A hyperbolic locus $\mathcal{H} \subset SL(2,R)^n$ is a connected open set such that for all $x\in\mathcal{H}$, $\{x_i\}_1^n$ is a uniformly hyperbolic set of matrices.  In $SL(2,R)^2$, the geometry of the loci was studied in Avila, Bochi, and Yoccoz's 2008 work. In this talk, some of the details of the geometry in higher dimensions will be discussed, as well as the relevance with Schrodinger operators. 

Spherical Designs are very special points on the sphere with the property that the average of a low-degree polynomial over the points is the same as the global average of the polynomial on the sphere. As it turns out, the definition can be suitably interpreted to make sense on a finite combinatorial Graph as well.  The arising structures are breathtakingly pretty (many pictures will be shown). They can be interpreted as the analogue of Platonic bodies in graphs. Graphs can have many more symmetries than Euclidean space and, correspondingly, some of these point structures are remarkably symmetric. This is also naturally related to Extremal Combinatorics where classical Theorems (the Erdos-Ko-Rado Theorem or the Deza-Frankl theorem)  suddenly turn into beautiful special cases.   If we only consider the hypercube graph {0,1}^d, we naturally encounter problems from coding theory.   A probabilistic interpretation tells us new things about the speed with which random walks on the graph become random.   There will be pictures, a survey of recent results by C. Babecki, K. Golubev, D. Shiroma, R. Thomas and many, many open problems.

https://sites.uci.edu/inverse/