AMS MRC conference Homotopical Combinatorics

Transfer systems are a new combinatorial object that exhibit surprising connections between abstract homotopy theory, equivariant topology, and combinatorics. About ten years ago, Blumberg and Hill defined the related "indexing systems'' as the central algebraic object controlling twisted multiplications that naturally arise in the study of equivariant cohomology theories. Rubin and Balchin--Barnes--Roitzheim independently recast this notion in a much simpler framework, characterizing indexing systems in terms of transfer systems, as a particular kind of weak subposet of the lattice of subgroups of a finite group G , ordered by inclusion. Work of Ormsby--Osorno and teams of collaborators has shown how the natural generalization of this notion to an arbitrary poset has fascinating combinatorial properties, and Balchin--MacBrough--Ormsby have further connected this to abstract homotopy theories on posets. Each of these connections provides exciting results which can be transferred and reinterpreted in the other fields, yielding unexpected new structure and theorems. This MRC will introduce participants to this burgeoning new area, bringing together researchers with interests in combinatorics, algebraic topology, and abstract homotopy theory. The field is rife with open problems, including basic questions about the structure of transfer systems, combinatorics problems associated to counting transfer systems for natural families of posets, identifying connections with other combinatorial structures, and applying the language of model categories to recast and reform these questions.