Recommendations: Graduate Level Texts and Notes
Inspired by the following Twitter thread:
Yo math tweeps, what is an absolutely standard textbook in your field that'd be accessible to advanced undergrads and/or newbie grad students?
— D. Zack Garza (∂² ⋘ 0) (@dzackgarza) October 11, 2020
Multiple Areas
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Qiaochu Yuan has some reading recommendations
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This blog also has a large list of recommendations, sorted by topic
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As does this one
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There is a similar community wiki on MSE
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You can find ranked recommendations on this question on MO
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UC Berkeley has a bibliography of books used by class.
Analysis
Real Analysis
Gerald B. Folland, Real Analysis: Modern Techniques and Applications
- Links to some homeworks and solutions at UCSD
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
Atiyah and MacDonald, Introduction to Commutative Algebra
Representation Theory
- Gaitsgory, Course Notes on Geometric Representation Theory
- Mcgerty, Notes on Lie Groups/Algebras
- Shoshany, Notes on Lie Groups
- From a physicist’s perspective
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
- Recommended by Daniel Litt, link to notes
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
- Gathmann, Notes
- Vakil, Rising Sea Notes
Robin Hartshorne, Algebraic Geometry
Eisenbud and Harris, The Geometry of Schemes
- Standard! Notes here: The Geometry of Schemes
Mumford, The Red Book of Varieties and Schemes
Number Theory
Algebraic Number Theory
J. Neukirch, Algebraic Number Theory
- Large number of exercises here
Cassels and Fröhlich, Algebraic Number Theory
J. Milne, Algebraic Number Theory
- Not a textbook: actually notes
Uncategorized
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, 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
- Second Steps in Algebraic Topology
- Slightly out-of-date
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
- Dwyer-Spalinski, Homotopy Theories and Model Categories
- Hovey-Shipley-Smith, Symmetric Spectra
- Hovey, Spectra and symmetric spectra in general model categories
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
- Notes: Lectures on Symplectic Geometry
- Skip chapters 4, 5, 25, 26, 30
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
- Justin Roberts, Knots Knotes
Rolfsen, Knots and Links
Livingston, Knot Theory
Colin Adams, The Knot Book
Turner, Five Lectures of Khovanov Homology
- Arxiv, link
Bar-Natan, On Khovanov’s categorification of the Jones polynomial
- Arxiv, link
J. Kock, Frobenius algebras and 2D TQFTs
Osvath and Szabo, Grid Homology for Knots and Links
Geometric Group Theory
Here are 4 books that all my grad students read:
— Andrew Putman (@AndyPutmanMath) October 11, 2020
1. Brown's "Cohomology of groups"
2. Serre's "Trees" (followed up w/ Scott's article "Topological methods in group theory")
3. Witte-Morris's "Introduction to arithmetic groups"
4. Farb-Margalit's "Primer on mapping class groups"
Mosher-Tangora, Cohomology Operations and Applications in Homotopy Theory
Serre, Trees
- Recommendation from Andrew Putman: follow up with Scott’s Topological Methods in Group Theory
