Algebraic Geometry Seminar

There are many attempts to find a "deeper base" for arithmetic than the natural numbers, incorporating more combinatorics. One approach is to replace the natural number $n$ with the $q$-analogue $(n{)}_q:=1+q+\cdots +q^{n-1}$; such $q$-analogues have recently been linked to prismatic cohomology and formal group laws. Another is to replace the category of finite sets by the category of finite $G$-sets for a group $G$, then decategorify. I will discuss recent work linking these two approaches, which in particular clarifies some long-standing issues with the symbol ``$f^{(n{)}_q}$''. In technical language, our construction takes a $\lambda$-ring equipped with a compatible formal group law and produces an $S^1$-Tambara functor.