**Abstract:**

After delivering a short manifesto on the nature of contemporary homotopy theory — no longer a subfield of topology! — I will introduce two homotopical structures amenable to combinatorial study. *Transfer systems* are central to equivariant derived algebraic geometry, but admit an elementary description as ideal-like sub-posets of lattices. *Model structures* underpin the entirety of homotopy theory, but it is a recent discovery that they arise as special intervals in the lattice of transfer systems when the underlying category is a poset. Both transfer systems and model structures exhibit striking connections with Catalan-type enumerative combinatorics, and this talk can be viewed as an invitation for combinatorialists to contribute to homotopy theory by applying their tools in this arena. Portions of this work are joint with Balchin, Franchere, Hafeez, MacBrough, Marcus, Osorno, Qin, Roitzheim, and Waugh.

**Note: This talk begins with a pre-seminar (aimed at graduate students) at 4:00–4:30. The main talk starts at 4:40.**

**Join Zoom Meeting: https://washington.zoom.us/j/ 91547335974Meeting ID: 915 4733 5974**