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Brendan Fong and David I. Spivak
Introductory · PDF
Introduces categorical structures through applications, including order, monoidal structure, databases and compositional systems.
Details and sources Brendan Fong and David I. Spivak · Author manuscript · Introductory · English
PDF
Complete author-deposited PDF opens on arXiv.
Introduces categorical structures through applications, including order, monoidal structure, databases and compositional systems.
Preparation: Comfort with mathematical notation and proofs; elementary sets and functions.
Use: A concrete first route into category theory and its applications.
Version: arXiv v3, October 12, 2018; title page updated October 16, 2018 · Checked 2026-10-13
Marc Levine
Research · PDF
Introduces pure motives, mixed-motive constructions and motivic cohomology in a compact course.
Details and sources Marc Levine · Long lecture notes · Research · English
PDF
Introduces pure motives, mixed-motive constructions and motivic cohomology in a compact course.
Preparation: Algebraic geometry; cohomology; algebraic cycles.
Use: Bridge from cycles to advanced motivic references.
Version: ICTP lecture manuscript; 61 pages · Checked 2026-10-11
Peter Scholze
Research · PDF
Studies abstract structures governing operations on sheaf-like categories.
Details and sources Peter Scholze · Long lecture notes · Research · English
PDF
Studies abstract structures governing operations on sheaf-like categories.
Preparation: Higher categories; sheaf theory; derived categories.
Use: Organize functorial tools used across modern geometry.
Version: WS 2022–23; updated October 2025 · Checked 2026-10-11
Jacob Lurie
Research · PDF
A research reference for spectral schemes, stacks and their sheaves. The author explicitly warns that the draft is unfinished and contains broken references.
Details and sources Jacob Lurie · Unfinished book draft · Research · English
PDF
A research reference for spectral schemes, stacks and their sheaves. The author explicitly warns that the draft is unfinished and contains broken references.
Preparation: Higher Topos Theory; Higher Algebra; schemes.
Use: Selective reference, not a first introduction.
Version: February 2018; author labels it under construction · Checked 2026-10-11
Full-text source checked: 2026-11-03
Found via — References: Spectral Algebraic Geometry
Zeibniz
Graduate · Free to read
An applied seminar on graph-based clustering and its spectral interpretation, connecting matrix eigenvectors, graph Laplacians and statistical learning.
Read online Save to bookshelf Details and sources Zeibniz · Course notes · Graduate · Chinese
Online text
Online lecture text; chapters can be read separately. 在线讲义，可按章节阅读。
Original title: 谱聚类
An applied seminar on graph-based clustering and its spectral interpretation, connecting matrix eigenvectors, graph Laplacians and statistical learning. 关于图上聚类及其谱解释的应用专题，将矩阵特征向量、图拉普拉斯和统计学习联系起来。
Preparation: Linear algebra; elementary graphs and probability. 线性代数、初等图论和概率。
Use: Read after eigenvalues and orthogonal diagonalization; compare the derivations with von Luxburg’s tutorial. 学过特征值和正交对角化后阅读，并与von Luxburg的教程比较推导。
Version: Fall 2020 seminar 2020年秋季讨论班 · Checked 2026-11-03
Jarod Alper
Research · PDF
A systematic development of algebraic stacks and moduli, with substantial geometric examples. The linked version is an author draft, not a claim of a final published edition.
Details and sources Jarod Alper · Book draft · Research · English
PDF
A systematic development of algebraic stacks and moduli, with substantial geometric examples. The linked version is an author draft, not a claim of a final published edition.
Preparation: Schemes, sheaves, commutative algebra.
Use: Study stacks after a first graduate algebraic geometry course.
Version: January 5, 2026 draft · Checked 2026-10-11
Gregory F. Lawler
Graduate · PDF
A book-length route from discrete martingales to stochastic calculus, with applications and supplementary rigorous material.
Details and sources Gregory F. Lawler · Author book draft · Graduate · English
PDF
The complete PDF opens publicly from the author’s university resource page.
A book-length route from discrete martingales to stochastic calculus, with applications and supplementary rigorous material.
Preparation: Probability; calculus; measure theory for the supplementary sections.
Use: Work through a stochastic-calculus course at applied or mathematical depth.
Version: Author draft; revised February 15, 2023 · Checked 2026-10-13
Ajay Chandra and Hendrik Weber
Research · PDF
A lecture course on singular SPDEs, regularity structures and their particle-system motivation.
Details and sources Ajay Chandra and Hendrik Weber · Lecture notes · Research · English
PDF
The complete PDF opens publicly from the author-submitted arXiv archive.
A lecture course on singular SPDEs, regularity structures and their particle-system motivation.
Preparation: Stochastic analysis; distributions; PDEs.
Use: Bridge from introductory SPDEs to renormalized equations.
Version: arXiv v3, July 11, 2017 · Checked 2026-10-13
Nathanaël Berestycki
Intermediate · PDF
Countable-state Markov chains and martingales, with exercises and branching-process applications.
Details and sources Nathanaël Berestycki · Lecture notes · Intermediate · English
PDF
The complete PDF opens publicly from the author’s university resource page.
Countable-state Markov chains and martingales, with exercises and branching-process applications.
Preparation: Undergraduate probability; linear algebra.
Use: Bridge from elementary probability to stochastic-process courses.
Version: February 9, 2026 · Checked 2026-10-13
Wolfgang Lück
Research · PDF
Relates covering-space L²-invariants to three-manifold topology, linking Betti numbers and torsion with geometric structure.
Details and sources Wolfgang Lück · Research survey · Research · English
PDF
The complete author-deposited PDF opens in the arXiv archive.
Relates covering-space L²-invariants to three-manifold topology, linking Betti numbers and torsion with geometric structure.
Preparation: Algebraic topology; covering spaces; three-manifolds; functional analysis helpful.
Use: Orient research reading on L²-invariants and manifold groups.
Version: arXiv v2, June 26, 2021 · Checked 2026-10-13
Jiří Lebl
Advanced · PDF
A course-length tour of pseudoconvexity, CR geometry and the ∂̄-problem. Broad exposure rather than an exhaustive reference, with exercises used throughout.
Details and sources Jiří Lebl · Textbook · Advanced · English (legacy metadata)
PDF
A course-length tour of pseudoconvexity, CR geometry and the ∂̄-problem. Broad exposure rather than an exhaustive reference, with exercises used throughout.
Preparation: One-variable complex analysis; real analysis.
Legacy entry retained from October 8, 2026
Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik
Advanced · Free to read
Develops the categorical setting for representation theory. A specialist continuation rather than an introductory algebra text; link to the authorized author-hosted edition.
Read online Save to bookshelf Details and sources Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik · Monograph · Advanced · English (legacy metadata)
Online text
Develops the categorical setting for representation theory. A specialist continuation rather than an introductory algebra text; link to the authorized author-hosted edition.
Preparation: Representation theory; category theory.
Legacy entry retained from October 8, 2026
Dan Sloughter
Intermediate · PDF
Develops multivariable differentiation as affine approximation, then integration, change of variables, line integrals and Green's theorem.
Details and sources Dan Sloughter · Textbook · Intermediate · English
PDF
Author's page links the complete concatenated 260-page PDF and states a Creative Commons licence.
Develops multivariable differentiation as affine approximation, then integration, change of variables, line integrals and Green's theorem.
Preparation: Single-variable calculus and elementary vectors.
Use: Study geometric explanations alongside a multivariable calculus course.
Version: Author-hosted notes; date not stated · Checked 2026-10-13
Kannan Soundararajan
Research · PDF
Surveys value distributions, moments, large values and central values of zeta and L-functions, with probabilistic perspectives and research questions.
Details and sources Kannan Soundararajan · Research survey · Research · English
PDF
Full author-submitted PDF opened publicly in the arXiv archive.
Surveys value distributions, moments, large values and central values of zeta and L-functions, with probabilistic perspectives and research questions.
Preparation: Graduate analytic number theory, complex analysis and probability.
Use: A research-level overview after a first course on zeta and Dirichlet L-functions.
Version: arXiv v1, 6 December 2021; survey intended for ICM 2022 · Checked 2026-10-13
Brad Osgood
Introductory · PDF
An example-driven introduction from Stanford's EE261 course. Useful for Fourier intuition and applications; follow with a measure-theoretic treatment for deeper analysis.
Read PDF Save to bookshelf Details and sources Brad Osgood · Course notes · Introductory · English (legacy metadata)
PDF
An example-driven introduction from Stanford's EE261 course. Useful for Fourier intuition and applications; follow with a measure-theoretic treatment for deeper analysis.
Preparation: Calculus, complex numbers and linear algebra.
Legacy entry retained from October 8, 2026
Charles Howard Hinton
Introductory · Free to read
Geometric models and visual exercises explore higher-dimensional space alongside early twentieth-century speculation.
Details and sources Charles Howard Hinton · Historical popular mathematics · Introductory · English
Online text
Complete HTML at Project Gutenberg.
Geometric models and visual exercises explore higher-dimensional space alongside early twentieth-century speculation.
Preparation: Elementary geometry; treat physical speculation in historical context.
Use: Practice spatial imagination while reading historical popular mathematics.
Version: 1906 second edition; Project Gutenberg transcription · Checked 2026-10-13
J. Peter May
Graduate · PDF
Develops operadic approaches to iterated loop spaces.
Details and sources J. Peter May · Research monograph · Graduate · English
PDF
Develops operadic approaches to iterated loop spaces.
Preparation: Algebraic topology; loop spaces.
Use: Study classical foundations of operads in topology.
Version: Lecture Notes in Mathematics 271, 1972 · Checked 2026-10-11
Charles A. Weibel
Graduate · PDF
A long introduction spanning low-dimensional groups and higher algebraic K-theory.
Details and sources Charles A. Weibel · Textbook · Graduate · English
PDF
A long introduction spanning low-dimensional groups and higher algebraic K-theory.
Preparation: Rings; modules; homological algebra; topology.
Use: Main course text and reference for K-theory.
Version: 2013 author edition · Checked 2026-10-11
William McCallum and Bjorn Poonen
Research · PDF
Explains p-adic integration for rational points on higher-genus curves, with examples and a bad-reduction appendix.
Details and sources William McCallum and Bjorn Poonen · Research survey · Research · English
PDF
Full PDF opened publicly on the author or university host.
Explains p-adic integration for rational points on higher-genus curves, with examples and a bad-reduction appendix.
Preparation: Algebraic curves; Jacobians; p-adic numbers.
Use: Learn an effective Diophantine method.
Version: Author-hosted notes; date not stated · Checked 2026-10-13
Bhargav Bhatt and Jacob Lurie
Research · PDF
Provides a stack-theoretic interpretation of prismatic crystals and cohomology.
Details and sources Bhargav Bhatt and Jacob Lurie · Research paper · Research · English
PDF
Provides a stack-theoretic interpretation of prismatic crystals and cohomology.
Preparation: Prisms; formal schemes; derived stacks.
Use: Companion to Absolute Prismatic Cohomology.
Version: 2022; latest version on arXiv · Checked 2026-10-11
Bhargav Bhatt and Peter Scholze
Research · PDF
Develops a topology suited to stronger sheaf-theoretic constructions on schemes.
Details and sources Bhargav Bhatt and Peter Scholze · Foundational research paper · Research · English
PDF
Develops a topology suited to stronger sheaf-theoretic constructions on schemes.
Preparation: Étale topology; sheaves; homological algebra.
Use: Background for contemporary cohomological geometry.
Version: Astérisque 369, 2015 · Checked 2026-10-11
Full-text source checked: 2026-11-03
Found via — References: The Pro-Étale Topology for Schemes
Ben Andrews and Christopher Hopper
Graduate · PDF
A course-length introduction to Ricci flow and geometric evolution equations.
Details and sources Ben Andrews and Christopher Hopper · Textbook / author manuscript · Graduate · English
PDF
A course-length introduction to Ricci flow and geometric evolution equations.
Preparation: Riemannian geometry; parabolic PDE.
Use: Prepare for geometric-analysis research.
Version: Author manuscript; book published 2011 · Checked 2026-10-11
Ravi Vakil
Advanced · PDF
A broad, exercise-rich development of schemes and their geometry. The October 21, 2025 author edition closely matches the published text; substantial algebra is essential.
Details and sources Ravi Vakil · Textbook / author edition · Advanced · English (legacy metadata)
PDF
A broad, exercise-rich development of schemes and their geometry. The October 21, 2025 author edition closely matches the published text; substantial algebra is essential.
Preparation: Commutative algebra; topology; category language.
Legacy entry retained from October 8, 2026
July 27, 2024 archive · CC BY-NC-ND 3.0; unchanged noncommercial copy.
The Stacks Project Authors
Advanced · Free to read
A detailed searchable reference with stable theorem tags. Use the Homology, Derived Categories and related chapters; it is not a gentle sequential first course.
Read online Save to bookshelf Details and sources The Stacks Project Authors · Reference work · Advanced · English (legacy metadata)
Online text
A detailed searchable reference with stable theorem tags. Use the Homology, Derived Categories and related chapters; it is not a gentle sequential first course.
Preparation: Homological algebra and category theory.
Legacy entry retained from October 8, 2026
Alexander Grothendieck
Graduate · PDF
An early algebraic treatment of Chern classes, useful for comparing the algebraic and topological constructions.
Details and sources Alexander Grothendieck · Research paper · Graduate · French
PDF
Public full text; checked against the title and contents. 已核对题名和目录，可公开阅读全文。
Original title: La théorie des classes de Chern
An early algebraic treatment of Chern classes, useful for comparing the algebraic and topological constructions. 陈类的早期代数处理，适合与拓扑中的构造对照阅读。
Preparation: Algebraic vector bundles; Chow groups. 代数向量丛与Chow群。
Use: Read alongside the Chern-class chapter of an intersection-theory text. 可配合相交理论教材中陈类的章节阅读。
Version: 1958; Bulletin de la SMF 86, 137–154 1958年；Bulletin de la SMF第86卷，137—154页 · Checked 2026-11-03
Found via — References: La théorie des classes de Chern