Summary

Homotopical combinatorics is an emerging field of mathematics at the intersection of homotopy theory, category theory, and combinatorics. Its methods are relatively elementary, but its theorems have deep implications in homotopy theory. Thanks to its many open problems, homotopical combinatorics is an attractive field for early-career researchers as well as faculty at undergraduate-serving institutions looking for research problems suitable for undergraduates.

The main objects of study in homotopical combinatorics are transfer systems, partial orders on finite posets which satisfy certain axioms. There are many combinatorial problems involving transfer systems, such as enumerating all possible transfer systems or compatible pairs of transfer systems on the poset of subgroups of a finite group. As another example, transfer systems on categories associated with poset lattices correspond to weak factorization systems, which give rise to model structures studied in abstract homotopy theory. These topics naturally lead to further questions in combinatorics, such as the enumeration of orthogonal factorization systems on categories which do not arise from posets.

In the first part of this seminar, we will introduce transfer systems and survey some recent work on their enumerative and algebraic combinatorics. In the second part of the seminar, we will discuss the motivation from equivariant and abstract homotopy theory behind transfer systems, such as N_∞ operads and weak/orthogonal factorization systems. Throughout the seminar, we hope to produce lecture notes, compile exercises, and prepare a list of open problems.

Seminar goals

• Learn about transfer systems, the main object of study in homotopical combinatorics, and their applications in equivariant homotopy theory and model category theory
• Survey recent results enumerating particular types of transfer systems, including transfer systems for certain cyclic groups and compatible pairs of transfer systems
• Write a set of notes which could serve as a self-contained introduction to homotopical combinatorics
• Create a list of open problems in homotopical combinatorics suitable for early-career researchers and undergraduate research projects

A more detailed weekly outline of topics and references is forthcoming.

Participant expectations

Participants will take turns giving presentations and/or writing lecture notes. The seminar organizers will make sure that presenters have written source material upon which to base their presentations and notes. Participants are expected to attend regularly.

Participants are encouraged to skim the source material being presented each week. They are also encouraged to participate in the private shared online workspace.

As a general rule, students are expected to obtain 1-2 independent study credits from their home universities for their participation in the seminar. This requirement is waived on an individual basis for students whose home universities do not have formal independent study opportunities.

Organizers

The seminar is organized by Scott Balchin (Queen’s University Belfast) and J.D. Quigley (University of Virginia). They are assisted by graduate student assistants to be named.

Application

Please submit the application form to apply. Applications received by December 15 will receive full consideration. Contact J.D. Quigley or Dan Isaksen for inquiries about applications.

Schedule

The seminar meets regularly on Wednesdays at 11:00am eastern time, starting January 15 and ending April 30.