This project belongs to the field of Algebraic Topology, in a broad sense, and with applications going both ways between algebra and topology. While originally, algebraic topology meant the study of homotopy types of topological spaces by the means of algebraic invariants assigned to them functorially, the past decade has seen a proliferation of applications of homotopical methods in various areas of mathematics, such as notably algebraic geometry, representation theory and aspects of mathematical physics. The project is concerned more specifically with applications of methods of homotopy theory to the understanding of fusion systems both finite and compact modeling conjugacies in finite groups and Lie groups. In this context we attack several important open problems, some of which are of interest in group theory.

We pursue the program for the classification of homotopy classes of maps between classifying spaces, also with the aim of developing complex homotopy representation theory. In a more algebraic setting, we aim to understand systematically exotic examples of fusion systems and construct new ones. As a new line of research we will start the study of stable homotopy theory of p-local compact groups, which is the starting point for extending statements on generalized cohomology theories for Lie groups.

The techniques of localisation and cellularisation belong properly both to algebraic topology and to algebra. We aim to investigate to which extent homotopy idempotent functors applied to classifying spaces detect algebraic structure. A more geometric aspect of the project goes in the direction of properties of the mapping class group, the group of isotopies of an orientable surface, and its relationship with the invariants of homology spheres.

We develop the theory of homotopy combinatorics, where the homotopy theory of groupoids and infinity-groupoids is applied to problems in enumerative and algebraic combinatorics, with special emphasis on applications to quantum field theory. In particular we develop a unifying framework for incidence algebras, combinatorial Hopf algebras, inductive data types, and related structures, uncovering common patterns in a wide range of application areas.