MRC Conference Week 3b: June 30-July 6, 2024
Homotopical Combinatorics
Organizers:
- Andrew Blumberg, Columbia University
- Michael Hill, UCLA
- Kyle Ormsby, Reed College
- Angélica Osorno, Reed College
- Constanze Roitzheim, University of Kent
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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.
One of the exciting features of the program is that there are few prerequisites. Familiarity with abstract homotopy theory or with modern methods of algebraic topology will allow deeper engagement with some of the potential problems, but is not required, and much of the subject can be approached purely combinatorially. Before the workshop, relevant readings will be provided to help provide background, and an online collaboration platform will be used to start discussing material and to begin building community. At the workshop, participants can expect to work in teams on research programs, to engage with lectures from senior faculty participants about aspects of homotopical combinatorics, and to have open feedback sessions for further discussion.
The primary focus of the workshop is supporting early-career researchers, including advanced graduate students, postdocs, and pre-tenure faculty. As such, there will also be professional development sessions, the topics of which will be driven by participant interest and need. We especially encourage members of traditionally excluded groups to apply.
Applications will be accepted on MathPrograms.org through Thursday, February 15, 2024 (11:59 p.m. EST).
