Books

There are some books which I consider fundamental in understanding certain undergraduate branches of mathematics. These books helped me a lot during my college years, and I wish I could have read these books earlier than I did (I didn’t read them completely, but I hope I will…).

The following three books are very good for analysis. They are very well explained and structured. At the end of each chapter there are many exercises and challenging problems.
Fourier Analysis: An Introduction – Elias M. Stein & Rami Shakarchi
Complex Analysis – Elias M. Stein & Rami Shakarchi
Real Analysis: Measure Theory, Integration, and Hilbert Spaces – Elias M. Stein & Rami Shakarchi

A great book for understanding the beginnings of linear algebra.
Linear Algebra done Right – Sheldon Axler

This book contains more than you need for a beginner in group, ring, field theory. It also contains a Galois Theory chapter. Has lots of problems at the end of every section.
Abstract Algebra – David S. Dummit (Author), Richard M. Foote

A book which treats geometry from an algebraic point of view. It’s very interesting and has challenging exercises. It contains facts about geometric transformations, affine and projective mappings, conics, cuadrics, etc.
Geometry – Michele Audin

Two famous problem books by V. Prasolov, one about plane geometry and another one for space geometry, are one of the best collections of problems structured on well defined sections, with very concise, but complete proofs. Don’t know the actual name of the book, but search Google for “Prasolov Geometry”.

Rudin’s books are very well written. They offer a great deal of information about analysis. Three volumes, Principles of Mathematical Analysis, Real and Complex Analysis and Functional Analysis. There are interesting problems after each chapter of the books. In my opinion, these are not to be read by beginners in the field. They are better to be read after knowing the basics in every aspect of the three parts of the analysis the books treat. For someone who is new in the field everything will be happening very fast. It would be better to read the first three books presented in the beginning of the page before trying the Rudins.

I was fascinated about one domain in mathematics with surprising applications in combinatorics and physics, namely Ergodic Theory. As an introduction in this domain, you can try Paul Halmos – Introduction to ergodic theory, Recurrence in ergodic theory and combinatorial number theory – Harry Furstenberg, Dynamical systems and ergodic theory – Mark Pollicott, Michiko Yuri. Some of the surprising applications of ergodic theory are Van der Waerden’s Theorem and Szemeredi’s Theorem.

Some other interesting objects of mathematics are -adic numbers, which were introduced as a completion of , just like the reals, but with a different norm. They have applications in number theory and physics. One unique application of the theory of -adic numbers is constructing a coloring of with three colors such that every line contains exactly two colors. Some sources say there is no other known method to do this without using -adic number theory. As a text for an introduction in the field you can read -adic Numbers: An Introduction – Fernando Quadros Gouvea.

For those who like problem solving, like me, you can check out Paul Erdos’ book, Proofs from the book. It contains famous problems with amazing proofs. Also, see the book containing the problems from Miklos Schweitzer contest, Contests in Higher Mathematics: Miklos Schweitzer Competitions, 1962-1991 (Problem Books in Mathematics). Another great text with solved problems given in phd admission exams is Problems and solutions in Mathematics, edited by Li Ta-Tsien. Also, take a look at A Hilbert Space Problem Book, by Paul Halmos, and Problems and Theorems in Linear Algebra by Prasolov.

For differential geometry, there are a variety of options. Here are my first books about this: A Comprehensive Introduction to Differential Geometry, vol 1-5, by Michael Spivak, Differential Geometry: a first course in curves and surfaces by Theodore Shifrin, An Introduction to Differentiable Manifolds and Riemannian Geometry, William M. Boothby, Manifolds, Tensor Analysis, and Applications, Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu.

For the study of functional analysis, beside of Rudin’s book, another classic in the field is Haim Brezis’ Functional Analysis, which has a brand new edition in 2011, with lots of exercises and problems at the end of every chapter. Another book which I found useful was Methods of Modern Mathematical Physics, vol 1. Functional Analysis, Michael Reed, Barry Simon.