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Investigating Bott periodicity

Sets of Mutually Anticommuting Linear Complex Structures

• 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

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.

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.