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Ideas to Understand

What is a perspective?

Fiber bundles, vector bundles.

• 影片: Daniel Chan. Vector bundles

Tensor

• 影片: Pavel Grinfeld. Tensor Calculus.
• 影片: Daniel Chan. Tensor products
• 影片: Daniel Chan. Tensor products of modules
• Tai-Danae Bradley. What's a Tensor Product?
• 影片: Tensors Bernd Sturmfels, Mateusz Michalek
• 书: Pavel Grinfeld. Introduction to Tensor Analysis and the Calculus of Moving Surfaces.

Algebraic geometry

• 维基百科: Moduli space
• 维基百科: Algebraic stack
• 维基百科: Scheme
• 维基百科: Grothendieck topology & Site
• 影片: Richard Borcherds. Galois theory, Algebraic geometry, Group theory Number theory

Sheaves.

• Classical algebraic geometry
  • 影片: Algebraic varieties Max Planck
• Grothendieck. Elements of Algebraic Geometry I
• 影片: Algebraic Geometry in the Time of Co-Vid
• 影片: Taylor Dupuy. Stacks.

Algebraic topology

• Norman Wildberger. Introduction to Algebraic Topology

Dimension

• 维基百科: N-sphere

Representation of topos

• Notes: Olivia Caramello. An Introduction to Grothendieck toposes 影片: Part 2 of 4

What is composition of perspectives?

Category theory

• 书: Tai-Danae Bradley. What is Applied Category Theory?
• 书: Emily Riehl. Category Theory in Context Kan extensions.
• 书: Tom Leinster. Basic Category Theory.
• 书: Brendan Fong, David I. Spivak. Seven Sketches in Compositionality: An Invitation to Applied Category Theory
• 书: Lawyere, Schanuel. Conceptual mathematics: An Introduction to Category Theory.
• 书: Steven Roman. An Introduction to the Language of Category Theory
• 影片: Programming with Categories Fong, Milewski, Spivak. Lecture 8: Adjunctions.
• 影片: Applied Category Theory Spivak, Fong.
• 影片: Seven Sketches in Compositionality download and increase volume
• 影片: Steven Roman. An Introduction to Category Theory
• 影片: Daniel Chan. Category theory
• 影片: Richard Southwell. Category theory for Beginners
• 影片: Math Proofsable
• Daniel Murfet. Foundations for Category Theory
• Evan Patterson. Categorical logic. List of resources.

Explanation

• Alexander Campbell. An Exegesis of Yoneda Structures.

Adjoint functors

• Applied Category Theory Course
• MS: Why should I care about adjoint functors
• Daniel Murfet lecture notes: Adjunctions
• Six functor formalism
  • Fausk, Hu, May. Isomorphisms Between Left and Right Adjoints

Yoneda Lemma and Lenses

• Daniel Murfet lecture notes: The Yoneda Lemma
• nLab: Lens Database views.
• Bartosz Milewski. Lenses, Stores and Yoneda.
• Bartosz Milewski. Understanding Yoneda.

String diagrams

• Amar Hadzihasanovic. Category theory and diagrammatic reasoning. 4. String diagrams and algebraic theories.
• 书: Topology: A Categorical Approach

Higher category theory

• Emily Riehl. ∞-Category Theory for Undergraduates.

Homotopy type theory

• Shulman. Homotopy type theory: The logic of space.

Statistics

What is a division of everything?

Chain complexes, homology and cohomology

• 维基百科: Chain complex
• Jean Gallier, Jocelyn Quaintance. A Gentle Introduction to Homology, Cohomology, and Sheaf Cohomology.
• 影片: Daniel Chan. Homological algebras
• 影片: Harpreet Bedi. Homology
• Daniel Murfet lectures on Groups, Categories & Homological Algebra
  • Beginnings of Homological algebra, complexes and their cohomology
  • The long exact sequence in homology
  • The snake lemma (Hilton & Stammbach)

Short and long exact sequences

• 维基百科: Exact sequence - examples:
  • Integers modulo two
  • {$0\rightarrow L^2 \xrightarrow{grad} \mathbb{H}^3 \xrightarrow{curl} \mathbb{H}^3 \xrightarrow{div} L^2 \rightarrow 0$}
    • 维基百科: Gradient, 维基百科: Curl, 维基百科: Divergence

Infinite three-cycle

• Understand the canonical way of transforming a short exact sequence to a long exact sequence. See: 维基百科: Derived functor
• 维基百科: Zig-zag lemma
• 维基百科: Mayer–Vietoris_sequence
• Supplement: The Long Exact Homology Sequence and Applications

What is the eight-cycle?

Bott periodicity

• Topology in Condensed Matter

Quadratic forms

Clifford algebras

• John Baez. Clifford algebras
• 书: Gabriel Weinreich. Geometrical vectors.
• 维基百科: Classification of Clifford algebras
• 影片: Urjit A. Yajnik. Theory of groups for physics applications. See: Transcripts.
  • LN: Nr.44 Urjit Yajnik. Representation of Lorentz Group and Clifford Algebra
• Article: Nikos Salingaros. On the classification of Clifford algebras and their relation to spinors in n dimensions. 1982
• 书: Clifford Algebras and the Classical Groups
• 书: H. Blaine Lawson, Marie-Louise Michelsohn. Spin Geometry. (PMS-38) 1989.
• 书: Pertti Lounesto. Clifford algebras and spinors.
• Andreas Dress. P: Distance Geometry and Geometric Algebra
• 影片: Balázs Csikós. Vector Fields on Spheres and Clifford Algebras

Wedderburn theorem on division algebras

• Jonathan Bloom. An Introduction to Wedderburn Theory & Group Representations.
• Matej Brešar. An elementary approach to Wedderburn’s structure theory.
• 书: Knapp. Advanced Algebra
  • Advanced Algebra. II. Wedderburn-Artin Ring Theory

Octonions

• 书: Conway, Smith. On quaternions and octonions. Amazon
• 书: Susumo Okubo. Introduction to Octonion and Other Non-Associative Algebras in Physics

N-spheres

• 维基百科: Hopf fibration Understand it.
• 维基百科: Hopf invariant and Adams's theorem.

What is the eightfold way?

维基百科: Snake lemma

What is a geometry?

SL(2,C) and Möbius transformations.

• 影片: Introduction to Complex Analysis
  • 影片: Möbius Transformations, Part 1
  • 影片: Möbius Transformations, Part 2
• Terrence Tao courses Complex analysis, real analysis, linear algebra, random matrices, topology, probability.
• 维基百科: Linear fractional transformation
• 维基百科: Modular group
• 维基百科: Modular form
• 书: Ramūnas Garunkštis. Modulinių formų įvadas.

Geometries Differences between affine, projective, conformal, symplectic geometry.

• 维基百科: Affine geometry
  • Videos: Daniel Chan. The algebra-geometry duality
  • 影片: Daniel Chan. Affine algebraic geometry
• 维基百科: Projective geometry
  • 影片: Daniel Chan. Projective geometry
  • 影片: Daniel Chan. Invariant theory
  • 影片: Richard Southwell. Geometry
• 维基百科: Conformal geometry
• 维基百科: Symplectic geometry
  • 影片: Tobias Osborne. Symplectic geometry & classical mechanics.
  • 影片: Helmut Hofer. First Steps in Symplectic Dynamics.
  • P: Barney Bramham, Helmut Hofer. First Steps Towards a Symplectic Dynamics
  • Josh Powell. Symplectic geometry in classical mechanics.
  • 影片: Hamiltonian systems and symplectic geometry Excellent introduction.
  • 维基百科: Tangent space
  • 维基百科: Exterior algebra
  • 维基百科: Exterior derivative
  • 维基百科: Differential form
  • 维基百科: Partition of unity
  • 维基百科: Stokes' theorem
  • 维基百科: Lie derivative
  • 维基百科: Symplectomorphism
  • 维基百科: Darboux's theorem
  • 维基百科: Frobenius theorem
  • 维基百科: Carathéodory–Jacobi–Lie theorem
  • 维基百科: Weyl algebra - Symplectic Clifford algebra
  • 书: V.I.Arnold. Mathematical Methods of Classical Mechanics.
  • 书: M.Nakahara. Geometry, Topology and Physics.
  • 书: A.Cannas da Silva. Lectures on Symplectic Geometry.

Classical Lie groups.

• 影片: Jonathan Evans. Lie groups and Lie algebras.
• 影片: Richard Borcherds. Lie groups.
• 书: Predrag Cvitanovic. Birdtracks, Lie’s, and Exceptional Groups.
• Keith Conrad. Complexification.
• 维基百科: Sylvester's Law of Inertia
• Espacios Vectoriales Topológicos. Mini Curso: Grothendieck... 1/6 (Fernando Zalamea)
• 影片: Jacob Lurie. Lie Algebras and Homotopy Theory.
• 维基百科: Whitehead bracket
• VU skaitykloje: Sigurdur Helgason. Differential geometry, Lie groups, and symmetric spaces.
• Sigurdur Helgason's course readings.
• Sigurdur Helgason. Sophus Lie, the mathematician.
• Sigurdur Helgason. Sophus Lie and the Role of Lie Groups in Mathematics.
• Sigurdur Helgason. A Centennial: Wilhelm Killing and the Exceptional Groups.
• Coxeter. Regular Polytopes. Chapter 11 on Lie theory.
• Cartan. La géométrie des groupes simples.

What is going beyond oneself?

Study finite fields and interpret {$F^{1^n}$}.

Study points at infinity and how they relate to coordinate systems.

What is the foursome?

Yoneda lemma.

Universality

• MIT Course. Random Matrix Theory and Its Applications.

Yates index set theorem.

What are the four geometries?

Relate quaternions and Mobius transformations.

• 影片: 3 Blue 1 Brown. Visualizing quaternions (4d numbers) with stereographic projection

Hopf algebras and real forms

• Grinberg, Reiner. Hopf algebras in combinatorics.
  • 影片: Federico Ardila. Hopf Algebras and Combinatorics, Index and Overview by Sara Billey

Lie theory

• 书: Brian Hall. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction.
• Robert Gilmore. Lie Groups, Physics, and Geometry: An Introduction for Physicists, Engineers and Chemists
• Roger Carter, Graeme Segal, Ian Macdonald. Lectures on Lie groups and Lie algebras

What is duality?

Representation theory

• Amritanshu Prasad. Representation Theory: A Combinatorial Point of View
• 影片: Representation Theory Max Planck

What is measurement?

Quantum physics

• Tobias Osborne. Quantum Field Theory. Notes
• Tobias Osborne. Notes on Relativistic Quantum Field Theory.
• Tobias Osborne. Advanced Quantum Physics Many particle systems. Lecture notes
• Brian Hall. Quantum Mechanics for Mathematicians.
• Roger Penrose. Road to Reality.
• 影片: Leonard Susskind. The Theoretical Minimum. Quantum mechanics.
• 影片: Baron Zwiebach. MIT. Quantum Physics III. Course
• MIT Open Courseware. Physics Department.

What is entropy?

Ergodic theory

• 影片: Ergodic theory Max Planck - Tatjana Eisner

Walks

• 影片: Random walks
• MIT Course: Universal random structured in 2D including walks on trees.
• MIT Course: Infinite random matrix theory.
• MIT Course: Random walks and diffusion.

How to think of sequences, hierarchies and networks?

• 影片: John Baez. Mathematics in the 21st Century

How to think of automata?

• Daniel Murfet's references on computation