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学习数学

Math Notebook: Investigations

Priorities: What to learn

• What are the fundamental math principles?
  • Study the theorems of plane geometry, universal hyperbolic geometry, etc.
  • Organize the most important math theorems and analyze their content in terms of more basic ideas.
• What are the elementary math concepts?
  • Study how important math concepts relate more basic concepts.
  • Relate basic concepts and more advanced concepts to duality.
• What is a perspective?
  • Short and long exact sequences.
  • Fiber bundles, vector bundles.
  • Homology and cohomology.
  • Tensors.
  • Sheaves.
• What is a geometry?
  • SL(2,C) and Mobius transformations.
  • Differences between affine, projective, conformal, symplectic geometry.
  • Classical Lie groups.
• 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.
  • Yates index set theorem.

Overview: Math to learn and use

• Understand the ways of figuring things out
  • binomial theorem: Polytopes, Coxeter groups, homology, Gauss-Bonet theorem, Euler's characteristic, Grassmanian
  • four classical Lie groups, Lie theory, Exceptional Lie groups, Triality
  • four geometries (affine, projective, conformal, symplectic) and how, in logic, they relate level and metalevel. Geometry
    • relation between discrete and continuous projective geometry
    • Symplectic geometry, Lagrangian mechanics, Hamiltonian mechanics
      • the use of variables
      • circle folding, origami and what their operations mean geometrically
  • Numbers: Reals, Complexes, Quaternions, Octonions, Cayley-Dickson construction, Associativity, duality-breaking
  • SL(2,C) and Möbius transformations as the basis for six transformations (reflection, shear, rotation, dilation, squeeze, translation). Visual complex analysis.
    • Network theory, Set theory axioms.
• Understand how math results unfold
  • math answers
    • A theory of what "equivalence" variously means
    • Foundations: Set theory, Models, Proof theory, Automata theory
  • duality and logic as the relationship between the concious and the unconscious
    • Langlands program, Poincare/Serre duality
    • Brownian motion for modeling the unconscious
    • Symmetric functions for understanding the representation theory of symmetric groups
    • Category theory: Voevodsky, Homotopy type theory, Eduardo Ochs
  • theorems
    • geometry theorems
    • Galois theory
    • Fundamental Theorem of Algebra
    • Spectral sequence in functional analysis.
    • Riemann-Zeta function hypothesis in number theory.
    • P vs NP
    • Qiaochu Yuan's reading list
• Understand perspectives
  • fiber bundles, vector bundles
  • sheaves in algebraic geometry
  • Grothendieck toposes, Caramello's work
  • Tensor, Triviality, Linear algebra
• Understand the divisions of everything
  • finite exact sequences
  • Bott periodicity, clock shifts in Clifford algebras, Geometric algebra.
    • higher order homotopy groups for sphere
  • Jacobi identity and universal hyperbolic geometry
  • Yates index theorem and recursive function theory
  • Yoneda lemma
  • Logic
• Understand representations
  • Six functor formalism
• Understand topologies
  • Mandelbrot set, Catalan for the properties of life
• Understand the eightfold way
  • Snake lemma, homology and cohomology
    • a unifying perspective on cohomology (or has Lurie already achieved this?)
• Model how God goes beyond himself
  • Make sense of the field with one element
    • Finite fields, Riemann hypothesis, modeling infinity with characteristic q
  • simplex -1, center and totality, Polytopes
• entropy as the basis for prayer: Information geometry
• Completely characterize an area of math such as plane geometry or chess

Overviews of Math and Its History

• Récoltes et Semailles, Part 1, Alexander Grothendieck. Also, translation into Spanish and other works.
  • Notes on the Life and Work of Alexander Grothendieck by Piotr Pragacz
• A View of Mathematics, Alain Connes
• Eugenia Cheng. Mathematics, Morally.
• Thurston. On Proof and Progress.

General resources for studying math

• Math books
  • Are there other nice math books close to the style of Tristan Needham?
• Websites
  • Kevin Brown collection of expositions of math

Fivesome: Classification of Sheffer polynomials

• Understand how Sheffer polynomials relate to one-parameter unitary groups, adjoint operators and Stone's theorem.

Foursome: Yoneda lemma.

• Understand the Yoneda lemma better.
• Understand Kan extensions and how they relate to the Yoneda lemma.
• Relate the univalence axiom and the Yoneda lemma.

Conceptions: Adjunctions

• Understand how adjunctions relate to adjoint operators.
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