Chiu-Chu Liu

Riemannian geometry, principal fiber bundles, positive energy theorem

3 EASY midterms and a medium final, weekly problem sets and she drops the lowest 2.

Differential Topology(manifolds, tangents, derivations, vector and tangent bundles, tensors, vector fields), Riemannian Geometry(metrics, curvature, lie groups, geodesics)

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.

I had her for calculus III, and she was way better than I could have expected. Advantages: Very easy midterms (all 3 of them). Every midterm was identical to the practice midterm, with some numbers changed here and there. As long as you understood the questions on those practice midterms, you were guaranteed to do well (read: essentially guaranteed A/A+). Straightforward problem sets, all problem sets have answers available online. The level of the problem sets was always the same level of difficulty as the textbook taught, so there was always an example problem in the textbook very similar to the ones on the problem sets. Final was also not that hard. Final was especially short, so really no trouble to finish, but it did have a couple questions that required a level of understanding further than just having the right equations memorized. Disadvantages: Quiet lecturer, not engaging, and made mistakes on the board frequently. But calc III, especially the level required from this class, is something you can teach yourself. Going to lecture did not give me any extra understanding that the book did not lay out clearly. Conclusion: perfect if you want to a gpa boost and you're comfortable skipping lecture and just learning from the book. I got an A but barely studied for any of the exams. Also nothing is curved in this class so prepare to push yourself to get every part of every pr…

Need to have proper background to take the class

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

Smooth manifolds, fiber bundles (chiefly vector bundles like the tangent bundle and the tensor bundles), local operations like Lie derivative and exterior derivative, connections on vector bundles, Riemannian geometry like the Levi-Civita connection and the different curvature tensors, Riemannian geometry and differential geometry of Lie groups, some focus on quotient manifolds and their geometry.

Manifolds, vector bundles, connections, geodesics, Lie groups

Basic definitions in Riemannian geometry and differential geometry along with some basic theorems and their proofs.