There are several great text books in geometric analysis and PDE, as well as some great online lecture notes and survey articles (which are freely available). The list below is by no means complete.

• Books:
  • "The Cauchy Problem in General Relativity" by H. Ringstrom is the book Prof. Pollack recommends for material related to his mini-course.
  • "A Course in Minimal Surfaces" by T. Colding and W. Minicozzi is the book Prof. Schoen recommends for material related to his mini-course
  • "Riemannian Manifolds: An Introduction to Curvature" by J. M. Lee is a fantastic introduction to Riemannian manifolds and the curvature tensor.
  • "Riemannian Geometry: A Modern Introduction" by I. Chavel is another great introduction.
  • "Lectures in Differential Geometry" by R. Schoen and S. T. Yau covers many topics that are very difficult to find elsewhere (e.g. eigenvalues of the Laplacian on surfaces, conformal invariants of locally conformally flat manifolds).
  • "Elliptic Partial Differential Equations of Second Order" by D. Gilbarg and N. Trudinger is the standard reference for second order, elliptic PDE.
  • "Maximum Principles in Differential Equations" by M. Protter and H. Weinberger is a fantastic introduction to maximum principles, and only requires that the reader know calculus.
• Online lecture notes and survey articles:
  • Richard Schoen wrote this excellent survey article about mean curvature in general relativity (published in the Proceedings of the Clay Summer School, ed. by D. Hoffman, AMS, 2005, pp. 113--136).
  • Richard Schoen also recommend "Survey lectures on minimal submanifolds" by Leon Simon, in Study 103, Annals of Mathematical Studies, ed. by E. Bombieri, 1983, pp. 3--52.
  • Daniel Pollack has written (with collaborators) two excellent survey articles, which you can find here and here.
  • These lecture notes by Peter Li are excellent.
  • Jeff Viaclovsky has written some nice lecture notes on Riemannian geometry and on PDE
  • Toby Colding has also written some nice lecture notes on PDE

I will post slides of the lectures as they become available.

• introductory lectures: A crash course in Riemannian geometry

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