Mathematics Library

Logic and Proof

Jeremy Avigad, Joseph Hua, Robert Y. Lewis and Floris van Doorn · Online textbook · Introductory · English

Online text

Complete 24-chapter online textbook; introductory chapter and copyright attribution opened.

Combines natural deduction and logical semantics with informal proofs, sets, induction and interactive Lean examples.

Preparation: School mathematics; willingness to learn symbolic notation.

Use: Compare informal arguments with formal derivations and checked proof scripts.

Version: Online documentation 3.18.4; copyright 2017; no revision date stated · Checked 2026-10-13

Markov Chains and Mixing Times

David A. Levin and Yuval Peres, with contributions by Elizabeth L. Wilmer · Textbook · Graduate · English

PDF

Studies finite Markov chains, coupling, spectral methods and mixing estimates.

Preparation: Probability; linear algebra; elementary graph theory.

Use: Bridge to probabilistic combinatorics.

Version: Second edition, 2017 · Checked 2026-10-11

MATH 8030 Introduction to Homological Algebra

Pete L. Clark · Course notes and exercises · Graduate · English

PDF

Connects tensor and Hom functors, module classes, chain complexes and derived functors.

Preparation: Rings, modules and tensor products.

Use: Work through a first homological algebra course with exercises.

Version: Author-hosted notes; date not stated · Checked 2026-10-13

Mathematical Analysis

于品 (Pin Yu) · Course notes · Intermediate · Chinese

Online text

Online lecture text; chapters can be read separately.

Original title: 数学分析

A three-semester course: real numbers, limits and calculus lead into measure, Fourier analysis, distributions and introductory PDE. Later parts need more preparation than the opening chapters.

Preparation: School mathematics; proof-writing for the first semester; linear algebra before later semesters.

Use: Begin with the real-number and convergence chapters; return to the later volumes when studying measure and distributions.

Version: 2018–2020 three-semester course · Checked 2026-11-03

Mathematical Discussions

Timothy Gowers · Expository essay collection · Introductory · English

Online text

Full text on the author’s website or archive.

Dialogues and explanations explore why definitions and proofs arise in algebra, analysis, geometry and logic.

Preparation: School mathematics; willingness to work through proofs.

Use: Supplement undergraduate courses with conceptual explanations.

Version: Author’s online collection; individual essays have their own histories · Checked 2026-10-13

Mathematical Method

Jesper Lützen and Ian Kiming · Course notes · Introductory · Danish

PDF

Complete 199-page course book opened from the department's official archive.

Original title: Matematisk Metode

English title: Mathematical Method. Introduces logic, proof strategies, induction and the analysis–synthesis approach to mathematical problem solving.

Preparation: School algebra and geometry; basic mathematical notation.

Use: Practise writing proofs and analysing problems before taking abstract undergraduate courses.

Version: October 17, 2008, as printed on the title page · Checked 2026-10-13

Mathematical Methods in Quantum Mechanics

Gerald Teschl · Textbook · Graduate · English

PDF

The AMS permits one stored copy for personal use; do not redistribute.

Self-adjoint operators, spectral theory and scattering are developed alongside Schrödinger operators.

Preparation: Real analysis; linear algebra; differential equations.

Use: Connect operator theory with mathematical physics.

Version: Second edition, 2014; authorized author PDF · Checked 2026-10-11

Mathematical Recreations and Essays

W. W. Rouse Ball · Recreational mathematics book · Introductory · English

PDF

Complete PDF at Project Gutenberg.

Puzzles, games and mathematical essays explore arithmetic, geometry, ciphers and recreational constructions.

Preparation: School algebra and geometry.

Use: Explore classical problems and the history of recreational mathematics.

Version: Historical edition digitized as Project Gutenberg 26839 · Checked 2026-10-13

Mathematics in Lean

Jeremy Avigad and Patrick Massot · Online textbook · Intermediate · English

PDF

Complete 216-page PDF opened on the authors' Lean-community project site; introductory HTML explains the exercise workflow.

Teaches Lean 4 and Mathlib through proofs about sets, elementary number theory, algebraic structures, linear algebra, limits and integration.

Preparation: Experience with ordinary mathematical proofs; undergraduate algebra and analysis for later chapters.

Use: Turn familiar mathematical arguments into checked proofs using the companion exercises.

Version: Release v4.19.0; PDF title page dated October 12, 2026 · Checked 2026-10-13

Mathematics in the Age of AI

Terence Tao · Public-lecture essay · Intermediate · English

PDF

Free PDF at the author or authorized archive.

Tao considers how mathematical practice and collaboration might respond to research-capable AI tools.

Preparation: General mathematical interest.

Use: Explore one researcher’s perspective on emerging tools.

Version: 2026 ICM public-lecture essay · Checked 2026-10-13

Maurer–Cartan Methods in Deformation Theory: The Twisting Procedure

Vladimir Dotsenko, Sergey Shadrin and Bruno Vallette · Author manuscript · Research · English

PDF

Full author manuscript on arXiv; front matter permits personal viewing and download.

Develops deformation and gauge methods for dg Lie algebras, homotopy algebras and operads, including twisting constructions.

Preparation: Homological algebra, Lie algebras and basic operads.

Use: Study deformation theory through a systematic account of twisting.

Version: arXiv v2, February 23, 2023 · Checked 2026-10-13

Measure and Integration

Mark Veraar and Emiel Lorist · Lecture notes · Intermediate · English

PDF

The complete 65-page PDF opened on the author's host.

Sigma-algebras, measure construction and Lebesgue integration lead to product measures, Lp spaces and Fourier-series applications.

Preparation: Undergraduate real analysis and metric spaces.

Use: Study a compact measure-theory course with convergence theorems and extension arguments.

Version: Author-hosted notes; date not stated · Checked 2026-10-13

Measure, Integration & Real Analysis

Sheldon Axler · Textbook · Advanced · English (legacy metadata)

PDF

A graduate treatment of measure and integration with connections to functional analysis and Fourier analysis. Suitable after proof-based analysis, not elementary calculus.

Preparation: Undergraduate real analysis; linear algebra.

Legacy entry retained from October 8, 2026

Median Algebras

Brian H. Bowditch · Research monograph draft · Research · English

PDF

The author’s preprint page links the full 280-page manuscript.

A systematic treatment of betweenness, convexity and walls, extending to median metric spaces, cube complexes and coarse median structures.

Preparation: Metric topology; group theory; geometric group theory helpful.

Use: Study median structures as tools in large-scale geometry.

Version: First draft March 10, 2022; revised May 12, 2024 · Checked 2026-10-13

Metric Spaces

Christian Berg; 1997 revision by Bergfinnur Durhuus · Course notes · Intermediate · Danish

PDF

University department archive supplies the complete 96-page course book, including the revision preface and exercises.

Original title: Metriske rum

English title: Metric Spaces. Connects distance and continuity with completeness, compactness and the contraction theorem, ending with differential-equation applications.

Preparation: Single-variable analysis, basic linear algebra and elementary proofs.

Use: Bridge concrete limits to metric-space arguments through the accompanying exercises.

Version: 1997 edition; revision preface July 1997 · Checked 2026-10-13

Microlocal Analysis and Evolution Equations: Lecture Notes from 2008 CMI/ETH Summer School

Jared Wunsch · Lecture notes · Research · English

PDF

The complete 92-page author-submitted manuscript opened on arXiv.

Wave propagation, Fourier-integral operators, parametrices, wave traces and scattering calculus are introduced through evolution equations.

Preparation: Graduate analysis, Fourier transforms and basic PDE.

Use: Bridge a first PDE course to propagation of singularities and geometric scattering.

Version: 2008 summer-school lectures; arXiv v3, July 2023 · Checked 2026-10-13

Model Theory of Valued Fields

David Marker · Course notes · Research · English

PDF

Combines valuation theory with quantifier elimination, algebraically closed valued fields and Ax–Kochen–Ershov methods.

Preparation: Model theory, fields and basic valuation theory.

Use: Study a sustained application of model theory to valued fields.

Version: Fall 2018 course notes · Checked 2026-10-13

Modular Forms, a Computational Approach

William A. Stein, with an appendix by Paul E. Gunnells · Textbook · Graduate · English

PDF

Turns modular-form constructions into explicit algorithms and computations.

Preparation: Linear algebra; number theory; modular forms basics.

Use: Connect theoretical modular forms with computational practice.

Version: 2007 author-hosted book · Checked 2026-10-11

Modular Forms: A Classical Introduction

Kimball Martin · Long lecture notes · Graduate · English

PDF

Full PDF opened publicly on the author or university host.

Introduces classical modular forms and their arithmetic through examples and structural theory. The later part is a developing draft.

Preparation: Complex analysis, abstract algebra and elementary number theory.

Use: A classical entry point before modular curves and automorphic representations.

Version: PDF dated 23 July 2025; first part originated in a Spring 2011 course · Checked 2026-10-13

Modular Functions and Modular Forms

J. S. Milne · Course notes · Intermediate · English (legacy metadata)

PDF

Introduces arithmetic modular forms with a geometric emphasis. A route from complex analysis and algebra toward arithmetic geometry.

Preparation: Complex analysis; groups and fields.

Legacy entry retained from October 8, 2026

Noncommutative Algebra

Pete L. Clark · Course notes · Graduate · English

PDF

Develops semisimple rings and modules, the Wedderburn–Artin theorem and radical methods.

Preparation: Rings, modules and group representations.

Use: Continue beyond commutative undergraduate ring theory.

Version: 2011 summer lecture series; author-hosted notes · Checked 2026-10-13

Notes on Analytic Number Theory

Kiran S. Kedlaya · Course notes · Advanced · English (legacy metadata)

Online text

A graduate course on primes, analytic methods and sieve ideas. The freely readable HTML format supports selective study; complex analysis is an essential preparation.

Preparation: Complex analysis; elementary number theory.

Legacy entry retained from October 8, 2026

Notes on Basic 3-Manifold Topology

Allen Hatcher · Unfinished book draft · Graduate · English

PDF

Introduces prime and torus decompositions and special classes of three-manifolds. This is an unfinished initial portion, not a complete textbook.

Preparation: Point-set topology; fundamental groups; manifolds.

Use: Enter low-dimensional topology.

Version: 2023 revision; 75-page early book draft · Checked 2026-10-11

Notes on Basic Homological Algebra

Andrew Baker · Short notes (15 pages) · Introductory · English (legacy metadata)

PDF

A compact entry into homological methods for readers already comfortable with modules. Useful as course notes; less expansive than a comprehensive graduate textbook.

Preparation: Rings, modules and exact sequences.

Legacy entry retained from October 8, 2026

Notes on class field theory

Kiran S. Kedlaya · Online lecture notes · Graduate · English

Online text

Official complete course/text index and linked reading material opened without login.

Develops reciprocity from abelian extensions of the rationals through cohomology, local fields and adeles.

Preparation: Galois theory; algebraic number theory.

Use: Bridge local and global reciprocity.

Version: Author-hosted notes; date not stated · Checked 2026-10-13