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–Schützenberger; –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.