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[[File:Linmechfieldsfolds.jpg|thumb|alt=Linear algebra, Mechanics, Relativity and Fields, Differential Geometry|The starter pack to physics and differential geometry.]]
[[File:Read.jpg|thumb|A graphic showing the list's dependencies. Click to enlarge.]]


The starter pack to physics and differential geometry.
This list of books provides the most direct and rigorous route to understanding differential geometry, the mathematical language of physics. Each selection thoroughly addresses its subject matter. The list does not need to be read linearly or only one book at a time. It is encouraged to go between books and/or read several together to acquire the necessary language and understand the motivations for each idea. The greatest hurdles are the motivation to learn and developing an understanding of the language of mathematics.


== Philosophy ==
See the image on the right for a visual representation of its dependencies.
Our point of view is that the texts typically used in physics and especially mathematics degree tracks are window dressing for the real job of being a mathematical physicist or even an engineer. Excellent texts meet a certain standard we set here; Texts should be concise to respect the reader's time and occupations, interdisciplinary, at least relating mathematical tools between areas of mathematics:
* Later volumes of Landau on emergent/less fundamental physics refer to more engineering applications and numerical analysis (vols. 5-10)
* The numerical analysis texts make an effort to discuss the geometry of tensor spaces, or preserve geometric structures in numerical integration
* Lang's algebra text contains examples and applications in geometry and number theory throughout
* Vaisman emphasizes the typically algebro-geometric method of sheaves in a differential geometry setting and to develop the theory of multiple sorts of manifolds
* We choose physics texts which connect to heavy mathematical machinery such as curvature and covariant derivatives in general relativity or symplectic/variational geometry in mechanics
* The algebraic topology texts are not "pure" either - focusing on applications to differential or algebraic geometry, and many more. Β 


Thus, the structure of this book list will be centered around core topics in theoretical physics which are already given direct connection to technology and reality, and the mathematics that follows from the theory rather than simply chasing popular formalisms. Future additions and pages will make an effort to connect with more project-based and experimental content, as our goal of demonstrating computational aspects (PDEs, Representation Theory, Numerical Analysis) have been fully satisfied and the supplementary material nearly completed. This pertains to our last criterion that there should be some elementary aspects in a text - showing the translation of the abstract machinery into basic computations to make the relationships with other areas even more transparent. We make no claim to complete coverage, especially since one needs to actively engage in experiment and new theoretical calculations to get further into a topic and this is a life's work. We do try to present essential mathematics and promising new directions.
Also see this [[Watch|list of video lectures]].


== Related Lists ==


Fredric Schuller's [[Watch|video lectures]] concisely summarize various algebraic and geometric constructions that commonly appear in theoretical physics.
A further set of texts extending this one, but working with the same basics has been produced leading all the way up and through gauge field theory, quantum mechanics, algebraic geometry, and quantum field theory [http://sheafification.com/the-fast-track/ here]
Β 
A [https://sheafification.com/the-fast-track/ related set of texts] works with the same basics to lay a path through gauge field theory, quantum mechanics, algebraic geometry, and quantum field theory.


== List Structure ==
== List Structure ==


Calculus is not in the pictured starter pack because it is found more often in high school curricula, while linear algebra (''despite being core to "applied mathematics" topics such as engineering, numerical computing, and statistics'') is often missing at the required level of rigor. Thus, we suggest looking at any '''Basic Mathematics''' to quickly fill in your gaps and as a source of quick and dirty computational techniques used universally.
The '''Royal Road to Differential Geometry and Physics''' is the list's core. While on that track, you should refer to the '''Fill in Gaps''' and '''Backbone''' sections as needed or as you desire.
Β 
The decalogy by '''Landau''' are the list's core. While on that track, you should start dipping into the texts listed under the Landau volumes to enhance your perspective on repeated readings. Books in the top rows are generally more basic and can be read at the same time, giving important insight into the structure of the subjects.


The '''General Mathematics''' section covers the knowledge that would be acquired in standard (but basic) graduate courses on the different areas of mathematics that later develop into modern topics, and should be developed alongside Landau.
The '''Fill in Gaps''' section covers the knowledge acquired in a strong high school mathematics education. Refer to it as needed, or begin there to develop your core skills.


The '''Aspirational''' section contains some of the big ideas, which may be repeated from earlier but deserve emphasis. These are the triumphs of mathematics, peaks that everyone deserves to climb.
The '''Backbone''' section supports the ideas in the '''Royal Road'''. Refer to it to strengthen your understanding of the ideas in the main track and to take those ideas further.


== Basic Mathematics ==
== Fill in Gaps ==
<div class="flex-container" style="clear: both;">
<div class="flex-container" style="clear: both;">
{{BookListing
{{BookListing
Line 37: Line 25:
| title = === Basic Mathematics ===
| title = === Basic Mathematics ===
| desc = Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.
| desc = Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.
}}
{{BookListing
| cover = Shilov Linear Algebra Cover.jpg
| link = Linear Algebra (Book)
| title = === Linear Algebra ===
| desc = Linear algebra of linear equations, maps, tensors, and geometry by Georgi Shilov.
}}
}}
{{BookListing
{{BookListing
Line 48: Line 30:
| link = Calculus (Book)
| link = Calculus (Book)
| title = === Calculus ===
| title = === Calculus ===
| desc = Overview of single and multi-variable calculus with applications to differential equations and probability by Tom Apostol.
| desc = Overview of Calculus by Tom Apostol.
}}
}}
</div>
</div>
== Landau ==
== Royal Road to Differential Geometry and Physics ==
<div class="flex-container">
<div class="flex-container">
{{BookListing
| cover = Lawvere Sets for Mathematics Cover.jpg
| link = Sets for Mathematics (Book)
| title = === Sets for Mathematics ===
| desc = Categorical approach to set theory by F. William Lawvere.<br>
'''Backbone reference:'''
* [[{{FULLPAGENAME}}#Set Theory and Metric Spaces|Set Theory and Metric Spaces]]
* [[{{FULLPAGENAME}}#Foundations of Analysis|Foundations of Analysis]]
}}
{{BookListing
| cover = Shilov Linear Algebra Cover.jpg
| link = Linear Algebra (Book)
| title = === Linear Algebra ===
| desc = Overview of linear algebra by Georgi Shilov.
}}
{{BookListing
{{BookListing
| cover = Landau Course in Theoretical Physics V1 Cover.jpg
| cover = Landau Course in Theoretical Physics V1 Cover.jpg
| link = Mechanics (Book)
| link = Mechanics (Book)
| title = === Mechanics ===
| title = === Mechanics ===
| desc = Classical mechanics of particles by Lev Landau.<br>
| desc = Classical mechanics of physics by Lev Landau.<br>
<div class="flex-container" style="clear: both;">
'''Prerequisite:'''
{{BookListing
* [[{{FULLPAGENAME}}#Calculus|Calculus]]
| cover = Mechmath.jpg
'''Backbone reference:'''
| link = Mechanics (Book)#Applications
* [[{{FULLPAGENAME}}#Ordinary Differential Equations|Ordinary Differential Equations]]
| title =
| desc = Symplectic geometry and other mathematical Structures of Classical Mechanics
}}
</div>
}}
}}
{{BookListing
{{BookListing
Line 71: Line 64:
| link = The Classical Theory of Fields (Book)
| link = The Classical Theory of Fields (Book)
| title = === The Classical Theory of Fields ===
| title = === The Classical Theory of Fields ===
| desc = Classical field theory of electromagnetism and general relativity by Lev Landau.<br>
| desc = Physics by Lev Landau.<br>
<div class="flex-container" style="clear: both;">
'''Prerequisite:'''
{{BookListing
* [[{{FULLPAGENAME}}#Linear Algebra|Linear Algebra]]
| cover = Fieldsmath3.jpg
| link = The Classical Theory of Fields (Book)#Applications
| title =
| desc = Differential/Riemannian geometry and other mathematical Structures in Relativistic Field Theory
}}
}}
</div>
}}
{{BookListing
| cover = Landau Quantum Mechanics.jpg
| link = Quantum Mechanics (Book)
| title = === Quantum Mechanics ===
| desc = Quantum Mechanics of particles, atoms, molecules by Landau and Lifshitz<br>
<div class="flex-container" style="clear: both;">
{{BookListing
{{BookListing
| cover = Quantmathv3.jpg
| cover = Bishop Tensor Analysis Cover.jpg
| link = Quantum Mechanics (Book)#Applications
| link = Tensor Analysis on Manifolds (Book)
| title = Β 
| title = === Tensor Analysis on Manifolds ===
| desc = Generalized functions, spectra of self-adjoint operators, and other mathematical Structures in Quantum Mechanics
| desc = Tensor analysis by Richard Bishop and Samuel Goldberg.<br>
}}
'''Prerequisite:'''
</div>
* [[{{FULLPAGENAME}}#Linear Algebra|Linear Algebra]]
'''Backbone reference:'''
* [[{{FULLPAGENAME}}#Principles of Mathematical Analysis|Principles of Mathematical Analysis]]
* [[{{FULLPAGENAME}}#Topology: A Categorical Approach|Topology: A Categorical Approach]]
}}
}}
{{BookListing
{{BookListing
| cover = Landau 4 Quantum Electrodynamics cover.jpg
| cover = Sternberg Differential Geometry Cover.jpg
| link = Quantum Electrodynamics (Book)
| link = Lectures on Differential Geometry (Book)
| title = === Quantum Electrodynamics ===
| title = === Lectures on Differential Geometry ===
| desc = Quantum Electrodynamics by Landau, written by Berestetskii, Lifshitz, and Pitaevskii.<br>
| desc = Differential geometry by Shlomo Sternberg.<br>
<div class="flex-container" style="clear: both;">
'''Prerequisite:'''
{{BookListing
* [[{{FULLPAGENAME}}#Linear Algebra|Linear Algebra]]
| cover = QFTmathv2.jpg
'''Backbone reference:'''
| link = Quantum Electrodynamics (Book)#Applications
* [[{{FULLPAGENAME}}#Principles of Mathematical Analysis|Principles of Mathematical Analysis]]
| title =
* [[{{FULLPAGENAME}}#Topology: A Categorical Approach|Topology: A Categorical Approach]]
| desc = Axiomatic and Geometric Structures in Quantum Field Theory
}}
</div>
}}
}}
{{BookListing
{{BookListing
| cover = Landau statistical physics.jpg
| cover = Vaisman Cohomology and Differential Forms Cover.jpg
| link = Statistical Physics (Book)
| link = Cohomology & Differential Forms (Book)
| title = === Statistical Physics ===
| title = === Cohomology & Differential Forms ===
| desc = Statistical Physics by Landau and Lifshitz.<br>
| desc = Cohomology and differential forms by Isu Vaisman.<br>
<div class="flex-container" style="clear: both;">
'''Backbone reference:'''
{{BookListing
* [[{{FULLPAGENAME}}#Algebra: Chapter 0|Algebra: Chapter 0]]
| cover = Statmechmath.jpg
* [[{{FULLPAGENAME}}#Algebra|Algebra]]
| link = Statistical Physics (Book)#Applications
| title =
| desc = Symplectic statistical mechanics, lattice, and stochastic quantization methods
}}
}}
</div>
</div>
}}
Β 
{{BookListing
== Backbone ==
| cover = Landau 6 fluid mechanics cover.jpg
<div class="flex-container">
| link = Fluid Mechanics (Book)
| title = === Fluid Mechanics ===
| desc = Fluid Mechanics by Landau and Lifshitz.<br>
<div class="flex-container" style="clear: both;">
{{BookListing
{{BookListing
| cover = Fluidsmath.jpg
| cover = Kaplansky Set Theory and Metric Spaces Cover.jpg
| link = Fluid Mechanics (Book)#Applications
| link = Set Theory and Metric Spaces (Book)
| title =
| title = === Set Theory and Metric Spaces ===
| desc = Topological hydrodynamics, statistical methods of turbulence, cloud physics, geometric and tensor numerical analysis
| desc = Set theory and metric spaces by Irving Kaplansky.
}}
</div>
}}
{{BookListing
| cover = Landau 7 elasticity cover.jpg
| link = Theory of Elasticity (Book)
| title = === Theory of Elasticity ===
| desc = Theory of Elasticity by Landau and Lifshitz.
}}
}}
{{BookListing
{{BookListing
| cover = Landau 8 electrodynamics of continuous media cover.jpg
| cover = E Landau Foundations of Analysis Cover.jpg
| link = Electrodynamics of Continuous Media (Book)
| link = Foundations of Analysis (Book)
| title = === Electrodynamics of Continuous Media ===
| title = === Foundations of Analysis ===
| desc = Electrodynamics of Continuous Media by Landau, Lifshitz, and Pitaevskii.
| desc = Analysis, intro to numbers, by Edmund Landau.
}}
}}
{{BookListing
{{BookListing
| cover = Landau 9 statistical physics part 2 cover.jpg
| cover = Rudin Principles of Mathematical Analysis Cover.jpg
| link = Statistical Physics part 2 - quantum theory (Book)
| link = Principles of Mathematical Analysis (Book)
| title = === Statistical Physics part 2 ===
| title = === Principles of Mathematical Analysis ===
| desc = Statistical Physics part 2 by Landau and Lifshitz.<br>
| desc = Mathematical analysis by Walter Rudin.
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Condensedmath.jpg
| link = Statistical Physics part 2 - quantum theory (Book)#Applications
| title =
| desc = High Tc superconductivity, holography and string theory in condensed matter, emergent gauge fields/Chern-Simons, random matrix theory, the Quantum Hall Effect
}}
</div>
}}
{{BookListing
| cover = Landau 10 physical kinetics cover.jpg
| link = Physical Kinetics (Book)
| title = === Physical Kinetics ===
| desc = Physical Kinetics by Landau and Lifshitz.<br>
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Kineticsmath.jpg
| link = Physical Kinetics (Book)#Applications
| title =
| desc = Open quantum systems, stochastic differential equations, plasma physics, combustion, cloud physics, and stochastic processes on manifolds
}}
</div>
}}
</div>
Β 
== General Mathematics ==
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Lawvere Sets for Mathematics Cover.jpg
| link = Sets for Mathematics (Book)
| title = === Sets for Mathematics ===
| desc = Categorical approach to set theory by F. William Lawvere.
}}
}}
{{BookListing
{{BookListing
Line 192: Line 126:
| title = === Ordinary Differential Equations ===
| title = === Ordinary Differential Equations ===
| desc = Ordinary differential equations by Vladimir Arnold.
| desc = Ordinary differential equations by Vladimir Arnold.
}}
{{BookListing
| cover = Bradley Bryson Terrilla Topology A Categorical Appoach Cover.jpg
| link = Topology: A Categorical Approach (Book)
| title = === Topology: A Categorical Approach ===
| desc = Topology by Tai-Danae Bradley, Tyler Bryson, Josn Terrilla. [https://topology.mitpress.mit.edu/ Click here for the Open Access version.]
}}
}}
{{BookListing
{{BookListing
Line 206: Line 146:
}}
}}
{{BookListing
{{BookListing
| cover = Brown groupoids cover.jpg
| cover = Aluffi Algebra Chapter 0 Cover.jpg
| link = Topology and Groupoids (Book)
| link = Algebra Chapter 0 (Book)
| title = === Topology and Groupoids ===
| title = === Algebra Chapter 0 ===
| desc = Topology and Groupoids by Ronald Brown. [https://groupoids.org.uk/pdffiles/topgrpds-e.pdf Click here for the Open Access version.]
| desc = Algebra by Paolo Aluffi. Easier than Lang's, but less direct.
}}
{{BookListing
| cover = Sternberg Differential Geometry Cover.jpg
| link = Lectures on Differential Geometry (Book)
| title = === Lectures on Differential Geometry ===
| desc = Differential geometry by Shlomo Sternberg.
}}
{{BookListing
| cover = Gelfand Generalized Functions vol 1 cover.png
| link = Generalized Functions (Book Series)
| title = === Generalized Functions: Properties and Operations ===
| desc = Generalized Functions: Properties and Operations by Israel Gel'fand and Georgi Shilov.
}}
{{BookListing
| cover = Gelfand Generalized Functions vol 2 cover.png
| link = Generalized Functions (Book Series)
| title = === Generalized Functions: Spaces of Fundamental and Generalized Functions ===
| desc = Generalized Functions: Spaces of Fundamental and Generalized Functions by Israel Gel'fand and Georgi Shilov.
}}
}}
{{BookListing
{{BookListing
Line 234: Line 156:
| title = === Algebra ===
| title = === Algebra ===
| desc = Algebra by Serge Lang. The most direct approach to the subject.
| desc = Algebra by Serge Lang. The most direct approach to the subject.
}}
{{BookListing
| cover = Bott and Tu Differential Forms in Algebraic Topology.jpg
| link = Differential Forms in Algebraic Topology (Book)
| title = === Differential Forms in Algebraic Topology ===
| desc = Differential Forms in Algebraic Topology by Raoul Bott and Loring Tu.
}}
{{BookListing
| cover = Fulton-Harris Representation Theory cover.jpg
| link = Representation Theory (Book)
| title = === Representation Theory ===
| desc = Representation Theory by William Fulton and Joe Harris.
}}
{{BookListing
| cover = Manin algebra v cover.jpg
| link = Algebra V: Homological Algebra (Book)
| title = === Algebra V: Homological Algebra ===
| desc = Algebra V: Homological Algebra by Sergei Gelfand and Yuri Manin.
}}
{{BookListing
| cover = Hartshorne Algebraic Geometry cover.jpg
| link = Algebraic Geometry (Book)
| title = === Algebraic Geometry ===
| desc = Algebraic Geometry by Robin Hartshorne.
}}
{{BookListing
| cover = Manin number theory cover.jpg
| link = Introduction to Modern Number Theory (Book)
| title = === Introduction to Modern Number Theory ===
| desc = Introduction to Modern Number Theory by Yuri Manin and Alexei Panchishkin.
}}
{{BookListing
| cover = Vaisman Cohomology and Differential Forms Cover.jpg
| link = Cohomology & Differential Forms (Book)
| title = === Cohomology & Differential Forms ===
| desc = Cohomology and differential forms by Isu Vaisman. Sheaf theoretic description of the cohomology of real, complex, and foliated manifolds.
}}
{{BookListing
| cover = May A Concise Course in Algebraic Topology cover.jpg
| link = A Concise Course in Algebraic Topology (Book)
| title = === A Concise Course in Algebraic Topology ===
| desc = A Concise Course in Algebraic Topology by Peter May.
}}
</div>
== Aspirational ==
Here are some more awesome books.
=== Quantum Fields Beyond Landau ===
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Dewitt global qft 1 cover.jpg
| link = The Global Approach to Quantum Field Theory (Book Series)
| title = === The Global Approach to Quantum Field Theory ===
| desc = The Global Approach to Quantum Field Theory by Bryce DeWitt.
}}
{{BookListing
| cover = Nima grassmannian scattering cover.jpg
| link = Grassmannian Geometry of Scattering Amplitudes (Book)
| title = === Grassmannian Geometry of Scattering Amplitudes ===
| desc = Grassmannian Geometry of Scattering Amplitudes by Nima Arkani-Hamed, Jacob Bourjaily, Freddy Cachazo, Alexander Goncharov, Alexander Postnikov, Jaroslav Trnka .
}}
</div>
=== Mathematics ===
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Brown NAT cover.jpg
| link = Nonabelian Algebraic Topology (Book)
| title = === Nonabelian Algebraic Topology ===
| desc = Nonabelian Algebraic Topology by Ronald Brown, Philip Higgins, and Rafael Sivera.
}}
{{BookListing
| cover = Kacvertex cover.jpg
| link = Vertex Algebras for Beginners (Book)
| title = === Vertex Algebras for Beginners ===
| desc = Vertex Algebras for Beginners by Victor Kac.
}}
{{BookListing
| cover = Frenkelvertex cover.jpg
| link = Vertex Algebras and Algebraic Curves (Book)
| title = === Vertex Algebras and Algebraic Curves ===
| desc = Vertex Algebras and Algebraic Curves by Edward Frenkel and David Ben-Zvi.
}}
{{BookListing
| cover = HHR Kervaire cover .jpg
| link = Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem (Book)
| title = === Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem ===
| desc = Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem by Michael A. Hill, Michael J. Hopkins, Douglas C. Ravenel.
}}
{{BookListing
| cover = Mcduff jholomorphic cover.jpg
| link = J-holomorphic Curves and Symplectic Topology (Book)
| title = === J-holomorphic Curves and Symplectic Topology ===
| desc = J-holomorphic Curves and Symplectic Topology By Dusa McDuff and Dietmar Salamon.
}}
{{BookListing
| cover = Fukaya intersection cover.jpg
| link = Lagrangian Intersection Floer Theory: Anomaly and Obstruction (Book Series)
| title = === Lagrangian Intersection Floer Theory: Anomaly and Obstruction ===
| desc = Lagrangian Intersection Floer Theory: Anomaly and Obstruction by Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono.
}}
{{BookListing
| cover = Hermann Geometric Computing Science cover.jpg
| link = Geometric Computing Science (Book)
| title = === Geometric Computing Science ===
| desc = Geometric Computing Science, Interdisciplinary Mathematics XXV by [[Robert Hermann]].
}}
}}
</div>
</div>
== Honorable Mentions ==
The following are some other good books, which are either redundant or otherwise didn't fit into the main collection of texts.(Olver PDEs, Coxeter books to be inserted)
{{SHORTDESC:The starter pack to physics and differential geometry.}}


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