Infinitesimal objects in algebraic geometry have a rich structure, but they are difficult to study due to the failure of classical Lie theory in the algebraic context, especially in characteristic p. This failure can be attributed to the fact that Lie algebras only capture first-order infinitesimal behavior, a limitation which vanishes when we shift our focus to a new kind of infinitesimal object: formal groups. In this talk, I will describe the basic theory and examples of formal groups, as well as how they give rise to a surprising and deep connection between algebraic geometry and algebraic topology: chromatic homotopy theory.
