The group of symmetries (or homeomorphisms) of a surface is an essential object in mathematics and science. It has connections to algebra, algebraic geometry, complex dynamics, number theory, and even fluid dynamics. In this talk, we discuss ways to measure the amount of mixing of a homeomorphism, combinatorial models to study the group of homeomorphisms, and studying the action of the group of homeomorphisms on hyperbolic spaces.

The talk will be devoted to a new invariant of the combinatorial type of a simplicial complex, the double cohomology of its moment-angle complex, which arose in toric topology, and its applications in topological data analysis. Its introduction was motivated by the generalization of the concept of persistent homology barcodes obtained in the framework of toric topology using the so-called persistent Tor-algebras, however, no prior knowledge of topological data analysis is assumed. On the one hand, this invariant can be defined "algebraically" as the secondary cohomology of the moment-angle complex of a given simplicial complex with respect to the differential we introduce on the Koszul algebra. On the other hand, it also allows for a "geometric" description that does not use toric topology and is based on knowledge of the combinatorial structure of all full subcomplexes of a simplicial complex. This new combinatorial invariant in some cases turns out to be significantly more convenient for calculations than the usual moment-angle complex cohomology. I will show examples of calculations of our invariant, as well as several theorems that allow simplifying calculations in a more general situation. The talk is based on joint works with A. Bahri, T. E. Panov, J. Song and D. Stanley.