Recommendations: Graduate Level Texts and Notes

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Inspired by the following Twitter thread:

Multiple Areas


Analysis

Real Analysis

Gerald B. Folland, Real Analysis: Modern Techniques and Applications

Walter Rudin, Real and Complex Analysis

  • Useful as a general reference, but there are more useful techniques in other books.

Stein and Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces

  • Has more useful techniques than Rudin.
  • Doesn’t include $L^p$ spaces or convexity.

Lieb-Loss, Real Analysis

  • Good source for $L^p$ spaces, convexity, and Fourier analysis.

Stein and Shakarchi, Fourier Analysis

  • Very elementary.

Schilling, Measures, Integrals, and Martingales

Royden, Real Analysis

Complex Analysis

Taylor, Complex Analysis

Simon, Complex Analysis

Stein and Shakarchi, Complex Analysis

Lars Ahlfors, Complex Analysis

Conway, Functions of one complex variable I

Functional Analysis

Conway, A Course in Functional Analysis

Differential Equations

Evans, Partial Differential Equations

  • Good source for Sobolev spaces.

V. Arnold, Ordinary Differential Equations

Probability

S. Ross, A First Course in Probability (Prentice-Hall)

Shiryayev, Probability.

Feller, An Introduction To Probability Theory And Its Applications

Durrett, Probability: Theory And Examples


Algebra

General/Introductory

Dummit and Foote, Abstract Algebra

  • Standard reference, encyclopaedic!

Hungerford, Algebra

Isaacs, Algebra

M. Artin, Algebra

Commutative Algebra

Altman-Kleiman, A Term of Commutative Algebra

https://www.mi.fu-berlin.de/en/math/groups/arithmetic_geometry/teaching/exercises/Altman_-Kleiman---A-term-of-commutative-algebra-_2017_.pdf

https://www.mi.fu-berlin.de/en/math/groups/arithmetic_geometry/teaching/exercises/Altman_-Kleiman---A-term-of-commutative-algebra-_2017_.pdf

Atiyah and MacDonald, Introduction to Commutative Algebra

Representation Theory

J-P. Serre, Linear Representations of Finite Groups

Humphreys, Introduction to Lie algebras and Representation Theory

Pramod Achar, Unreleased Geometric Representation Theory Text

Nicolas Libedinsky, Gentle Introduction to Soergel Bimodules

Hall, Lie Groups, Lie Algebras, and Representations

Kirillov, Introduction to Lie Groups and Lie Algebras

Homological

Peter J. Hilton and Urs Stammbach, A Course in Homological Algebra

  • Recommended by Dan Nakano

Kirillov, Lie Groups and Lie Algebras

  • Recommended by Daniel Litt

Algebraic Geometry

General/Introductory

Robin Hartshorne, Algebraic Geometry

Eisenbud and Harris, The Geometry of Schemes

Mumford, The Red Book of Varieties and Schemes


Number Theory

Uncategorized

J. Neukirch, Algebraic Number Theory

  • Large number of exercises here

Cassels and Fröhlich, Algebraic Number Theory

J.-P. Serre, A Course in Arithmetic

Silverman, The Arithmetic of Elliptic Curves

Marcus, Number Fields

  • Covers quadratic fields

J.-P. Serre, Local fields

F. Lorenz, Algebra Volume II: Fields with structure, algebras and advanced topics

J. Milne, Algebraic Number Theory

  • Not a textbook: actually notes

J. Milne, Class Field Theory

  • Not a textbook: actually notes

Weil, Basic Number Theory

Saban Alaca and Kenneth Williams, Introductory Algebraic Number Theory

  • Covers quadratic fields

Valenza, Fourier Analysis on Number Fields


Topology

Algebraic Topology

Hatcher, Algebraic Topology

Standard reference.

Peter May, A Concise Course in Algebraic Topology

Dodson and Parker, A User’s Guide to Algebraic Topology

  • Covers more advanced topics than a usual course: some sheaf theory, bundles, characteristic classes, obstruction theory
  • Appendices on algebra, topology, manifolds/bundles, and tables of homotopy groups

Glen Bredon, Topology and Geometry

  • Blends differential and algebraic topology, can be disorienting as a first pass

Milnor, Topology from the Differentiable Viewpoint (Princeton)

  • Classic reference.

Bott and Tu, Differential Forms in Algebraic Topology (Springer)

  • Classic reference.

Massey, A Basic Course in Algebraic Topology

Homotopy Theory

Bott-Tu, Differential Forms in Algebraic Topology

Griffiths-Morgan, Rational Homotopy Theory and Differential Forms

Mosher-Tangora, Cohomology Operations and Applications in Homotopy Theory

Manifolds

Milnor, Topology from the differentiable viewpoint.

Differential Geometry and Topology

Manfredo P. Do Carmo, Riemannian Geometry

Manfredo P. Do Carmo, Differential Geometry of Curves and Surfaces

Guillemin and Pollack, Differential Topology

John M. Lee, Introduction to Smooth Manifolds

Milnor, Morse Theory

Pollack, Differential Topology

Milnor, Lectures on h-Cobordism

Frank Warner, Foundations of Differentiable Manifolds and Lie Groups (?)

Guillemin, Stable Mappings and Their Singularities

Symplectic Geometry/Topology

Dusa McDuff, Introduction to Symplectic Topology

Eliashberg, From Stein to Weinstein and Back

Cannas da Silva, Lectures on Symplectic Geometry

Complex Geometry

Claire Voisin, Hodge Theory and Complex Algebraic Geometry, Volumes I and II

Daniel Huybrechts, Complex Geometry An Introduction

Griffiths-Harris, Principles of Algebraic Geometry

Carlson, Period Mappings and Period Domains

Rick Miranda, Algebraic Curves and Riemann Surfaces

F. Kirwan, Complex Algebraic Curves

Voison, Hodge Theory and Complex Algebraic Geometry I

Knot Theory

Livingston, Knot Theory

Colin Adams, The Knot Book

Turner, Five Lectures of Khovanov Homology

Bar-Natan, On Khovanov’s categorification of the Jones polynomial

J. Kock, Frobenius algebras and 2D TQFTs

Geometric Group Theory

Mosher-Tangora, Cohomology Operations and Applications in Homotopy Theory

Serre, Trees

Witte-Morris, Introduction to arithmetic groups

Farb-Margalit, Primer on mapping class groups


Physics

Uncategorized

Mark Srednicki, Quantum Field Theory

Pierre Deligne, Quantum Fields and Strings: A Course for Mathematicians

Howard Georgi, Lie Algebras in Particle Physics

Kusse, Mathematical Physics


Combinatorics

Uncategorized

Stanley, Enumerative Combinatorics Vol 1

Bruce Sagan - Springer, The Symmetric Group

Doug West, Introduction to Graph Theory


Logic

Set Theory

Kunen, Set Theory: An introduction to independence proofs

Model Theory

Tent and Ziegler, A Course in Model Theory


Unsorted Recommendations

Sipser, Introduction to the Theory of Computation

Murray, Mathematical Biology

Mac Lane-Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory

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