Goals: This course covers both harmonic and $J$-holomorphic maps. One goal is to include new approaches to the issue of bubbling and conformal invariance.    Course Syllabus

Prerequisites: A working knowledge of (i) Riemannian geometry and (ii) elliptic PDE at the level of Evans' Chapters 5 and 6.

Textbook: Riemannian Geometry and Geometric Analysis,, by Jurgen Jost.

Preliminary list of topics:

1. Harmonic maps.
2. Symplectic geometry and topology.
3. J-holomorphic maps.
4. Riemann surfaces and complex curves.
5. Gromov-Witten moduli spaces.
6. Analysis results.
7. Bubbling and Gromov compactness.
8. GW invariants.
9. Harmonic map heat flow.

Books on the same material:

• Riemannian Geometry and Geometric Analysis, by Jurgen Jost.
• The analysis of harmonic maps and their heat flows, by Fanghua Lin and Changyou Wang.
• J-holomorphic curves and quantum cohomology, D. McDuff and D. Salamon.
• J-holomorphic curves and symplectic topology, by D. McDuff and D. Salamon.