In this talk, we introduce the area of homotopical combinatorics, that is, under-standing homotopic theoretic data using combinatorial data. This area arose fromthe work of Barnes, Balchin and Roitzheim and Rubin. They showed the equi-variant commutative operads, $N∞$ operads, which capture the data of homotopy commutativity in $G$- ring spectra, can be understood in terms of certain subgroup lattices of the group, namely, $G$-transfer systems.
We will first give the necessary background on equviariant operads and $G$- transfer systems via Tambara functors. We then present our joint work with KristenMazur, Angelica Osorno, Constanze Roitzheim, Danika Van Niel, and Valentina Za-pata Castro on compatible G transfer systems over the group $G = C_{p^r q^s}$ for primes $p$ and $q$ discuss its implications in equivariant stable homotopy theory.
