Juan Maldacena reviews the Hartle-Hawking no boundary geometry in the context of slow roll inflation. He provides an analytic approximation to the geometry and explains the rationale behind the proposal. The no boundary geometry predicts a positive curvature for the universe, which disagrees with observations. Maldacena discusses various proposals to resolve this discrepancy, including slow roll eternal inflation, the ill-definition of the path integral due to the non-renormalizability of gravity or the unbounded below nature of the action, the possibility that the universe is not in the Hartle-Hawking state, the consideration of a tunneling wavefunction, the application of a selection principle, and the importance of quantum corrections. He emphasizes that the no boundary proposal is a natural extension of the successful inflationary prediction for primordial curvature perturbations but faces challenges when applied to the overall spatial curvature of the universe.Juan Maldacena reviews the Hartle-Hawking no boundary geometry in the context of slow roll inflation. He provides an analytic approximation to the geometry and explains the rationale behind the proposal. The no boundary geometry predicts a positive curvature for the universe, which disagrees with observations. Maldacena discusses various proposals to resolve this discrepancy, including slow roll eternal inflation, the ill-definition of the path integral due to the non-renormalizability of gravity or the unbounded below nature of the action, the possibility that the universe is not in the Hartle-Hawking state, the consideration of a tunneling wavefunction, the application of a selection principle, and the importance of quantum corrections. He emphasizes that the no boundary proposal is a natural extension of the successful inflationary prediction for primordial curvature perturbations but faces challenges when applied to the overall spatial curvature of the universe.