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Jiří Lebl
Introductory · PDF
Introduces ODEs with computation, modeling and examples. A useful first encounter, but its engineering orientation should be supplemented by proof-based analysis.
Details and sources Jiří Lebl · Textbook · Introductory · English (legacy metadata)
PDF
Introduces ODEs with computation, modeling and examples. A useful first encounter, but its engineering orientation should be supplemented by proof-based analysis.
Preparation: Calculus; basic linear algebra.
Legacy entry retained from October 8, 2026
Jacob Lurie
Graduate · PDF
Develops metric nonpositive curvature and the Cartan–Hadamard theorem; the author explicitly excludes several Riemannian and negative-curvature topics.
Details and sources Jacob Lurie · Lecture notes · Graduate · English
PDF
Develops metric nonpositive curvature and the Cartan–Hadamard theorem; the author explicitly excludes several Riemannian and negative-curvature topics.
Preparation: Metric spaces; topology.
Use: Focused geometric bridge.
Version: Undated author manuscript, 34 pages · Checked 2026-10-11
Semyon Dyatlov
Graduate · PDF
Proves stable and unstable manifold theorems, with geodesic-flow and dispersing-billiard examples.
Details and sources Semyon Dyatlov · Expository notes · Graduate · English
PDF
The complete PDF opens publicly from the author-submitted arXiv archive.
Proves stable and unstable manifold theorems, with geodesic-flow and dispersing-billiard examples.
Preparation: Multivariable analysis; differential geometry; differential equations.
Use: Learn the geometric machinery behind hyperbolic systems.
Version: arXiv v1, May 29, 2018 · Checked 2026-10-13
Allen Hatcher
Introductory · PDF
An undergraduate introduction to the language of topological spaces. A compact bridge before algebraic topology.
Details and sources Allen Hatcher · Lecture notes · Introductory · English
PDF
Full author/institution PDF; check source terms before any redistribution.
An undergraduate introduction to the language of topological spaces. A compact bridge before algebraic topology.
Preparation: Proof-writing, sets and elementary real analysis.
Version: Author notes; publication year not stated on the download page · Checked 2026-10-11
John Franks
Intermediate · PDF
A one-quarter introduction moves from Riemann and regulated integrals to Lebesgue integration, convergence theorems, L² and Fourier series; the measure construction is supplied in an appendix.
Details and sources John Franks · Lecture notes · Intermediate · English
PDF
The complete 118-page author-submitted manuscript opened on arXiv.
A one-quarter introduction moves from Riemann and regulated integrals to Lebesgue integration, convergence theorems, L² and Fourier series; the measure construction is supplied in an appendix.
Preparation: A proof-based undergraduate real-analysis course.
Use: Bridge from first real analysis to measure-theoretic integration.
Version: arXiv v3, August 10, 2009 · Checked 2026-10-13
Izabella Łaba and Caleb Marshall
Intermediate · Free to read
A coordinated honours sequence joins multivariable limits, differentiation, optimization and integration with curves, surfaces, vector calculus and an introduction to differential forms.
Read online Save to bookshelf Details and sources Izabella Łaba and Caleb Marshall · Textbook · Intermediate · English
Online text
The author archive offers both full course volumes and separate solutions; both course PDFs returned HTTP 200.
A coordinated honours sequence joins multivariable limits, differentiation, optimization and integration with curves, surfaces, vector calculus and an introduction to differential forms.
Preparation: Single-variable calculus; willingness to learn rigorous proofs.
Use: Bridge computational calculus to proof-oriented multivariable analysis. The two sequential volumes are counted together.
Version: Both volumes typeset August 7, 2026 · Checked 2026-10-13
John K. Hunter
Graduate · PDF
Develops weak derivatives, Sobolev spaces and elliptic equations.
Details and sources John K. Hunter · Long lecture notes · Graduate · English
PDF
Develops weak derivatives, Sobolev spaces and elliptic equations.
Preparation: Real analysis; measure theory; basic functional analysis.
Use: Bridge classical PDE calculations to weak solutions.
Version: 2014 university-hosted notes · Checked 2026-10-11
Bruce Kleiner and John Lott
Research · PDF
Expands the Ricci-flow and surgery arguments behind three-dimensional geometrization, with detailed explanations and appendices.
Details and sources Bruce Kleiner and John Lott · Research exposition · Research · English
PDF
The complete author-deposited PDF opens in the arXiv archive.
Expands the Ricci-flow and surgery arguments behind three-dimensional geometrization, with detailed explanations and appendices.
Preparation: Riemannian geometry; geometric analysis; prior Ricci flow.
Use: Read alongside Perelman’s papers after learning the basic evolution estimates.
Version: arXiv v5, February 19, 2013 · Checked 2026-10-13
Gregory F. Lawler
Graduate · PDF
A concise measure-theoretic foundation course connecting probability to graduate real analysis.
Details and sources Gregory F. Lawler · Lecture notes · Graduate · English
PDF
The complete PDF opens publicly from the author’s university resource page.
A concise measure-theoretic foundation course connecting probability to graduate real analysis.
Preparation: Measure spaces; Lebesgue integration; convergence; Fourier analysis.
Use: Bridge from real analysis to graduate probability.
Version: March 21, 2016 · Checked 2026-10-13
Danny Calegari
Graduate · PDF
A topological route from symplectic linear algebra and Hamiltonian flows to holomorphic curves and the outline of Floer homology.
Details and sources Danny Calegari · Course notes · Graduate · English
PDF
The author’s university page links this freely readable PDF.
A topological route from symplectic linear algebra and Hamiltonian flows to holomorphic curves and the outline of Floer homology.
Preparation: Smooth manifolds; differential forms; algebraic topology.
Use: Prepare for analytic and Floer-theoretic treatments.
Version: March 5, 2025 revision; Winter 2022 course · Checked 2026-10-13
Evan Dummit
Intermediate · Free to read
Ten parts move from integer arithmetic and congruences to cryptography, approximation, elliptic curves, quadratic rings and geometry of numbers.
Read online Save to bookshelf Details and sources Evan Dummit · Lecture-note collection · Intermediate · English
Online text
Official complete course/text index and linked reading material opened without login.
Ten parts move from integer arithmetic and congruences to cryptography, approximation, elliptic curves, quadratic rings and geometry of numbers.
Preparation: Proof-writing and school algebra; abstract algebra for later parts.
Use: Use as a foundation-to-upper-undergraduate sequence.
Version: Spring/Fall 2022 collection on the author's notes page · Checked 2026-10-13
Emmanuel Kowalski
Graduate · PDF
Interweaves rings of integers and ideals with Dirichlet series and prime distribution; the author identifies this as a developing draft.
Details and sources Emmanuel Kowalski · Lecture notes · Graduate · English
PDF
Full PDF opened publicly on the author or university host.
Interweaves rings of integers and ideals with Dirichlet series and prime distribution; the author identifies this as a developing draft.
Preparation: Graduate algebra; real and complex analysis.
Use: Connect algebraic and analytic arithmetic.
Version: Version of October 7, 2026 · Checked 2026-10-13
Andrew V. Sutherland
Graduate · Free to read
Graduate course notes on number fields, local fields and class-field-theoretic methods. The linked collection provides individual lecture PDFs.
Details and sources Andrew V. Sutherland · Long lecture notes · Graduate · English
Online text
The official lecture index supplies the full course as separate PDFs; this is not a single merged PDF.
Graduate course notes on number fields, local fields and class-field-theoretic methods. The linked collection provides individual lecture PDFs.
Preparation: Abstract algebra; commutative algebra (course corequisite).
Use: Work through a graduate algebraic-number-theory sequence.
Version: Fall 2025 complete course collection · Checked 2026-10-11
Andrew V. Sutherland
Graduate · Free to read
Builds automorphic forms from Fuchsian groups, then studies Hecke operators, trace formulas, modular Jacobians and Galois representations.
Details and sources Andrew V. Sutherland · Lecture-course collection · Graduate · English
Online text
Official complete course/text index and linked reading material opened without login.
Builds automorphic forms from Fuchsian groups, then studies Hecke operators, trace formulas, modular Jacobians and Galois representations.
Preparation: Complex analysis; algebra; topology; number theory.
Use: Bridge classical modular forms and arithmetic applications.
Version: MIT 18.786, Spring 2024 · Checked 2026-10-13
Evan Dummit
Graduate · PDF
Develops arithmetic over finite-field polynomial rings and function fields, then valuations, divisors, Riemann–Roch and the Weil conjectures for curves.
Details and sources Evan Dummit · Long lecture notes · Graduate · English
PDF
Full PDF opened publicly on the author or university host.
Develops arithmetic over finite-field polynomial rings and function fields, then valuations, divisors, Riemann–Roch and the Weil conjectures for curves.
Preparation: Abstract algebra and elementary number theory; familiarity with commutative algebra is helpful.
Use: A function-field bridge from algebraic number theory to the arithmetic of curves.
Version: 2021, version 1.00; Fall 2021 Math 7315 course · Checked 2026-10-13
Bernhard Keller
Research · PDF
Surveys differential graded enhancements, derived Morita theory and invariants beyond triangulated categories.
Details and sources Bernhard Keller · Research survey · Research · English
PDF
Complete author-deposited PDF opens on arXiv.
Surveys differential graded enhancements, derived Morita theory and invariants beyond triangulated categories.
Preparation: Derived categories and homological algebra.
Use: Enter the language of dg categories and derived invariants.
Version: arXiv v5, June 19, 2006 · Checked 2026-10-13
Mikko Salo
Research · PDF
Treats geodesic X-ray transforms, wave-equation boundary measurements and inverse conductivity on Riemannian backgrounds.
Details and sources Mikko Salo · Expository course notes · Research · English
PDF
The complete 60-page manuscript opened at users.jyu.fi.
Treats geodesic X-ray transforms, wave-equation boundary measurements and inverse conductivity on Riemannian backgrounds.
Preparation: Riemannian geometry; Sobolev spaces; microlocal basics.
Use: Move from Euclidean inverse problems to geometric ones.
Version: Based on Copenhagen and Montreal minicourses, 2021 · Checked 2026-10-13
Wilhelm Schlag
Research · PDF
A focused exposition of spectral methods for wave decay, including low-energy behavior and examples involving trapping.
Details and sources Wilhelm Schlag · Survey · Research · English
PDF
The complete 22-page author-submitted manuscript opened on arXiv.
A focused exposition of spectral methods for wave decay, including low-energy behavior and examples involving trapping.
Preparation: Spectral theory, Fourier analysis and linear dispersive PDE.
Use: Connect resolvent information with pointwise decay estimates.
Version: arXiv v2, June 2021 · Checked 2026-10-13
William P. Thurston
Intermediate · PDF
Thurston reflects on understanding, proof and communication, drawing on foliations and three-dimensional geometry.
Details and sources William P. Thurston · Mathematical essay · Intermediate · English
PDF
Free PDF at the author or authorized archive.
Thurston reflects on understanding, proof and communication, drawing on foliations and three-dimensional geometry.
Preparation: General interest in mathematics; topology helps with examples.
Use: Compare formal proof with broader mathematical understanding.
Version: 1994 author preprint; Bulletin of the AMS 30 · Checked 2026-10-13
Terence Tao
Intermediate · Free to read
Advice and linked essays address mathematical exposition, revision and communication with readers.
Read online Save to bookshelf Details and sources Terence Tao · Expository guidance collection · Intermediate · English
Online text
Full text on the author’s website or archive.
Advice and linked essays address mathematical exposition, revision and communication with readers.
Preparation: Experience reading mathematical arguments.
Use: Develop clearer mathematical writing.
Version: Author’s online collection · Checked 2026-10-13
The Open Logic Project
Graduate · PDF
A modular collection spanning logic, computability, set theory and model theory. Chapters can be selected for a course.
Details and sources The Open Logic Project · Textbook · Graduate · English
PDF
A modular collection spanning logic, computability, set theory and model theory. Chapters can be selected for a course.
Preparation: Proof-writing; elementary logic.
Use: Broaden a foundations reading programme.
Version: Rolling complete build; distinct from the legacy Calgary textbooks · Checked 2026-10-11
Martin Markl
Research · PDF
Reviews operads, PROPs and their algebras, including structures with several inputs and outputs.
Details and sources Martin Markl · Research survey · Research · English
PDF
Complete author-deposited PDF opens on arXiv.
Reviews operads, PROPs and their algebras, including structures with several inputs and outputs.
Preparation: Homological algebra, tensor categories and basic operads.
Use: Compare operadic and PROP formalisms before specialized research.
Version: arXiv v3, February 11, 2006 · Checked 2026-10-13
Gerald Teschl
Intermediate · PDF
Connects rigorous ODE theory to stability, dynamical systems and chaos. A bridge to graduate dynamics; the author permits personal-use downloads, not unrestricted redistribution.
Details and sources Gerald Teschl · Textbook / authorized online edition · Intermediate · English (legacy metadata)
PDF
Connects rigorous ODE theory to stability, dynamical systems and chaos. A bridge to graduate dynamics; the author permits personal-use downloads, not unrestricted redistribution.
Preparation: Real analysis; linear algebra; calculus.
Legacy entry retained from October 8, 2026
Kiran S. Kedlaya
Research · PDF
Develops nonarchimedean differential modules and their analytic structure.
Details and sources Kiran S. Kedlaya · Research monograph · Research · English
PDF
Cambridge permission covers the posted first edition. The author expressly asks readers not to redistribute it.
Develops nonarchimedean differential modules and their analytic structure.
Preparation: Nonarchimedean fields; differential equations; algebra.
Use: Research preparation in p-adic differential modules.
Version: Authorized first-edition PDF, 2010; second edition 2022 is distinct in access, not counted again · Checked 2026-10-11
Bhargav Bhatt
Research · PDF
A recent course approaching p-adic comparison theory and its geometric foundations.
Details and sources Bhargav Bhatt · Long lecture notes · Research · English
PDF
A recent course approaching p-adic comparison theory and its geometric foundations.
Preparation: Schemes; cohomology; p-adic fields.
Use: A guided route toward contemporary arithmetic geometry.
Version: Fall 2025 course notes · Checked 2026-10-11