Back stable Schubert calculus

We study the back stable Schubert calculus of the infinite flag variety. Our main results are: - a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part; - a novel definition of double and triple Stanley symmetric functions; - a proof of the positivity of double Edelman-Greene coefficients generalizing the results of Edelman-Greene and Lascoux-Schutzenberger; - the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form of the Edelman-Greene insertion algorithm; - the construction of the Peterson subalgebra of the infinite nilHecke algebra, extending work of Peterson in the affine case; - equivariant Pieri rules for the homology of the infinite Grassmannian; - homology divided difference operators that create the equivariant homology Schubert classes of the infinite Grassmannian.