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Linear algebra, Mechanics, Relativity and Fields, Differential Geometry
The starter pack to physics and differential geometry.

The starter pack to physics and differential geometry.

Philosophy[edit]

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.

Thus, the structure of this book list will be centered around core topics in theoretical physics which have 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.

Related Lists[edit]

Fredric Schuller's video lectures concisely summarize various algebraic and geometric constructions that commonly appear in theoretical physics.

A 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[edit]

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 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 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.

Basic Mathematics[edit]

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Basic Mathematics

Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.

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Linear Algebra

Linear algebra of linear equations, maps, tensors, and geometry by Georgi Shilov.

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Calculus

Overview of single and multi-variable calculus with applications to differential equations and probability by Tom Apostol.

Landau[edit]

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Mechanics

Classical mechanics of particles by Lev Landau.

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Symplectic geometry and other mathematical Structures of Classical Mechanics

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The Classical Theory of Fields

Classical field theory of electromagnetism and general relativity by Lev Landau.

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Differential/Riemannian geometry and other mathematical Structures in Relativistic Field Theory

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Quantum Mechanics

Quantum Mechanics of particles, atoms, molecules by Landau and Lifshitz

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Generalized functions, spectra of self-adjoint operators, and other mathematical Structures in Quantum Mechanics

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Quantum Electrodynamics

Quantum Electrodynamics by Landau, written by Berestetskii, Lifshitz, and Pitaevskii.

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Axiomatic and Geometric Structures in Quantum Field Theory

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Statistical Physics

Statistical Physics by Landau and Lifshitz.

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Symplectic statistical mechanics, lattice, and stochastic quantization methods

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Fluid Mechanics

Fluid Mechanics by Landau and Lifshitz.

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Topological hydrodynamics, statistical methods of turbulence, cloud physics, geometric and tensor numerical analysis

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Theory of Elasticity

Theory of Elasticity by Landau and Lifshitz.

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Electrodynamics of Continuous Media

Electrodynamics of Continuous Media by Landau, Lifshitz, and Pitaevskii.

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Statistical Physics part 2

Statistical Physics part 2 by Landau and Lifshitz.

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High Tc superconductivity, holography and string theory in condensed matter, emergent gauge fields/Chern-Simons, random matrix theory, the Quantum Hall Effect

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Physical Kinetics

Physical Kinetics by Landau and Lifshitz.

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Open quantum systems, stochastic differential equations, plasma physics, combustion, cloud physics, and stochastic processes on manifolds

General Mathematics[edit]

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Sets for Mathematics

Categorical approach to set theory by F. William Lawvere.

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Ordinary Differential Equations

Ordinary differential equations by Vladimir Arnold.

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Complex Analysis

Complex analysis by Lars Ahlfors.

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Applications of Lie Groups to Differential Equations

Applications of Lie Groups to Differential Equations by Peter Olver.

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Topology and Groupoids

Topology and Groupoids by Ronald Brown. Click here for the Open Access version.

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Lectures on Differential Geometry

Differential geometry by Shlomo Sternberg.

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Generalized Functions: Properties and Operations

Generalized Functions: Properties and Operations by Israel Gel'fand and Georgi Shilov.

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Generalized Functions: Spaces of Fundamental and Generalized Functions

Generalized Functions: Spaces of Fundamental and Generalized Functions by Israel Gel'fand and Georgi Shilov.

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Algebra

Algebra by Serge Lang. The most direct approach to the subject.

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Differential Forms in Algebraic Topology

Differential Forms in Algebraic Topology by Raoul Bott and Loring Tu.

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Representation Theory

Representation Theory by William Fulton and Joe Harris.

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Algebra V: Homological Algebra

Algebra V: Homological Algebra by Sergei Gelfand and Yuri Manin.

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Algebraic Geometry

Algebraic Geometry by Robin Hartshorne.

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Introduction to Modern Number Theory

Introduction to Modern Number Theory by Yuri Manin and Alexei Panchishkin.

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Cohomology & Differential Forms

Cohomology and differential forms by Isu Vaisman. Sheaf theoretic description of the cohomology of real, complex, and foliated manifolds.

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A Concise Course in Algebraic Topology

A Concise Course in Algebraic Topology by Peter May.

Aspirational[edit]

Here are some more awesome books.

Quantum Fields Beyond Landau[edit]

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The Global Approach to Quantum Field Theory

The Global Approach to Quantum Field Theory by Bryce DeWitt.

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Grassmannian Geometry of Scattering Amplitudes

Grassmannian Geometry of Scattering Amplitudes by Nima Arkani-Hamed, Jacob Bourjaily, Freddy Cachazo, Alexander Goncharov, Alexander Postnikov, Jaroslav Trnka .

Mathematics[edit]

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Nonabelian Algebraic Topology

Nonabelian Algebraic Topology by Ronald Brown, Philip Higgins, and Rafael Sivera.

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Vertex Algebras for Beginners

Vertex Algebras for Beginners by Victor Kac.

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Vertex Algebras and Algebraic Curves

Vertex Algebras and Algebraic Curves by Edward Frenkel and David Ben-Zvi.

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Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem

Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem by Michael A. Hill, Michael J. Hopkins, Douglas C. Ravenel.

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J-holomorphic Curves and Symplectic Topology

J-holomorphic Curves and Symplectic Topology By Dusa McDuff and Dietmar Salamon.

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Lagrangian Intersection Floer Theory: Anomaly and Obstruction

Lagrangian Intersection Floer Theory: Anomaly and Obstruction by Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono.

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Geometric Computing Science

Geometric Computing Science, Interdisciplinary Mathematics XXV by Robert Hermann.

Honorable Mentions[edit]

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)