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.