2019-2020年值得期待的10本书

写在前面：这里所记录的是关于数学、物理“基础”的一些书目，而此“基础”是 foundation 而不是fundamentals。这有两个意思：如果您是一个数学家或物理学家，则您多半并不会对这方面的学问感兴趣。如果您是一个哲学家，则您多半会因为基础不够而看不懂。但以上问题都是某些奇怪的学术体制或价值观念导致的，如果您确实要读，我会说这些书的门槛是零。

1. New Spaces for Mathematics and Physics 编者 Gabriel Catren, Mathieu Anel

本书主要讲数学和物理中出现的新“空间”观念。在读之前可以先看看：A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry（Cartier 著）。可以这样总结日前的空间观：一条路是Connes的提议，它基于C^{\star}-代数，进而去做非交换几何（谱几何）。另一条路是Grothendieck的创见，它基于拓扑斯理论，进而去做函子（代数）几何。如果读者了解量子物理，就知道相应也有两种做量子场论的方法：一种是基于算子代数，称作AQFT；另一种是基于范畴学，称作FQFT，以拓扑场论为例，可以关注Atiyah-Segal，Baez-Dolan，Lurie。

本书编者Anel已经更新了本书的目录： http://mathieu.anel.free.fr/mat/doc/Anel-Catren-NewSpacesTOC.pdf 最终出版的版本与以下的会议目录有一些差异。

如果未来一年只能看一本书，我会选这本。

会议目录：

1. Smoothness and singularities

Schemes (Pierre Cartier)

see also

[1] Mumford, D. (1999). The red book of varieties and schemes: includes the Michigan lectures (1974) on curves and their Jacobians (Vol. 1358). Springer Science & Business Media.

[2] Manin, Y. I. (2018). Introduction to the Theory of Schemes. Springer.

Synthetic differential geometry - new methods for old spaces (Anders Kock)

see also

[1] Bell, J. L., & Bell, J. L. (1998). A primer of infinitesimal analysis. Cambridge University Press.

[2] Moerdijk, I., & Reyes, G. E. (2013). Models for smooth infinitesimal analysis. Springer Science & Business Media.

[3] Lavendhomme, R. (2013). Basic concepts of synthetic differential geometry (Vol. 13). Springer Science & Business Media.

[4] Kock, A. (2006). Synthetic differential geometry (Vol. 333). Cambridge University Press.

Lie (or differentiable) groupoids (Jean Pradine)

将(流形,Lie群,Lie代数)“升级为”(分叶结构,Lie广群,Lie广代数). Lie广代数是形式化经典力学的一个几何设定. 切丛和余切丛上的经典力学形式都可搬运到Lie广代数中，目前在Lie广代数中刻画经典力学有两种方案：prolongation以及Tulczyjew triple。

see also

[1] Mackenzie, K. C., & Mackenzie, K. C. (2005). General theory of Lie groupoids and Lie algebroids (Vol. 213). Cambridge University Press.

[2] Moerdijk, I., & Mrcun, J. (2003). Introduction to foliations and Lie groupoids (Vol. 91). Cambridge University Press.

Diffeologies (Patrick Iglesias-Zemmour)

see also

Iglesias-Zemmour, P. (2013). Diffeology (Vol. 185). American Mathematical Soc..

2. Spaces with categories of points

Geometric aspects of topos theory in relation with logical doctrines (André Joyal)

see also

[1] Reyes, M. L. P., Reyes, G. E., & Zolfaghari, H. (2004). Generic figures and their glueings: A constructive approach to functor categories. Polimetrica sas.

Sheaves and functors of points (Michel Vaquié)

see also

[1] C. Centazzo, E. M. Vitale, Sheaf theory , pp.311-358 in Pedicchio, Tholen (eds.), Categorical Foundations , Cambridge UP 2004.

[2] Masaki Kashiwara, Pierre Schapira, Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006)

[3] MacLane, S., & Moerdijk, I. (2012). Sheaves in geometry and logic: A first introduction to topos theory. Springer Science & Business Media.

Of course, Peter Johnstone and Manin-Gelfand.

Stacks and the Artin property (Carlos Simpson)

[1] Angelo Vistoli, Grothendieck topologies, fibered categories and descent theory MR2223406; math.AG/0412512 pp. 1–104 in Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, Fundamental algebraic geometry. Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc. 2005. x+339 pp. MR2007f:14001

[2] Olsson, Martin (2016), Algebraic spaces and stacks, Colloquium Publications, 62, American Mathematical Society, ISBN 978-1470427986

[3] Eidin, Dan (2003), "What is... a Stack?" (PDF), Notices of the AMS, 50 (4): 458–459.

[4] Fantechi, Barbara (2001), "Stacks for everybody" (PDF), European Congress of Mathematics Volume I, Progr. Math., 201, Basel: Birkhäuser, pp. 349–359, ISBN 3-7643-6417-3, MR 1905329

[5] Laumon, Gérard; Moret-Bailly, Laurent (2000), Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 39, Berlin, New York: Springer-Verlag, ISBN 978-3-540-65761-3

Note: schemes ⊆ algebraic spaces ⊆ Deligne–Mumford stacks ⊆ algebraic stacks (Artin stacks) ⊆ stacks.

The Geometry of Ambiguity: An introduction to the ideas of Derived Geometry (Mathieu Anel)

TBA

3. Non-commutative geometry

Geometry from statistical physics and information theory (Matilde Marcolli)

Geometry in triangulated categories (Maxim Kontsevich)

Quantum differential geometry (Shahn Majid)

4. Spaces up to homotopy

Spaces as infinity-groupoids (Timothy Porter)

Homological decomposition and motives (Denis-Charles Cisinski)

Homotopy type theory: the logic of homotopy spaces (Mike Shulman)

5. Spaces in classical & quantum mechanics

Derived stacks in symplectic geometry (Damien Calaque)

Higher pre-quantized geometry (Urs Schreiber)

Cobordisms and extended quantum field theories (Dan Freed)

Notion of space in topos quantum theory (Cecilia Flori)

6. Spaces in quantum gravity

Spin networks and spinfoams (Hanno Sahlmann)

Super-geometry (Mikhail Kapranov)

Twistor theory (Roger Penrose)

Stringy geometry and emergent space (Marcos Mariño)

7. Open questions

What experiments tell us about space and time (John Baez)

Time between real and imaginary: what geometries describe the Universe near the Big Bang? (Yuri Manin)

2. The Geometrical Foundations of Classical Field Theory 作者 James Weatherall

3. The Aims and Structure of Cosmological Theory 作者 James Weatherall, Chris Smeenk

还有一本讲义，Space, Time, and Geometry from Newton to Einstein, feat. Maxwell.

应该是Jim近年所作工作的一个总结，大意是为经典场论构造语义范畴。这方面我写了很多notes，如果有兴趣我可以多谈谈（多半感兴趣的人不多，因为大家现在都关心QFT）。

4. The Routledge Companion to the Philosophy of Physics 编者 Eleanor Knox & Alastair Wilson

看了一下网上已经有的章节，我认为此书的取材是比较广泛的。其实现在所谓 philosophy of physics （我尤其不喜欢这个称呼）已经有了一些handbook，例如 The Oxford Handbook of Philosophy of Physics 和 Elsevier Handbook 里头的 Philosophy of Physics。后者写的比较细，可以用来入门。

5. The Quantum Puzzle 作者 Robert Spekkens

对量子力学的基础问题感兴趣的朋友可以读，但我觉得量子基础的正确做法还是几何。可以看看这个会议纪要，New geometric concepts in the foundations of physics.

目录

Introduction

The nature of scientific theories

Operational quantum mechanics

Realist talk

Hidden variable theories: possibilities and constraints

Two Orthodoxies

The Copenhagen interpretation

Decoherence theory

The deBroglie-Bohm interpretation

Quantum logic

Consistent histories

Many worlds

Modal interpretations

Collapse theories

Outlook

6. MONOIDAL CATEGORIES A Unifying Concept in Mathematics, Physics, and Computing

7. Theoretical Computer Science for the Working Category Theorist

这两本书的作者都是 Noson S. Yanofsky，目标读者可能比较偏向理论计算机科学。当然，monoidal category 是量子物理的语义范畴上的正确结构，这点基本无疑。

8. Introduction to cpLogic and Logicism 作者 Gregory Landini

自从逻辑主义在1920年代失势之后，人们对它基本上就提不起什么兴趣。但我的感觉是罗素-怀特海所做的工作中有一部分还没被真正理解。这书还没出来，可先看看同作者的 Frege’s Notations: What They Are and How They Mean 以及 https://www.iep.utm.edu/russ-log/ ，此人是罗素研究的可靠作者。此外也可比较一下卡尔纳普对逻辑主义的看法：Carnap on the Foundations of Mathematics (Peter Koellner著)。

9. Foundations of Set Theory: The Search for New Axioms 作者 Peter Koellner, Hugh Woodin

Morris, Sean：Quine, New Foundations, and the Philosophy of Set Theory

10. Two-dimensional conformal field theory 作者 Yi-zhi Huang

这是我过去感兴趣的一个课题。

目录

1 Axioms for two-dimensional conformal field theories

2 Meromorphic conformal fields and vertex operator algebras

3 Operator product expansion and intertwining operator algebras

4 Braided tensor categories and vertex tensor categories

5 Modular invariance

6 Verlinde formula and modular tensor categories

7 Full and open-closed conformal field theories

8 Open Questions

11. Categories for Quantum Theory 作者 Chris Heunen and Jamie Vicary

范畴量子力学的教科书，已经出版

12. Handbook of Homotopy Theory 主编 Haynes Miller

目录拷贝自nLab：

Tyler Lawson, EnE_n-ring spectra and Dyer-Lashof operations, (pdf)

Benoit Fresse, Little discs operads, graph complexes and Grothendieck–Teichmüller groups, in Haynes Miller(ed.) Handbook of Homotopy Theory (arXiv:1811.12536)

Paul Balmer, A guide to tensor-triangular classification, (pdf)

Daniel Isaksen, Paul Arne Østvær, Motivic stable homotopy groups, (arXiv:1811.05729)

Nathaniel Stapleton, Lubin-Tate theory, character theory, and power operations, (arXiv:1810.12339)

Julia Bergner, A survey of models for (∞,n)(\infty,n)-categories (arXiv:1810.10052)

Wolfgang Lueck, Assembly Maps, (arXiv:1805.00226)

Mark Behrens, Topological modular and automorphic forms (arXiv:1901.07990)

Tobias Barthel and Agnès Beaudry, Chromatic structures in stable homotopy theory, (arXiv:1901.09004)

Gregory Arone, Michael Ching, Goodwillie Calculus, (arXiv:1902.00803)

Kirsten Wickelgren, Ben Williams, Unstable Motivic Homotopy Theory, (arXiv: 1902.08857)

David Ayala, John Francis, A factorization homology primer, (arXiv: 1903.10961)

Lars Hesselholt, Thomas Nikolaus, Topological cyclic homology, (arXiv:1905.08984)

David Gepner, An Introduction to Higher Categorical Algebra, (arXiv:1907.02904)

Søren Galatius, Oscar Randal-Williams, Moduli spaces of manifolds: a user’s guide, (arXiv:1811.08151)

Gijs Heuts, Lie algebra models for unstable homotopy theory, (arXiv:1907.13055)

13. The Philosophy of Duality: From Classical Mechanics to String Theory. 作者 De Haro, S. and J. Butterfield. Oxford University Press (forthcoming, 2020)

14. Higher Structures in M-theory 编者: Branislav Jurco, Christian Saemann, Urs Schreiber, Martin Wolf 可以在 https://arxiv.org/abs/1903.02807 中看到文集收录的文章。

15. Triangulated categories of mixed motives. D.-C. Cisinski & F. Déglise

16. The norm residue theorem in motivic cohomology. C. Haesemeyer and C. Weibel

17. Quantum Riemannian Geometry, E.J. Beggs and S. Majid, in press Grundlehren der mathematischen Wissenschaften, vol. 355, Springer (2019).

18. The Open Handbook of Formal Epistemology 编者 Richard Pettigrew, Jonathan Weisberg

已经在线出版 https://jonathanweisberg.org/post/open-handbook/

19. Modal Homotopy Type Theory, 作者 David Corfield, Oxford University Press, 2020-02.

20. Oxford Handbook of the History of Interpretations and Foundations of Quantum Mechanics, 编者 O. Freire (Oxford University Press, 2021).

21. P. Mnev, "Quantum Field Theory: Batalin–Vilkovisky Formalism and Its Applications," AMS University Lecture Series 72 (2019).

22. Loring W. Tu, Introductory Lectures on Equivariant Cohomology: (AMS-204) (Annals of Mathematics Studies). (2020.3)

23. Albert Schwarz, Mathematical Foundations of Quantum Field Theory. 2020.5.

彩蛋

23. Symmetry 作者 Dan Grayson

https://github.com/UniMath/SymmetryBook

只是一本介绍 univalent 数学的本科教科书，事实上我想写一本给幼儿园小孩的 univalent 数学绘本。