Math 547 (Yu) is an introduction to the primary concepts and techniques of Riemannian geometry: Riemannian metrics and connections, geodesics, curvature, and Jacobi fields. We will use these tools to prove the fundamental theorems relating curvature and topology: the theorems of Gauss-Bonnet, Cartan-Hadamard, Bonnet, and Cartan-Ambrose-Hicks. Prerequisite: MATH 546.

Math 548 (Pollack) will cover basic Lorentzian geometry (including causal structure and energy conditions for Lorentzian manifolds) and the foundations of General Relativity: the Einstein field equations and their variational derivation, exact solutions (e.g. Schwarzschild, Kerr, Reisner-Nordstrom, De Sitter/anti-De Sitter) and the initial value formulation. Prerequisite: MATH 547

Math 549 (Pollack) will focus on more advanced topics in Mathematical Relativity with an aim to get acquainted with active areas of current research. Possible topics will include: Gravitational collapse and black holes, the positive mass theorem and Penrose inequalities, structure of spacetime singularities, the (current) standard model for cosmology, gravitational waves. Prerequisite: MATH 548