John Morgan

deriving so much takes little integration some such clear lectures I took a horrendous AP Calculus BC class when I was still in high school, and I was terrified that my professor at Columbia would be equally incompetent, going off only examples from the book and leaving little concepts for clarification. However, my experience with this class was very much the opposite. Professor Cautis is indeed soft-spoken, but I sat in the back of the classroom for most of the semester and was able to hear him clearly. He does only go off examples, but I found his explanations of the thought processes behind calculus before delving into those lent me a much stronger understanding of the material than I had previously been taught. Some of the examples are obviously developed to lengthen the class time, as we often would get out ten or even twenty minutes late, but this is more of a testament of Cautis's ability to get to the point of each lecture rather than ramble on. As a new professor, he obviously has flaws in being stuck in that distant didactic lecture style, but he showed an obvious passion for mathematics and often showed it through the way he talked about the concepts. Also, his style was very appropriate for calculus, though still in the rough, especially at the beginning of the semester, when his lectures seemed more muddled, but he improved over the semester. The two midterms wer…

You should really already be familiar with some decent amount of algebraic topology and category theory before you take this course. Know what homotopy is, know what homology is, and maybe even try to know what cohomology is. This will give you the time to understand properly the long list of more advanced topics he also teaches. 1 Columbia University: Arts & Sciences Fall 2021 Course: MATHGR6307_001_2021_3-ALGEBRAICTOPOLOGYI : MATHGR6307_001_2021_3 - ALGEBRAIC TOPOLOGY I

Knowledge: A survey of basic differential geometry (smooth manifolds, differential forms, Riemannian manifolds)

While at first it may seem as though Morgan is a bit boring, as the semester goes on I came to appreciate his teaching. He goes over many examples with the class and is always willing for in-class discussion and explains well questions that people have. He's approachable and friendly outside of class. The material gets progressively harder throughout the semester (on a grander scale than most classes), such that the first third may seem easy, the second moderate, and the last, pretty impossible. But the whole class works that way, so the curve is good...although the class is curved around a C

I learned about manifolds, bundles, complex curves, de Rahm cohomology, and Riemannian geometry among other special topics in modern geometry.

1 problem set of ~10 problems/week that took at most an hour or two.

weekly homework assignments, mix of even and odd problems; 2 midterms; final worth 50%

Survey course in Modern Geometry - as advertised. Covered manifolds, differential geometry, complex geometry, with memorable forays into algebraic topology, category theory, and general relativity.

The problem sets are pretty long compared to those of the other first year graduate courses.

AVOID THIS CLASS. Be warned: this class is not for those who earnestly wish to learn geometry. The professor is not good at teaching and the course lacks any organizational structure. The only people who could understand the course were those who already took graduate level geometry (for whom this class would be unnecessary). On top of that, the most baffling thing was the interaction between the professor and an undergrad who would be on the computer for first half of the class and wrote only two lines of pure words for each problem set. The course almost had a take-home exam but the professor chose to listen to the babbling of an undergrad student who suggested oral exam (so that he can avoid the situation of not being able to write any math), which created additional burden for everyone else. 1 Columbia University: Arts & Sciences Fall 2021 Course: MATHGR6402_001_2021_3-MODERNGEOMETRY : MATHGR6402_001_2021_3 - MODERN GEOMETRY Instructor: John Morgan