Even though the field with one element, , is a meaningless concept, shadows of it have been apparent in multiple categorical analogies. More immediately, one can generalize multiple constructions from algebraic geometry over  to general commutative monoids, which behave like rings over this elusive . In this talk we define, via this analogy, schemes over , and consider zeta functions over  varieties. These results are taken from arXiv:math/0404185.

Zoom Meeting link:  https://ncsu.zoom.us/j/95923219557?pwd=T3MzUEZRazRaY2tBdjl1dFVJOG50QT09