Chromatic Counting

A story about chromatic homotopy can go as follows:

• counting discretely (i.e. via ${\pi }_0(FinSe{t}^{\simeq })$), is “easy”. This is arithmetic.
• counting homotopically (i.e. via $FinSe{t}^{\simeq }$) is “hard”. This is stable homotopy theory.
• To simplify this last one, we can count homotopically up to $p$-torsion. (then glue together with a rational part via fracture/recollement/“Hasse principle”)
• We notice that we can iterate this: once we arrive in the $p$-local world, we find there is more torsion to localize against. Furthermore, these $K(n)$ localizations are now quite “easy”.

In this vault we’ll record insights around how to count “things in real life” using chromatic homotopy theory. This means counting

• a bag of objects (chromatic entropy)
• a collection of identical particles in quantum mechanics ($\infty$-semi-additive Pauli principles)
• Cells in a $p$-space (Chromatic cardinality of Lurie-Yanovski)
• Cycles in quiver varieties (Higher Frobenii + E-theoretic quantum groups of Yang-Zhao)
• Lagrangian Manifolds (Floer homotopy of Abouzaid-Blumberg)