Below is a frequently updated list of textbooks, papers, surveys, and notes I am currently reading and an archived list of finished materials.
What I’m Currently Reading
- Book: Riemann’s Zeta Function (H. M. Edwards). A 300 page analysis on Riemann’s pivotal 1895 manuscript Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse and some devlopements after its publication. Definitely worth a read alongside the original paper (which an English translation is provided in the appendix).
Past Readings (Books)
- Book: Linear Algebra, Done Right (Axler). The best abstract linear algebra text I have come across. It is worthwhile, but not necessary, to have read a computation-based linear algebra text first to familiarize one’s self with the definitions and ideas. Most of the exercises are quite simple, and are worthwhile doing for someone with less mathematical maturity. It does not treat linear algebra in the most generality as Axler works over and only (but this will suffice for most things).
- Book: Introduction to Lie Algebras and Representation Theory (Humphreys). A standard reference for Lie algebras and representation theory. The first three chapters are quite easy to follow, but the fourth and fifth are more terse. The text can serve as a stand alone introduction to root systems if that is what the reader is looking for. A strong background in linear algebra is necessary, and it will help if the reader can translate between the language of linear algebra and abstract algebra quickly.
- Book: An Introduction to Manifolds (Tu). A phenomenal textbook for introducing smooth manifolds. Chapter serves as an overview for the remainder of the text and introduces the theory in by connecting it to vector calculus. Chapters 2, 3, 4, and 6 on manifolds, the tangent space, differential forms, and integration respectively are incredibly well-written and focus on introducing the general theory as much as how to perform associated computations. Chapter 5 on Lie groups and Lie algebras is very light and a deeper text should be consulted for further interest. It is best, but not necessary, to read Chapter 7 on de Rham Theory with some familiarity in homological algebra (see Dummit & Foote Chapter 17).
- Book – Understanding Analysis (Abbott). The only downside of this textbook is that you might fall prey to the idea that every mathematics textbook is written this well. The text coves the fundamentals of a first year undergraduate corse in analysis, but there is so much to be gained beyond the standard definitions and theorems. A bountiful amount of intuition is given in every chapter (especially about uniform continuity, Taylor series, and the Cantor set), and there are many historical remarks for the reader who likes the backstory of mathematical analysis. Many of the exercises are worthwhile doing albeit not exceptionally difficult, and some sections of the text are more DIY type where the reader is encouraged to do most of the work. The last chapter is dedicated to additional topics and provides good motivation for spaces.
Past Readings (Papers & Surveys)
- Survey: On Some Applications of Automorphic Forms to Number Theory (Bump-Friedberg-Hoffstein). This paper was suggested to me as a good survey of how analytic properties of Dirichlet (or multiple Dirichlet) series gives information about interesting number theoretic quantities . The paper is fairly light from a technical viewpoint, but mentions many modern techniques and heuristics to obtain analytic continuation of these series to some region of a complex domain.