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Overviewing Bott periodicity

I am exploring whether and how Bott periodicity may express the eight-cycle of mental states, the divisions of everyting.

Bott periodicity, considered broadly, refers to an eightfold periodicity of

• Morita equivalence classes of real Clifford algebras
• Classical families of compact symmetric spaces
• Topological insulators - ways that Hamiltonians can get along with time reversal (T) and charge conjugation (C) symmetry
• Ways that antiunitary operators (and, in the complex case, the unitary operators) commute with an irreducible unitary representation of a supergroup on a super Hilbert space
• Associative real super division algebras
• Real spin representations
• Homotopy groups of {$O(\infty)$}

These refer to entities over the real numbers. There also is a twofold Bott periodicity for entities over the complex numbers. Together they constitute what is known as the tenfold way, for example, of topological insulators.

Taken together these ten things form the Brauer-Wall monoid of {$\mathbb{R}$}.

Here are two academic presentations that I have given about this eight-cycle of divisions of everything.

• Time and Space as Representations of Decision Making
• Consciousness as the Social Awareness Schema of a Disembodying Mind

I will be studying how groups {$A_k=M(C_k)/i*M(C_{k+1})$} relate to John Baez's forgetful functor by which he defines symmetric spaces.

BottPeriodicity