From one perspective, Schubert calculus counts the intersections of a set of $T$-invariant subvarieties called Schubert varieties within the flag manifold $G/B$. These intersections also compute quantities important in representation theory, combinatorics and geometry, and the term `Schubert calculus’ is now used to describe an interrelated set of questions in a more general geometric, cohomological, or combinatorial context. In this talk, we answer a question in Peterson Schubert calculus. The Peterson variety is a subvariety of $G/B$ consisting of flags satisfying specified conditions related to a regular nilpotent linear operator, and is a special case of regular nilpotent Hessenberg varieties. Petersons are not $T$-invariant but are invariant under a one-dimensional torus $S \subset T$. In joint work with Brent Gorbutt, we proved the positivity of $S$-equivariant Schubert calculus for the Peterson variety in type $A_{n-1}$, resolving a conjecture of Tymoczko and Harada. To do this, we found an explicit, combinatorially-positive formula for the coefficients of the products. Along the way, we also discovered an unexpected combinatorial identity of binomial and multinomial coefficients. In addition to these results, I will discuss some geometric ideas behind the a current conjecture (joint with Mihalcea and Singh) about Peterson positivity in all Lie types.