(Week 2) tangent bundle and the other tensors, tangent bundle of mapping space, Riemannian/Finsler metric on Hilbert/Banach manifold. Reference: [M] §1.7, §1.8, §1.9.
(Week 3) vector field and the flow, Palais-Smale condition, critical point and ambient isotopy. Reference: [M] §1.10, §1.11, §1.12.
(Week 4) minimax principle, properties of the energy functional on L12(S1,M): property C and smoothness of critical state. Reference: [M] §1.12, §2.1, §2.2.
(Week 5) some homotopy theory, topology of the space of maps from circles, Fet-Lusternik theorem, second variational formula. Reference: [M] §1.5, §2.3.
(Week 7) bumpy metric for geodesic, energy and harmonic map. Reference: [M] §2.7, §4.1.
(Week 8) conformal invariance in 2D, energy and area functionals, Hopf differential, S2 case. Reference: [M] §4.1, §4.2.
(Week 9) the variation of the energy function in conformal structures: T2 and higher genus case. introduction to the α-energy of Sacks&Uhlenbeck. Reference: [M] §4.3, §4.4.
(Week 10) α-energy and Condition C, regularity of critical point, Bochner formula for harmonic maps. Reference: [M] §4.4, §4.6.
(Week 11) ε-regularity theorem, limit as α→1. Reference: [M] §4.6.
(Week 12) removable of singularities, bubbling and harmonic S2. Simons identity. Reference: [M] §4.6, §4.7. Simons 1968 paper.
(Week 13) index and nullity of minimal submanifolds in the spheres, the gap theorem of Simons. Reference: Simons 1968 paper.
(Week 14) minimal cones and their stability. Reference: Simons 1968 paper.
analysis of harmonic maps in higher dimensions: monotonicity, and sup-norm estimate. Reference: Schoen 1984 paper.
(Week 15) singular set of harmonic map: closedness and Hausdorff measure; second variational formula of energy: global property into negatively curved space; some local regularity estimates. Reference: Schoen 1984 paper.