This Week's Find
11/04/2024-11/10/2024, Fall 2024 Week 6
- Universal Enveloping Algebras, Nilpotent Lie Algebras
I fell in love with the expositions of the mastero Jean-Pierre Serre. His level of concision is unmatched, and reading it feels like him standing in front of me giving me a live lecture. And also he has so many relevant books that lies right on my interest: Lie Groups and Lie Algebras, A Course in Arithmetic, Local Fields, Galois Cohomology, and much more…
- Whitney’s Embedding Theorem, Smooth Approximation Theorem
- Hilbert’s Theorem 90, Ring of Integers of Cyclotomic Fields
Hilbert’s Theorem 90 gives an explicit description of the kernel of the norm map of a finite cyclic extensions of fields. Namely,
One of the graduate students here Zach Baguher gave a Graduate students seminar talk on Proving Theorems by Overkill. One of his examples was to show that there exists a nontrivial rational point on the unit circle, and the way he proved it was to show that there exists a point for every p-adics by Hensel’s Lemma and for the real field. I think we can similarly overkill this problem by Hilbert 90, which gives even more: an explicit description of the pythagorean triples.
- Solving group theory problems on the Algebra Qual
I learned a new trick in checking that a group is not simple. When we exhaust all information from Sylow Theorems, we can instead look at the normalizer
</math> here, which has order
and then we can apply the second isomorphism theorem and Frattini’s Argument to count elements and get a contradiction. The exercise that taught me this trick is about showing that every group of order 315 is the direct product of a group of order 5 and a semidirect product of a group of order 7 and a subgroup of order 9.
- Filling in gaps in category theory
My understanding in category theory had always been that of any pure mathematician outside of algebra-demanding fields like homotopy theory. I am familiar with common categorical constructs and definitions, and I speak the categorical language frequently, but I would never say that I am good at it. Reading the algebra class notes made me realize there were many constructs I didn’t know. I learned about comma categories, universal objects, universal objects, and cleared up my confusion on subtle things as small as “should I call this right or left, over or under?”. In future weeks, I hope to go over the last chapters of Riehl’s Category Theory in Context and do tons of exercise, so I can become better in proving things in abstract nonsense style.
p.s. I will add references to books and papers I read later. p.s^2. Thanks to John Baez for inspiring me to start blogging.