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We study the back stable Schubert calculus of the infinite flag variety. Ourmain results are: 1) a formula for back stable (double) Schubert classesexpressing them in terms of a symmetric function part and a finite part; 2) anovel definition of double and triple Stanley symmetric functions; 3) a proofof the positivity of double Edelman-Greene coefficients generalizing theresults of Edelman-Greene and Lascoux-Schutzenberger; 4) the definition of anew class of bumpless pipedreams, giving new formulae for double Schubertpolynomials, back stable double Schubert polynomials, and a new form of theEdelman-Greene insertion algorithm; 5) the construction of the Petersonsubalgebra of the infinite nilHecke algebra, extending work of Peterson in theaffine case; 6) equivariant Pieri rules for the homology of the infiniteGrassmannian; 7) homology divided difference operators that create theequivariant homology Schubert classes of the infinite Grassmannian.

Back stable Schubert calculus