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... (more)

This workshop will bring together researchers interested in a panoply of unusual configuration spaces, arising in applied fields or in plausible models, to look for similarities or creative tensions between them.

Classical configuration spaces in mechanics are spaces of tuples of material points in Euclidean 3D space, sometimes constrained by the mutual distances or just precluding coincidences of any two particles. The no-k-equal configuration spaces and their generalizations relax the no-coincidence conditions allowing some number (greater than 1) of points to coincide. Linkages and origami (hinged structures) represent another well-established class of configuration spaces. Despite many spectacular results on the topology and geometry of the configuration (or moduli) spaces of such mechanical constructions, many natural questions are hard and remain unanswered. In general, neither the topology (even for generic structures), nor singularities of the configuration spaces are known... (more)

While the semester program as a whole is inspired by the original view of topology as analysis situs fueled by applications in natural sciences and engineering, this workshop emphasizes the impact of topology and geometry on discrete structures.

Combinatorially inspired configuration spaces, such as arrangements of points, lines, hyperplanes, polytopes, and the like, provide intricate material and ongoing challenge for topological and geometric techniques. The latter have often gone through a process of adjustment towards their discrete, stratified objects, as in the case of discrete Morse theory or application of Fourier analysis. Notably, the recent solution of the log-concavity conjecture for matroids by Adiprasito, Huh and Katz was achieved by developing Hodge theory for combinatorial geometries which opens up most exciting perspectives on further applications. The construction of higher dimensional expanders is yet another promising direction. Inspired by the rich theory of graph... (more)