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This page lists books that people from Libera ##math found useful. Note that taste in books varies wildly, so we don’t officially endorse any of the books.

Additions are welcome! See the community page.

Other book lists: Chicago undergraduate mathematics bibliography, freenode ##math, EFnet ##math.

Contributors: greenbagels, int-e, Khwarizmi, rain1, savask

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  • Martin Aigner and Günter M. Ziegler, Proofs from THE BOOK. (Wikipedia)

    A book containing mostly elementary proofs from various areas that are just beautiful.

  • Evan Chen, An Infinitely Large Napkin.

    An accessable overview of mathematics as a whole. Covers a very wide range of topics for those wanting to get a big picture overview.

History

  • John Stillwell, Mathematics and Its History.

    This book broadly describes the development of different branches of math, with short profiles of mathematicians and optional exercises.

Foundations

  • Landau, Foundations of Analysis.

    The word “foundations” in the title should really be stressed, as the sole goal of this small book is to construct natural, rational, real and complex numbers from Peano axioms. This is perhaps a mathematical text requiring a bare minimum from the reader, in author’s words: “I will ask of you only the ability to read English and to think logically - no high school mathematics, and certainly no higher mathematics.”

    The main bulk of the text is mostly a list of definitions and theorems without additional commentary, which makes it closer to the script for a modern proof assistant rather than a normal textbook. Read it if you are a formalization enthusiast.

Analysis

Real Analysis

  • Charles Chapman Pugh, Real Mathematical Analysis.

    A comprehensive text with a large variety of problems; has many pictures, multiple constructions of R, metric topology, and extra sections on linear algebra, differential forms, and Lebesgue measure.

  • Gelbaum, Olmsted, Counterexamples in Analysis.

    In this book the authors collected a wealth of peculiar examples from real analysis, which often defy intuition. Nowhere differentiable continuous functions, space-filling curves, non-measurable sets - these and other examples are classified by the respective topics in the classical real analysis curriculum. An indispensable supplement for the real analysis student and an entertaining read for a mature mathematician.

Complex Analysis

  • Eberhard Freitag and Rolf Busam, Complex Analysis.

    Develops the theory of complex functions. The early chapters provide the foundational theorems of the subject and show how to use this toolbox to solve integrals. Later chapters develop the theory of elliptic functions and some special functions like Euler’s Gamma function.

Algebra

General Algebra

  • David S. Dummit, Richard M. Foote, Abstract Algebra, 3rd Edition.

    A very comprehensive textbook on algebra, including groups, and Galois theory. Very complete with lots of interesting exercises to consolidate the material.

  • Lang, Algebra.

    This is a thorough graduate course in algebra following a classical array of topics: groups, rings and modules, Galois theory, polylinear algebra, structure of bilinear forms and representation theory. Despite the first edition of this book being published in 1965, it reads as a modern text. For instance, Lang introduces basic category theory notions alongside groups in the first chapter, and tackles homology in the third.

    The book assumes almost no prior background besides basic set theory. Proofs are mostly rather detailed while straightforward generalizations are left for the reader. There are plenty of exercises at the end of each chapter, and some are quite challenging: first two editions have the famous “prove all theorems in a homological textbook without looking into proofs” exercise.

  • Paolo Aluffi, Algebra Chapter 0.

    An excellent comprehensive algebra textbook that uses category theory language throughout.

Group Theory

  • Gorenstein, Finite Groups.

    A comprehensive account of finite group theory with an eye towards the classification of finite simple groups. The variety of topics covered is extremely wide: structure of p-groups, fusion and transfer, character theory, normal p-complement theorems, etc. Contains expositions to landmark results such as the Hall-Higman theorem and the classification of CN-groups, among others.

    The book is essentially self-contained, but it presents a hard read for a novice. For example, classical theorems of Sylow and Jordan-Hölder are just stated in the introduction without proofs. An indispensable reference for a finite group theorist, not so much for anyone else.

Category Theory

Number Theory

  • Serre, A Course in Arithmetic.

    Despite its name this slim book presents a course in modern number theory. The first half is algebraic and covers results on finite and p-adic fields, quadratic reciprocity, as well as the classical material on quadratic forms over integers, rationals, reals and p-adics. The second half is analytic and explores Dirichlet’s theorem on arithmetic progressions and the theory of modular forms.

    The text contains no exercises, but it is almost balanced by the terseness of the exposition: proofs are short and often require a solid algebraic background from the reader. Do not read this as a first number theory book, but do return after you gain some knowledge of the subject.

Computer Science

Information Theory

Machine Learning

  • Emil Hvitfeldt, Julia Silge, Supervised Machine Learning for Text Analysis in R.

    Supervised Machine Learning for Text Analysis in R provides comprehensive instruction in statistical regression and classification using the modern Tidyverse ecosystem for R, including random forests, SVMs and deep learning models.