Resources

Below is a list of resources of textbooks, notes, and papers for certian subfields of interest. Comments are attached to each resource. If you have any trouble getting ahold of any of these sources feel free to reach me via my contact page here.

Algebra

  • Textbook – Dummit & Foote’s Abstract Algebra. This is one of the standard textbooks for abstract algebra. It has a bountiful ammount of exercises, but can be slightly verbose at times. It has a nice introduction to representation theory near the end.
  • Notes – P. Garrett’s Abstact Algebra. A collection of freely accessible notes by professor P. Garrett at the University of Minnesota. The sections on modules, eigenvalues, and tensor products are particularly well-written. There is also a dense set (pun-intended) of worked examples throughout the notes. If you are not doing the exercises I would suggest attempting the worked examples by yourself first before reading the solution.
  • Textbook – Axler’s Linear Algebra, Done Right. The best abstract linear algebra text I have come across. It is worthwile to have read a computation-based linear algebra text first to familarize one’s self with the defintions and ideas. This book can be read in about two weeks with enough mathematical maturity and was instructive for me to really learn linear algebra. It does not treat linear algebra in the most generality as Axler works over \mathbb{R} and \mathbb{C} only.

Topology

  • Textbook – Munkres’s Topology. A standard textbook for a point-set topology course. It serves as decent textbook for self study as well, and part II of the book is dedicated to algebraic topology. However, it uses non-standard topologies quite often in many examples and exercises which are usually not delt with beyond this text.
  • Textbook – Hatcher’s Algebraic Topology. A staple for studying algebraic topology. Many of the proofs are quite geometric in nature which aids intuition, but lacks formality in places. The appendix section is worth reading for a full in-depth investigation into the text. The text also contains many subsections of additional topics one might come across in the field.
  • Textbook – Tu’s An Introduction to Manifolds. A phenominal textbook for learning manifold theory. Prerequisits consist of point-set topology, basic analysis, and abstract algrebra (in particular linear algebra). Tu first covers the theory in \mathbb{R}^{n} (connecting many topics back to multivariable calculus), and then moves more generally to smooth manifolds. I split the text into chapters 1-3, 4-6, and 7.

Ananlysis

  • Textbook(s) – Rudin’s Principles of Mathematical Analysis, Real and Complex Analysis, Functional Analysis (Baby, Papa, Grandpa). I am grouping Rudin’s three texts on analysis together here. They are all quite light on the intuition behind the arguments presented, but most proofs are clean and quick. It is worthwile to mull over the proofs and see why they work. Many of the exercises are quite long and tricky, but Rudin does have solution manules which is a big pull to study from these texts.
  • Textbook – Ahlfor’s Complex Analysis. A staple for compelx analysis. It is much more verbose than Papa Rudin and includes many detailed pictures. The sections on comformaily are not great, and I would suggest one looks elsewhere. Most of the exercises, throught the book, are very instructive and require utilizing the theory introduced rather than tricks.

Number Theory

  • Textbook – Ireland & Rosen’s A Classicial Introduction to Modern Number Theory. The first few chapters are a great introduction to modular arithmetic. Contains introductory chapters to algebraic number theory, elltipic curves, Diophantine equations, and Dirichlet L-functions (I would look elsewhere for a more in-depth reading). Also has several refences to both useful and historic notes and texts at the end of of each chapter. If you are paitent, this is a great text. Chapter 10 contains one of the best intorductions to projective space that I have seen.
  • Textbook – Silverman & Tate’s Rational Points on Elliptic Curves. This is an introductory textbook to elliptic curves. Not all statements contain proofs and many proofs are not as formal. However, this book should be read to get the general setting of the theory, and very little backgound is needed
  • Textbook – Silverman’s The Arithmetic of Elliptic Curves. This is the big brother of Silverman & Tate’s Rational Points on Elliptic Curves. In other words, it is a formal introduction into elliptic curves. Chapter 1 does cover algebraic varities, but it is useful to be familar with them, and in particualr the Zariski topology beforehand, since Silverman introduces varities without the Zariski topology. A lot of the intution behind the results in this text comes from Silverman & Tate so keep that text on hand.
  • Textbook – Neukirch’s Algebraic Number Theory. While a very terse text, it is essentially a self contained volume of algebraic number theory and much can be gained from Chapters 1-3 alone (it is bacially an intorductory). Make sure your abstract algebra and linear algebra are well-refined before reading this text.