Mathematics Library

Iwasawa Theory

Romyar Sharifi · Lecture notes · Research · English

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Full PDF opened publicly on the author or university host.

Develops units, class groups, completed group rings and arithmetic in infinite p-adic towers.

Preparation: Algebraic number theory; local fields; module theory.

Use: Prepare for Iwasawa-theoretic research.

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

K-Theory and Elliptic Operators

Gregory D. Landweber · Expository lecture notes · Graduate · English

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The complete author-deposited PDF opens in the arXiv archive.

Develops topological K-theory and the Atiyah–Singer theorem, with equivariant extensions and appendices on Bott periodicity and spectral sequences.

Preparation: Algebraic topology; vector bundles; elementary differential operators.

Use: Bridge bundle theory to the topological formulation of index theory.

Version: arXiv v1, April 27, 2005; PDF title page also displays October 22, 2018 · Checked 2026-10-13

K-Theory and Topological Cyclic Homology of Henselian Pairs

Dustin Clausen, Akhil Mathew and Matthew Morrow · Research paper · Research · English

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Relates relative K-theory and topological cyclic homology with finite coefficients for henselian pairs.

Preparation: Algebraic K-theory; spectra; topological cyclic homology.

Use: Study the finite-coefficient TC input before the henselian-pair comparison.

Version: arXiv 1803.10897; inspected electronic version · Checked 2026-11-03

Found via — References: $K$-Theory and Topological Cyclic Homology of Henselian Pairs

Kerodon

Jacob Lurie · Online reference project · Graduate · English

Online text

An online reference for categorical and homotopical foundations, organized by stable tags.

Preparation: Category theory; simplicial sets.

Use: Selective study and precise citations.

Version: Continuously revised online text; checked October 11, 2026 · Checked 2026-10-11

Kuranishi Spaces and Symplectic Geometry, I–II

Dominic Joyce · Book draft · Research · English

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Develops Kuranishi foundations for symplectic applications. Both preliminary volumes are available; neither is represented as a final published book.

Preparation: Symplectic geometry; moduli; derived differential geometry.

Use: Advanced foundation for virtual geometric constructions.

Version: 2017 preliminary two-volume manuscript; counted once · Checked 2026-10-11

Large Cardinals

Asaf Karagila · Advanced course notes · Research · English

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Surveys large-cardinal hierarchies through ultrapowers, extenders, compactness, inconsistency bounds and choiceless principles.

Preparation: Axiomatic set theory, elementary embeddings and model-theoretic methods.

Use: Study large-cardinal techniques and their relationships before specialist papers.

Version: Updated April 13, 2026; Spring 2025 Leeds course · Checked 2026-10-13

Large Networks and Graph Limits

László Lovász · Author book manuscript · Research · English

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Develops graph homomorphisms, graphons and graphings as analytic models for large networks.

Preparation: Graph theory; linear algebra; probability; measure theory.

Use: Book-length preparation for graph-limit research.

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

Lecture Notes — Model Theory (Math 411)

Anand Pillay · Course notes · Advanced · English (legacy metadata)

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A concise graduate treatment beginning with structures and languages. The lecture-note format rewards familiarity with logical notation and independent proof work.

Preparation: First-order logic; abstract algebra.

Legacy entry retained from October 8, 2026

Lecture Notes on Cherednik Algebras

Pavel Etingof and Xiaoguang Ma · Course notes · Research · English

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Complete author-deposited PDF opens on arXiv.

Studies rational Cherednik algebras, category O, Dunkl operators and related representation-theoretic constructions.

Preparation: Graduate algebra, Lie theory and group representations.

Use: Enter Cherednik algebra research through a structured lecture course.

Version: arXiv v4, April 19, 2010; MIT Fall 2009 course · Checked 2026-10-13

Lecture Notes on Embedded Contact Homology

Michael Hutchings · Lecture notes · Research · English

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The complete PDF opens on the author’s or university’s public site.

Develops embedded contact homology, its index and differential, computations and capacities for symplectic embedding problems.

Preparation: Contact geometry; holomorphic curves; Floer homology background.

Use: Study ECH constructions and their four-dimensional applications.

Version: Notes based on 2012 lectures; PDF revision date not stated · Checked 2026-10-13

Lecture Notes on Equivariant Cohomology

Matvei Libine · Long lecture notes · Graduate · English

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Introduces equivariant cohomology, its geometric models and localization methods in a graduate-course format.

Preparation: Differential forms; Lie groups; cohomology.

Use: Develop the equivariant models before using the localization formulas.

Version: arXiv 0709.3615; inspected electronic version · Checked 2026-11-03

Found via — References: Lecture Notes on Equivariant Cohomology

Lecture Notes on Ergodic Theory

Omri Sarig · Course notes · Advanced · English (legacy metadata)

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A measure-theoretic path into ergodic theory. Best after measure and integration; the author warns that the notes have not been proofread and should be used carefully.

Preparation: Measure theory; basic probability.

Legacy entry retained from October 8, 2026

Lecture Notes on Multidimensional Complex Analysis

Harold P. Boas · Lecture notes · Graduate · English

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The complete 97-page course manuscript opened on the author's teaching site.

Several-variable power series and Hartogs phenomena lead to polynomial and holomorphic convexity, pseudoconvexity, the Levi problem, the d-bar equation and holomorphic mappings.

Preparation: First-year graduate real analysis and single-variable complex analysis.

Use: Bridge one-variable analysis to the analytic and geometric machinery of several complex variables.

Version: Draft of January 6, 2020; fall 2019 course · Checked 2026-10-13

Lecture Notes on the Dynamics of the Weil-Petersson Flow

Carlos Matheus · Lecture notes · Research · English

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The complete PDF opens publicly from the author-submitted arXiv archive.

Explains geometric ingredients of Weil-Petersson-flow ergodicity and discusses mixing.

Preparation: Riemann surfaces; differential geometry; ergodic theory.

Use: Study the geometric side of a research-level ergodicity theorem.

Version: arXiv v1, January 4, 2016 · Checked 2026-10-13

Lecture Notes on Thermodynamic Formalism for Topological Markov Shifts

Omri Sarig · Lecture notes · Research · English

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The complete PDF opens publicly from the author’s university resource page.

Pressure and equilibrium ideas for symbolic dynamics and countable Markov shifts.

Preparation: Measure-theoretic ergodic theory; symbolic dynamics; real analysis.

Use: Bridge into thermodynamic formalism for noncompact symbolic systems.

Version: May 5, 2009; Penn State Spring 2009 · Checked 2026-10-13

Lecture Notes on Tree-free Regularity Structures

Felix Otto, Kihoon Seong and Markus Tempelmayr · Lecture notes · Research · English

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The complete PDF opens publicly from the author-submitted arXiv archive.

A diagram-free approach to stochastic model estimates for singular quasilinear SPDEs using Malliavin calculus.

Preparation: SPDEs; Schauder theory; Malliavin calculus; regularity structures.

Use: Advanced continuation after introductory singular-SPDE lectures.

Version: arXiv v2, September 21, 2023 · Checked 2026-10-13

Lectures in Harmonic Analysis

Thomas H. Wolff; edited by Izabella Łaba and Carol Shubin · Lecture notes · Research · English

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The editor links the 85-page revised manuscript; the complete PDF returned HTTP 200.

Fourier-transform foundations lead to uncertainty, stationary phase, restriction, Hausdorff measures, distance sets and Kakeya problems.

Preparation: Measure theory, Lp spaces and mathematical maturity with proofs.

Use: Bridge Fourier-analysis coursework to restriction and geometric harmonic-analysis problems.

Version: Revised lecture manuscript, March 2002 · Checked 2026-10-13

Lectures on a Method in the Theory of Exponential Sums

Matti Jutila · Long lecture notes · Research · English

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Complete institutional PDF, authorized for individual use only. Copyright remains with TIFR; retypesetting may retain errors. This is not a redistribution licence.

A focused analytic-number-theory specialization in exponential-sum estimates. Enter after practicing analytic estimates and arithmetic sums.

Preparation: Analytic number theory; Fourier methods; complex analysis.

Version: TIFR Lectures on Mathematics 80; authorized retypeset historical edition — original date on title page · Checked 2026-10-11

Lectures on Advanced Analytic Number Theory

Carl Ludwig Siegel · Long lecture notes · Graduate · English

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A classical advanced treatment of analytic arithmetic. Historical notation may differ from current treatments. Individual-use access only; do not redistribute.

Preparation: Graduate number theory; complex analysis.

Use: Supplement modern analytic-number-theory courses.

Version: TIFR volume 23; retypeset digital edition (see PDF imprint for publication year) · Checked 2026-10-11

Lectures on Algebra

李文威 (Wen-Wei Li) · Author textbook · Intermediate · Chinese

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Public full text; checked against the title and contents.

Original title: 代数学讲义

A first-year algebra course joining linear algebra with the language of groups, rings and modules. The public electronic edition is distinct from the later publication layout.

Preparation: School algebra and proof-writing.

Use: Use the vector-space and matrix chapters first, then follow the algebraic structures.

Version: 2026-10-16 electronic edition · Checked 2026-11-03

Found via — References: 代数学讲义

Lectures on an Introduction to Grothendieck’s Theory of the Fundamental Group

J. P. Murre · Long lecture notes · Graduate · English

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Develops the algebraic fundamental group via finite étale covers. Historical notation may differ from current treatments. Individual-use access only; do not redistribute.

Preparation: Schemes; Galois theory.

Use: Classical preparation for étale geometry.

Version: TIFR volume 40; retypeset digital edition (see PDF imprint for publication year) · Checked 2026-10-11

Lectures on Analytic Geometry

Peter Scholze · Long lecture notes · Research · English

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Free PDF at the author or authorized archive.

Develops liquid vector spaces and analytic rings. The preface identifies its analytic-space definition as a historical stepping stone.

Preparation: Condensed mathematics; derived algebra; functional analysis.

Use: Study liquid techniques and compare with later analytic stacks.

Version: May 2026 revision of the winter 2019/20 course · Checked 2026-10-13

Full-text source checked: 2026-11-03

Found via — References: Lectures on Analytic Geometry

Lectures on Analytic Number Theory

Hans Rademacher · Long lecture notes · Graduate · English

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A classical analytic number theory course. Historical notation may differ from current treatments. Individual-use access only; do not redistribute.

Preparation: Complex analysis; arithmetic functions.

Use: Historical graduate reference.

Version: TIFR volume 2; retypeset digital edition (see PDF imprint for publication year) · Checked 2026-10-11

Lectures on Approximation by Polynomials

J. C. Burkill · Long lecture notes · Graduate · English

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Treats classical polynomial approximation. Historical notation may differ from current treatments. Individual-use access only; do not redistribute.

Preparation: Undergraduate real analysis.

Use: Focused bridge to approximation theory.

Version: TIFR volume 16; retypeset digital edition (see PDF imprint for publication year) · Checked 2026-10-11

Lectures on Calogero-Moser Systems

Pavel Etingof · Lecture notes · Research · English

PDF

Complete author-deposited PDF opens on arXiv.

Connects integrable systems with Hamiltonian reduction, deformation theory and representation-theoretic algebras.

Preparation: Lie theory, abstract algebra and some differential geometry.

Use: Explore interactions between integrable systems and representation theory.

Version: arXiv v4, December 21, 2009 · Checked 2026-10-13