eCHT REU 2023

The 2023 eCHT REU is an online Research Experience for Undergraduates supported by the NSF and led by Kyle Ormsby and Christy Hazel (with assistance from Scotty Tilton). Between June 12 and July 21, we will be researching problems in homotopical combinatorics, a field that studies the abstract homotopy theory of small categories.

Below you can find our schedule, a list of resources for the program, and a problem list. Further information (including Zoom and Jamboard links) is available to participants on our Zulip channel.

schedule

_{All times US Pacific Daylight (UTC-07:00). See Zulip for links. We’ll observe the Juneteenth (June 19) and Independence Day (July 4) holidays, so no meetings those days.}

weeks 1 and 2 (weekdays June 12 - 23)

• 9-10am: Lecture
• 10-11am: Problem session
• 11am-12pm: Lunch
• 12-1pm: Lecture
• 1-2pm: Discussion

weeks 3 through 6 (weekdays June 26 - July 21)

• 9-10am: Morning briefing
• 10-11am:
  • MWTh: Office hours (teams with Kyle or Christy)
  • TuF: Lab meeting (full group)
• 11am-1pm: Lunch / work in private or teams
• 1pm: Team check-in

resources

The following notes, papers, and books might be useful. Those marked with a ✹ are our core references.

papers

• Balchin-Barnes-Roitzheim, $N_\infty$-operads and associahedra
  • The original link between transfer systems on $C_{p^n}$ and Catalan combinatorics.
• Balchin-Bearup-Pech-Roitzheim, Equivariant homotopy commutativity for $G=C_{pqr}$
• Balchin-MacBrough-Ormsby, Composition closed premodel structures and the Kreweras lattice
• Balchin-MacBrough-Ormsby, Lifting $N_\infty$ operads from conjugacy data
  • Transfer systems for a non-Abelian group via subgroups up to conjugacy.
• ✹ Balchin-MacBrough-Ormsby, The combinatorics of $N_\infty$ operads for $C_{qp^n}$ and $D_{p^n}$
• ✹ Balchin-Ormsby-Osorno-Roitzheim, Model structures on finite total orders
  • Primary reference for the homotopical combinatorics program. Enumerates model structures on the category $0\to 1\to \cdots \to n$.
• Blumberg-Hill, Operadic multiplications in equivariant spectra, norms, and transfers
  • $N_\infty$ operads, norms, and indexing systems.
• Blumberg-Hill, Bi-incomplete Tambara functors
  • Foundations for functors admitting restriction, some transfers, and some norms.
• Chan, Bi-incomplete Tambara functors as $\Box$-commutative monoids
  • Compatible systems of transfer systems characterize bi-incomplete Tambara functors.
• ✹ Franchere-Ormsby-Osorno-Qin-Waugh, Self-duality of the lattice of transfer systems via weak factorization systems
  • The original link between transfer systems for $G$ and weak factorization systems for Sub($G$).
• Hafeez-Marcus-Ormsby-Osorno, Saturated and linear isometric transfer systems for cyclic groups of order $p^mq^n$
• Hill, Equivariant chromatic localizations and commutativity
  • The effect of Bousfiel localization on norms / transfer systems.
• Hill-Meng-Li, Counting compatible indexing systems for $C_{p^n}$
  • The numerology of incomplete Tambara functors for $C_{p^n}$ matches that of composition closed model structures on $[n]$.
• Rubin, Characterizations of equivariant Steiner and linear isometries operads
  • Saturated transfer systems, nice diagrams of transfer system lattices.
• Rubin, Categorifying the algebra of indexing systems
• Rubin, Combinatorial $N_\infty$ operads
  • Equivalence of $N_\infty$ operads up to homotopy with transfer systems (and indexing systems).

other writing

• ✹ Blog: Homotopical combinatorics – an introduction for combinatorialists
• Slides: Transfer Systems and Model Structures for Combinatorialists
• Slides: Homotopical Combinatorics (and video)

problems

Some open problems in homotopical combinatorics and transfer systems:

1. Explore the combinatorics of the recursive $\odot$ construction of transfer systems presented in BMO22a for new families of lattices/groups. Start with rectangular and Boolean lattices.
2. Enumerate compatible pairs of transfer systems (in the sense of Cha22 and HML23) for new families of groups. Use this information to further explore the structure of bi-incomplete Tambara functors BH22.
3. Employ catalytic generating function methods and the recurrences of BMO22a to determine a closed formula for the number of transfer systems on dihedral groups of order $2p^n$ and cyclic groups of order $qp^n$. Once closed formulas are in hand, seek bijective correspondences with known structures.
4. After identifying the lattice of transfer systems for any Abelian group $A$, follow the methods of BOOR23 and BMO22b to enumerate premodel structures, composition closed premodel structures, and Quillen model structures on Sub($A$).
5. Building off of Rub21b, begin to understand the relationship between saturated transfer systems and linear isometries operads for non-cyclic groups.
6. Leveraging recent work on the structure of transfer systems, extend the work of Hil19 to better understand the effect of various localizations on equivariant commutativity.
7. Explore the duality on transfer systems discovered by FOOQW22, extending it to non-Abelian groups and lifting it to operads.
8. Following BK11, understand the relationship between model structures on posets, relative posets, cofibrant objects in RelCat, and $(\infty,1)$-categories. Can we use model structures on posets to compute morphisms in certain $\infty$-categories?