一些有用的数学资源整理

这是我个人所做的一些资源整理，放了我学习过并认为比较好的*用于入门*的材料。当然，有很多材料没有出现在这个列表里，这可能是因为以下一些原因：我不知道/没看过；和我所选择的材料有较多的重叠；我主观上认为写得不是很好；过于深入，不适合用于入门；不在我想要分享的方向里；我忘了等等。这里的选择是很主观的，因此一些我所没有列出的材料可能刚好受到你的喜爱。当然我也欢迎完善的建议。

综合

• [Book?] ICM Proceedings (Good place to look if you don’t know what to do or learning what have been done)

代数

交换代数

• [Book] Commutative Algebra with a View towards Algebraic Geometry by David Eisenbud
• [Book] Commutative Ring Theory by Hideyuki Matsumura

丛代数

• [Book] Lecture Notes on Cluster Algebras by Robert J. Marsh
• [Video] Cluster Algebras by Bernhard Keller, Sergey Fomin, Idun Reiten, Dylan Thurston https://www.bilibili.com/video/BV1p4411h71B
• [Video] Cluster Algebra by Pavel Glashin https://www.bilibili.com/video/BV1FV411J7h6

量子群

• [Book] Quantum Groups by Christian Kassel

同调代数

• [Book] Methods of Homological Algebra by Sergei I. Gelfand and Yuri I. Manin

几何群论

• [Video] IHES Summer School 2019: Aspects of Geometric Group Theory https://www.bilibili.com/video/BV1Wt411c7pu/
• [Book] Geometric Group Theory by Cornelia Druţu and Michael Kapovich

其他

• [Book] Triangulated Categories edited by Thorsten Holm, Peter Jorgensen, and Raphaël Rouquier

微分几何

微分流形

• [Book] Introduction to Smooth Manifolds by John Lee (A little bit lengthy)

黎曼曲面和 Teichmüller space

• [Video] 黎曼曲面导论 by 齐震宇 https://www.bilibili.com/video/BV1Ss411J7ot
• [Book] Compact Riemann Surfaces by Jürgen Jost
• [Book] Teichmüller Theory and Applications to Geometry, Topology, and Dynamics by John H. Hubbard
• [Book] Moduli of Riemann Surfaces edited by Benson Farb, Richard Hain, and Eduard Looijenga

黎曼几何

• [Book] Riemannian Geometry by Manfredo Perdigao do Carmo
• [Book] Riemannian Geometry: A Modern Introduction by Isaac Chavel

复几何

• [Video] 复几何 by 齐震宇 https://www.bilibili.com/video/BV1Ys411J7Mb
• [Book] Complex Geometry: an Introduction by Daniel Huybrechts
• [Book] Hodge Theory and Complex Algebraic Geometry I & II by Claire Voison
• [Book] Calabi-Yau Geometry and Related Topics by Mark Gross, Daniel Huybrechts, Dominic Joyce

辛几何

• [Book] Lectures on Symplectic Geometry by Ana Cannas da Silva
• [Note] A beginner’s introduction to Fukaya categories by Denis Auroux, https://arxiv.org/abs/1301.7056
• [Book] Fukaya Categories and Picard-Lefschetz Theory by Paul Seidel
• [Book] Morse Theory and Floer Homology, Michèle Audin and Mihai Damian
• [Book] Holomorphic Curves in Low Dimensions by Chris Wendl

其他

• [Video] Ergodic Theory, Geometry and Dynamics by Curtis McMullen https://www.bilibili.com/video/BV1FT4y1Q7cD
• [Book] Riemannian Holonomy Groups and Calibrated Geometry by Dominic Joyce

拓扑

基础代数拓扑（同调论/同伦论）

• [Video] Algebraic Topology by 齐震宇 https://www.bilibili.com/video/BV1DB4y147L8
• [Note] Algebraic Topology by 李思 https://sili-math.github.io/AT2020/Algebraic-Topology.2020.pdf
• [Book] Algebraic Topology by Hatcher
• [Book] Algebraic Topology by Tammo tom Dieck (更加现代化和代数化)

微分拓扑

• [Book] Topology from the Differentiable Viewpoint by John Milnor
• [Book] Morse Theory by John Milnor

稳定同伦论

• [Video] Introduction to stable homotopy theory by Denis Nardin https://www.bilibili.com/video/BV1gy4y1i7GE
• [Book] Foundations of Stable Homotopy Theory by David Barnes and Constanze Roitzheim

无穷范畴和无穷代数

• [Video] Introduction to higher category theory by Tobias Dyckerhoff https://www.bilibili.com/video/BV1DA411N72o
• [Book] Introduction to Infinity-Categories by Markus Land
• [Video] ∞-categories and Higher Algebra by Achim Krause and Thomas Nikolaus https://www.bilibili.com/video/BV18p4y1r75g
• [Survey] An Introduction to Higher Categorical Algebra by David Gepner, https://arxiv.org/abs/1907.02904
• [Book] Higher Topos Theory and Higher Algebra by Jacob Lurie (Definitive references, but far too lengthy)

等变同调论

• [Book] Introductory Lectures on Equivariant Cohomology by Loring W. Tu

示性类

• [Book] Characteristic Classes by John W. Milnor and James D. Stasheff

K 理论

• [Note] Vector Bundles and K-Theory by Allen Hatcher https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf
• [Book] The K Book: an Introduction to Algebraic K-Theory by Charles A. Weibel
• [Book] The Local Structure of Algebraic K-Theory by Bjørn Ian Dundas, Thomas G. Goodwillie, Randy McCarthy
• [Video] Topological Cyclic Homology by Achim Krause and Thomas Nikolaus https://www.bilibili.com/video/BV1LK4y1j7zW

低维拓扑

• [Note] 4-dimensional topology by Ciprian Manolescu https://ymsc.tsinghua.edu.cn/%5F%5Flocal/6/41/86/F54C83B75671919DB1365230BA1%5F14A4EF5E%5F23465A.pdf?e=.pdf
• [Note] Introduction to Geometric Topology by Bruno Martelli, https://arxiv.org/abs/1610.02592
• [Book] Quantum Field Theory and Manifold Invariants edited by Daniel S. Freed, Sergei Gukov, Ciprian Manolescu, Constantin Teleman, and Ulrike Tillmann
• [Book] Quantum Invariants of Knots and 3-Manifolds by Vladimir G. Turaev

其他

• [Book] Sheaves in Topology by Alexandru Dimca
• [Video] A Crash Course in Topos by Andre Joyal https://www.bilibili.com/video/BV1QW411T7o6
• [Book] Sheaves in geometry and logic : a first introduction to topos theory, Saunders MacLane and Ieke Moerdijk
• [Video] Topological Quantum Matter by 王晴睿 https://www.bilibili.com/video/BV1N44y1Y7qa
• [Book] Operads in Algebra, Topology and Physics by Martin Markl, Steve Shnider and Jim Stasheff
• [Book] Tensor Categories by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik

代数几何

代数几何基础

• [Note] Foundations of Algebraic Geometry by Ravi Vakil http://math.stanford.edu/~vakil/216blog/FOAGaug2922public.pdf (Good for self-study than Hartshorne but a little bit lengthy)
• [Book] Algebraic Geometry by Robin Hartshorne

双有理几何

• [Video] Birational geometry of algebraic varieties by Dori Bejleri https://www.bilibili.com/video/BV1yp4y1r7DV/
• [Video] MSRI Derived Algebraic Geometry and Birational Geometry and Moduli Spaces https://www.bilibili.com/video/BV1G4411F7WA
• [Book] Introduction to the Mori Program by Kenji Matsuki (Easier to read then Kollar & Mori one)
• [Book] Birational Geometry of Algebraic Varieties by Janos Kollar & Shigefumi Mori (Standard reference but quit hard to read)

代数栈（algebraic stacks）

• [Note] Introduction to Algebraic Stacks by Kai Behrend https://personal.math.ubc.ca/~behrend/math615A/stacksintro.pdf
• [Video] Introduction to stacks and moduli by Jarod Alper https://www.bilibili.com/video/BV1br4y1T7Gk
• [Book] Algebraic Spaces and Stacks by Martin Olsson

几何不变量

• [Book] An Introduction to Invariants and Moduli by Shigeru Mukai
• [Book] Geometric Invariant Theory by D. Mumford, J. Fogarty, F. Wirwan

奇点理论

• [Book] Introduction to Singularities by Shihoko Ishii
• [Notes] Topics on singularities of algebraic varieties by Mircea Mustata http://www-personal.umich.edu/~mmustata/NotesIntroSing.pdf

模空间

• [Video] Deformation Theory by Robin Hartshorne https://www.bilibili.com/video/BV1Yt411p7js/
• [Note] Moduli of Higgs Bundles by Andrew Neitzke https://gauss.math.yale.edu/~an592/exports/higgs-bundles.pdf
• [Book] Lectures on Hilbert Schemes of Points on Surfaces by Hiraku Nakajima
• [Book] Hilbert Schemes of Points and Infinite Dimensional Lie Algebras by Zhenbo Qin
• [Book] The Geometry of Moduli Spaces of Sheaves by Daniel Huybrechts and Manfred Lehn
• [Book] Geometry of Algebraic Curves II by Enrico Arbarello, Maurizio Cornalba, Phillip A. Griffiths

计数几何

• [Video] Gromov-Witten/Donaldson Thomas theory and Birational/Symplectic invariants by Artan Sheshmani https://www.bilibili.com/video/BV1Gi4y1T7ed
• [Video] MSRI Enumerative Geometry Beyond Numbers https://www.bilibili.com/video/BV1a7411C7fj
• [Note] 13/2 ways of counting curves by R. Pandharipande and R. P. Thomas https://arxiv.org/abs/1111.1552
• [Book] An Invitation to Modern Enumerative Geometry by Andrea T. Ricolfi
• [Book] An Invitation to Quantum Cohomology by Joachim Kock and Israel Vainsencher
• [Book] Mirror Symmetry and Algebraic Geometry by David A. Cox and Sheldon Katz
• [Book] Calabi-Yau Varieties: Arithmetic, Geometry and Physics edited by Radu Laza, Matthias Schütt, Noriko Yui

模体上同调

• [Video] Motivic Theory https://www.bilibili.com/video/BV1x4411579M
• [Book] Lectures Notes on Motivic Cohomology by Carlo Mazza, Vladimir Voevodsky, and Charles Weibel
• [Book] Triangulated Categories of Mixed Motives by Denis-Charles Cisinski and Frédéric Déglise

导出代数几何

• [Note] Derived Algebraic Geometry by Bertrand Toën https://arxiv.org/abs/1401.1044
• [Note] An introduction to derive (algebraic) geometry by J. Eugster and J.P. Pridham http://arxiv.org/abs/2109.14594
• [Video] Derived Algebraic/Differential Geometry by Artan Sheshmani https://www.bimsa.cn/newsinfo/749630.html
• [Video] Derived Algebraic Geometry at ETH https://www.bilibili.com/video/BV1TJ41117aJ/
• [Video] Introduction to Derived Geometry by Tony Yue Yu http://bicmr.pku.edu.cn/content/show/70-2449.html Password: Geometry

其他

• [Video] Vertex Operator and Conformal Blocks by 归斌 https://www.bilibili.com/video/BV1fZ4y1b7hg
• [Note] D-modules in birational geometry by Mihnea Popa https://people.math.harvard.edu/~mpopa/notes/DMBG-posted.pdf
• [Note] Introduction to Noncommutative Crepant Resolution by Michael Wemyss https://www.maths.gla.ac.uk/~mwemyss/NCCR4.pdf
• [Video] Tropical Geometry by Bernd Sturmfels https://www.bilibili.com/video/BV1n54y1y7b1
• [Book] Tropical Geometry and Mirror Symmetry by Mark Gross
• [Book] Motivic Integration by Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag
• [Book] Intersection Theory by William Fulton
• [Book] Toric Varieties by David Cox, John Little, Hal Schenck
• [Book] K3 Surfaces by Daniel Huybrechts
• [Book] Fourier-Mukai Transformations in Algebraic Geometry by Daniel Huybrechts

表示论

李群和李代数

• [Book] Lie Groups Beyond an Introduction by Anthony W. Knapp
• [Book] Representation of Semisimple Lie Algebras in the BGG Category O by James E. Humphreys
• [Book] Infinite Dimensional Lie Algebras by Victor G. Kac

代数群

• [Video] 代数群介绍 by 王崧 https://www.bilibili.com/video/BV1KB4y1m7DY
• [Book] Algebraic Groups by J. Milne

有限代数表示

• [Book] Elements of the Representation of Associated Algebras I, Ibrahim Assem, Daniel Simson, and Andrzej Skowronski
• [Book] Quiver Representations and Quiver Varieties, Alexander Kirillov Jr.

自守表示

• [Book] An Introduction to Automorphic Representations by Jayce R. Getz, Heekyoung Hahn https://sites.duke.edu/heekyounghahn/files/2022/04/GTM.pdf

几何表示论

• [Video] Enumerative Geometry and Geometric Representation Theory by Andrei Okounkov (B站搬运不全) https://crei.skoltech.ru/cas/calendar/okounkov%5Flect20/
• [Book] Representation Theory and Complex Geometry by Neil Chriss and Victor Ginzburg
• [Book] Geometry of Moduli Spaces and Representation Theory edited by Roman Bezrukavnikov, Alexander Braverman, and Zhiwei Yun
• [Book] D-Modules, Perverse Sheaves and Representation Theory by Ryoshi Hotta, Kiyoshi Takeuchi, Toshiyuki Tanisaki
• [Book] Perverse Sheaves and Applications to Representation Theory by Pramod N. Achar

其他

• [Book] Homological Theory of Representations by Henning Krause
• [Book] Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations by Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov

数论

数论相关的资源有比较全的列表可以参考，当然这个列表是整理的网上资源，不包含书本等内容：https://langlandslearninggroup.github.io/video%5Flecture.html 补充：

• [Note] p-adic L-functions

https://chriswilliams1404.wixsite.com/website/course-on-p-adic-l-functions

• [Video] 西安交通大学2021数论暑期学校 https://space.bilibili.com/1090503015
  • 代数簇上的有理点
  • 遍历论及其在数论中的应用
  • 自守形式的解析理论

数学物理

• [Video] PSI Lectures https://pirsa.org/courses
• [Note] Lectures on Gauge Theory by David Tong http://www.damtp.cam.ac.uk/user/tong/gaugetheory/gt.pdf
• [Note] TASI Lectures on Solitons by David Tong http://www.damtp.cam.ac.uk/user/tong/tasi/tasi.pdf
• [Note] Lectures on Supersymmetry by David Tong http://www.damtp.cam.ac.uk/user/tong/susy/susy.pdf
• [Book] Quantum Field Theory for the Gifted Amateur by Tom Lancaster and Stephen J. Blundell
• [Note] A Solution to Peskin & Schroeder by Zhong-Zhi Xianyu https://zzxianyu.files.wordpress.com/2017/01/peskin%5Fproblems.pdf
• [Note] A mini-course on topological strings by Marcel Vonk https://arxiv.org/abs/hep-th/0504147
• [Book] Mirror Symmetry by Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Cumrun Vafa, Ravi Vakil, and Eric Zaslow
• [Book] Dirichlet Branes and Mirror Symmetry by Paul S. Aspinwall, Tom Bridgeland, Alastair Craw, Michael R. Douglas, Mark Gross, Anton Kapustin, Gregory W. Moore, Graeme Segal, Balázs Szendrői, and P.M.H. Wilson
• [Book] Tau Functions and their Applications by John Harnad and Ferenc Balogh
• [Video] Topological Recursion, from Enumerative Geometry to Integrability by Bertrand Eynard https://www.bilibili.com/video/BV1FW411L7nk/
• [Book] Counting Surfaces by Bertrand Eynard
• [Book] Factorization Algebras in Quantum Field Theory I, II by Kevin Costello and Owen Gwilliam
• [Book] Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction by S. Friedli and Y. Velenik
• [Book] Statistical Mechanics edited by Scott Sheffield and Thomas Spencer
• [Book] Moonshine Beyond the Monster by Terry Gannon

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