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Investigating Bott periodicity

Sets of Mutually Anticommuting Linear Complex Structures

• I have identified perspectives with linear complex structures.
• I can identify adding a perspective, as the operation +1, or going in the opposite direction around the Bott clock.
• I want to understand what is a shift in perspective?
  • I need to understand how the twosome relates two perspectives, a first and a second.
• I need to understand symmetric spaces.
• I need to understand how divisions of everything grow by adding one perspective at a time.
• I need to understand how divisions of everything are modeled by chain complexes and exact sequences.
• I need to look for such chain complexes / exact sequences in the Lie group embeddings.
• I need to understand how the eight perspectives collapse into one.
• I need to understand how that happens with the two kinds of Clifford algebras.
• I need to consider how that relates to the field with one element.
• I need to understand symmetric spaces in terms of various kinds of subspaces of spaces.
• Understand the products of mutually anticommuting linear complex structures {$J_1J_2\cdots J_k$}.
  • Calculate {$J_1J_2\cdots J_k$} as given in this table.
  • Study Stone, Chiu, Roy
  • Figure out what is a shift in perspective.
  • Study how a symmetric space describes the ways of adding a new anticommuting linear complex structure, thus a new perspective.
    • Study Eschenburg
• Draw intuition from the quantum symmetries, how they relate to the process of adding mutually anticommuting linear complex structures.
  • Moore. Quantum Symmetries and Compatible Hamiltonians
  • Bott Periodicity For Clifford Algebra Maniacs

Hopf Fibration

• perspective - circle, perspective on perspective - sphere, perspective on perspective on perspective - 3-sphere

Symmetric Spaces

• Understand how symmetric spaces variously express subspaces of vector spaces.
  • Moore. Quantum Symmetries and K-Theory.
  • Study how real structures (where {$T^2=+1$} or {$C^2=+1$}) and quaternionic structures (where {$T^2=-1$} or {$C^2=-1$}) are defined.
  • See the articles by Eschenburg.

Understand what is an isotropy group

• Understand what can be said about an isotropy group of an isotropy group.
• Specifically consider the case of {$O(4n)$}, {$U(2n)$} and {$Sp(n)$}.

Vector Bundles on Unit Spheres

Understand Bott periodicity in terms of vector bundles on unit sheres and how that relates to Clifford modules.

• Study Jost Eschenburg, Bernhard Hanke. Bott Periodicity, Submanifolds, and Vector Bundles
• Understand the clutching construction.

Octonions

Understand how the octonions organize Bott periodicity.

• Jost Eschenburg. Geometry of Octonions.
• John Baez. {$\mathbb{O}P^1$} and Bott Periodicity. Canonical line bundles {$L_\mathbb{R}$}, {$L_\mathbb{C}$}, {$L_\mathbb{H}$}, {$L_\mathbb{O}$} give elements {$ [L_\mathbb{R}]$}, {$ [L_\mathbb{C}]$}, {$ [L_\mathbb{H}]$}, {$ [L_\mathbb{O}]$} that generate, respectively, {$\widetilde{KO}(S^1)\cong\mathbb{Z}_2,\widetilde{KO}(S^2)\cong\mathbb{Z}_2,\widetilde{KO}(S^4)\cong\mathbb{Z},\widetilde{KO}(S^8)\cong\mathbb{Z}$}. The isomorphism {$\widetilde{KO}(S^n)\rightarrow \widetilde{KO}(S^{n+8})$} given by {$x\rightarrow [L_\mathbb{O}]$} yields Bott periodicity. The canonical octonionic line bundle over {$\mathbb{O}P^1$} generates Bott periodicity.

Krebs Cycle

• Understand the Krebs cycle and look for connections.
  • Compare the eightfold Nanorooms diagram with the ninefold Nick Lane diagram.
  • Consider how glycolysis may act through eight layers (as with geometry, deriving noncontradiction out of contradiction) and establish the eight-cycle.

Other mathematical expressions of the divisions of everything

Study how chain complexes and exact sequences carve up mental space.

• Threesome: Fiber bundle
• Foursome: Yoneda Lemma
• Fivesome: Helmholtz decomposition
• Sevensome: Snake Lemma

Gain intuition regarding the field with one element.

• Combinatorial interpretations.
• Buildings

Divisions of everything

• Systematize the examples of the three minds in the Theory Translator.