Elena Giorgi

Jacobi fields, second fundamental form, embeddings of manifolds, spaces of constant sectional curvature, variations of energy, mathematical GR -- Schwarzschild, Kerr black holes, trapped surfaces.

20% Weekly Psets (~3 hours), 30% Midterm, 50% Final

Jacobi Fields, Second fundamental forms, Hadamard, Variations, and most fun, General Relativity.

This class is very structured and follows the textbook closely. Lectures were clear and never confusing but I never felt engaged or excited about the material.

In this class we studied Jacobi fields, isometric immersions, completeness of Riemannian manifolds, spaces of constant curvature, and variations of energy of curves. We also got a crash course introduction to mathematical General Relativity, regarding the initial value formulation of the Einstein field equations, the causal structure of spacetime, null-structure equations, black holes, and wave-equations/energy estimates on black hole spacetimes.

Applying the concepts of connections and curvature of Riemannian manifolds studied in the first semester of this course. Jacobi Fields, Isometric Immersions, Geodesically Complete Manifolds, Spaces of Constant Curvature, Variations of Energy. The class also ended with an overview of Mathematical GR.

I continued to learn about the modern theory of geometry, particularly covering chapters 6-9 in do Carmo's book on Riemannian geometry and some additional content on mathematical GR.

Various techniques for solving many different types and cases of ODEs.

Jacobi fields, isometric immersions (second fundamental form and fundamental equations), Hopf-Rinow and Hadamard Theorem, spaces of constant curvature, variations of energy, Bonnet-Meyers and Synge-Weinstein, and general relativity (existence of black holes, null-structure equations, Kerr black holes)

I learnt how to solve first and higher order differential equations, how to apply Laplace transforms, and solving systems of first order linear equations