This thematic semester aims at exploring those areas of topology where the research challenges stem from scientific and engineering problem and computer experiments rather than the intrinsic development of the topology proper. In this context, topology is a toolbox of mathematical results and constructions which impacts and inspires developments in other areas. Born as a supporting discipline, aimed at creating a foundation of intuitive notions immensely useful in differential equations and complex analysis, algebraic topology remains indispensable in many disciplines.

Our goal here is to concentrate on relatively recent areas of research enabled, in particular, by the computational revolution in mathematical discovery.

The past decade has seen a surge of interest in applied topology, a vaguely defined area that became a focus of attraction for several communities neighboring mathematics proper - biology, engineering, computer science, data analysis, to name a few. The central lure of this emerging area is the feeling that the flexibility afforded by the passage from geometric constructions to analysis situs, analysis of (mutual) positions, can be a powerful force in applications. The meta-goal of this project is to contribute to the development of the original view of topology as analysis situs: the extraction of discrete, combinatorial invariants from continuous shapes.

Here is a sample of the topics to be covered: moduli spaces of linkages, hinges and origami; configuration spaces of hard disks and coverings; topological robotics; topology of random simplicial complexes and random varieties; topology in condensed matter; high-dimensional expanders; equipartitions and Tverberg type theorems.