton type. I shall survey recent work on the Pentagram Map and its generalizations, emphasizing its close ties with the theory of cluster algebras. I shall describe a higher-dimensional version of the Pentagram Map and, somewhat counter-intuitively, its 1-dimensional version. Time permitting, I shall also talk about configuration theorems of projective geometry related with the pentagram map.
Alberto Verjovsky (IMATE-Cuernavaca)
El teorema de uniformización de Koebe y Poincaré
Empezando con el teorema de la aplicación conforme de RIemann del disco a una región simplemente conexa y propia del plano complejo y la existencia de coordenadas isotérmicas, se explicará y dará una demostración del teorema de uniformización. Se darán ejemplos aplicaciones de este teorema fundamental.