# 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

# 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

## General/Introductory

- 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

## 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

- 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

## Comments