Lecture 1. l^2 and the shift in l^2. The Hardy space H^2. The shift in H^2. Inner and outer functions; Beurling’s decomposition. Beurling’s theorem. Abstract interpretation of Beurling’s theorem. Shifts in Hilbert spaces. Index of a shift. Equivalence of shifts. Lecture 2. Rota’s universality theorem. Relation with the invariant subspace problem: maximal shift-invariant subspaces. Shift in l^2(H^ 2). The Beurling-Lax theorem. The Beurling-Lax matrix of a shift-invariant subspace of l^2(H^2). Determinantal operators and determinantal subspaces. Lecture 3. Shift-invariant subspaces in l^2(H^2) of finite rank are infinite intersection of determinantal subspaces. Limit of shift-invariant subspaces. Shift-invariant subspaces in l^2(H^2) are limit of infinite intersection of determinantal subspaces. Maximal shift-invariant subspaces in finite direct sum of H^2 and in l^2(H^2).