[RLM] Re-Learning Mathematics 1 - Summary

Here I’d like to give a little summary of what I want to do with this project. The goals are three-fold.

The first is simply to feel more confident of my general mathematical abilities/background. I know that there will no doubt be times when my work means I will have to learn things from other fields, and I believe that putting in the work now to have the background to read those later will be worth it. The second is to prepare for courses. As a PhD student I will need to take required classes which I wish to be prepared for. The third is to expand my research horizon’s somewhat. There is a wide mathematical world out there, and just entering grad school there is no reason for me to be married to my current research interests. However, I don’t think it is likely that if I pivot that I will pivot so far.

With these goals in mind, I want to present some topics I would be interested in. However, it would be inappropriate to provide these without understanding what I feel I have on solid ground. After four years of undergrad, I feel super confident on my finite-dimensional linear algebra, especially on the computational side. I also feel happy with my real function theory/metric spaces. That is to say, I am happy with actually applying those tools in other fields, even if I don’t necessarily remember precisely why all the assumptions for L’Hopital’s rule are strictly necessary.

That being said, some courses/topics for this project are the following:

- ODEs with a focus on application to PDEs. I have in mind the first chapter of Tao’s nonlinear dispersive equations and Arnold’s geometric ODEs book

- General Geometry/Differential Geometry. I took a course with baby do Carmo that after working through papa do Carmo has made me realize that I don’t know that material at all and it might be nice to re-read/learn. I’m also interested in progressing with the Differential geometry perhaps in the direction of Petersen and onto Peter Li’s book of Geometric Analysis.

- Complex Analysis towards Riemann Surfaces. I took a nice course on Complex Analysis, and I am interested in working towards Riemann surfaces. Therefore I will be reading Stein & Shakarchi’s Complex Analysis and these Notes on Complex Analysis and Riemann Surfaces

- Basic Algebra. I feel as though I would not feel happy as a mathematician if I could not answer the basic questions from a grad-student’s general exams in algebra at least somewhat. I think that Artin’s Algebra will continue to be good for this. I also feel that I would like to start moving forward in my algebra background as I know that algebraic geometry is beautiful and exciting. Therefore I would love the chance to understand all of that, if only as an outsider.

- Differential Topology. This is a subject I never felt I took a class in, but underlies a lot of geometry and is just plain exciting to me. I have started skimming Guilleman and Pollack and it looks perfect in tone and exercise difficulty.

- Real Analysis. A subject which I never took in undergrad, but use all the time. Gonna be honest this feels a little bit like eating my veggies, but I’m happy to do it. References include but are not limited to Folland and Stein and Shakarchi 3/4.

- PDEs. I feel I have encountered/seen PDEs several times, but I am yet to put all of it together into a cohesive narrative. I can’t help but feel that Evans’ is my only option for this. Although, I found a cool alternative book for this that I may consider.

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