MATHG R6403

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

Riemannian geometry, principal fiber bundles, positive energy theorem

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

The first semester was a review of some basic differential topology and an introduction to Riemannian geometry (connections, geodesics, curvature); it also included some discussion of group actions on manifolds and Lie groups. Half of the second semester focused on further discussion of Riemannian geometry (Jacobi fields, second fundamental form, Hopf-Rinow, Hadamard, hyperbolic space), and the other half focused on principal bundles (including connections and curvature), some complex geometry (which featured in the discussion of unitary bundles), and general relativity--in particular, Witten's proof of the Positive Energy Theorem in the late 20th century.

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.

I would recommend this class to anyone who is interested in differential geometry for any reason; this material offers great insight into the world of geometry in general, so I might extend that to say that I recommend this course to anyone interested in any geometry-related subject. 1 Columbia University: Arts & Sciences Spring 2024 Course: MATHGR6403_001_2024_1-MODERNGEOMETRY : MATHGR6403_001_2024_1 - MODERN GEOMETRY Instructor: Chiu-Chu Liu

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.

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)

Great course, engaging lectures, and covered many topics that extended the results and foundations covered in semester one.