Bott Periodicity Models Consciousness? Preliminary Exploration
Channel: Math 4 Wisdom
Date: 26/11/2023

Does Bott periodicity have metaphysical significance?  The divisions of everything, the cognitive frameworks which define mental states in Wondrous Wisdom, likewise form an eight cycle.

I, Andrius Kulikauskas, present to John Harland my understanding of Bott periodicity, how to calculate the eightfold pattern using Clifford algebras, and what clues suggest that indeed this may express my model of the mental states on which the Unconscious, Conscious and Consciousness operate.
October 20, 2023. 

Transcript with slides and links
https://www.math4wisdom.com/wiki/Exposition/20231126BottPeriodicity

I hope this video helps you understand this challenging phenomenon and encourages you to reach out to us so that we can study together!  Thank you for leaving comments!

0:50 Bott periodicity (BP) was discovered by Raoul Bott and published in 1957-1959.
1:20 Homotopy theory is a tool kit in algebraic topology. Homotopy group of a circle.
3:10 Definition of stable orthogonal group.
4:35 Homotopy groups of O(∞).
7:30 Homotopy groups of spheres. 8-fold BP appears there.
11:00 In 1989, I discovered an eight-cycle of cognitive frameworks. In 2016, I learned of BP..
12:15 Divisions of everything. Their eightfold periodicity.  
13:55 Symmetric spaces can be read off a chain of 8 embeddings amongst orthogonal, unitary and symplectic groups.
16:25 Clifford algebras likewise manifest eightfold BP.
18:15 Quotients of Lie groups define symmetric spaces. John Baez's talk.  Definition of symmetric spaces.
22:05 Clifford algebras make this more concrete.
22:50 BP is a statement in K-theory and vector bundles.  
23:45 Grassmannian generalizes the binomial theorem, which generates the subsets of a set. Gaussian binomial coefficients count the subspaces of a vector space over a finite field. 
25:35 Grassmannian are manifolds of the subspaces of a vector space. BP organizes the different flavors of Grassmannians.
26:50 The binomial theorem applies to generators of Clifford algebras.  I think the pattern of groups Z_2, Z_2, 0, Z, 0, 0, 0, Z, refers to metaphysical flavors of choices.
27:50 Imagine an 8-track mind. Each choice is not independent because it affects how many more choices the mind can make.
28:45 Neurologically, the brain may have a map of this eightfold mind.
29:20 There is a two-fold unitary BP. 8+2=10. Topological insulators have a Tenfold Way relevant for quantum computing. Classification of spaces of Hamiltonians.
34:00 Why are there 4 classical Lie families?
35:00 How to organize all of math and its branches? Lie theory links algebra and analysis.
36:50 What are the symmetries of math itself? Root systems of classical Lie families relate counting forwards and backwards.
44:20 Inverse matrices expressed by transposes, conjugate transposes, quaternionic transposes. Symmetries of math itself.
46:30 How to calculate the pattern for BP? Paper by Attiyah, Bott, Shapiro. Exposition by Dexter Chua. Book "Fibre Bundles" by Dave Husemoller. 
47:35 How matrices fit together. Z_2 signifies that a 2x2 representation of complex numbers contains two 1x1 representations of real numbers. Grothendieck groups express this obtusely.
51:00 Z arises because R+R has two representations.52:40 0 arises when matrices have the same size.
53:15 Overviewing the eightfold pattern.
55:00 Clifford algebra extends the binomial theorem.
57:00 Binomial theorem here is asymmetric, thus the pseudoscalar is key. Anticommutativity.
58:25 Calculating Clifford algebras as matrix algebras using recursion relations. 
59:10 2x2 real matrices have two generators which square to +1 but their product squares to -1.
1:00:10 If one generator squares to +1 and the other to -1, then their product squares to -1. This also defines 2x2 matrices.
1:00:50 Grading distinguishes odd and even basis elements. 
1:02:00 Keep track of generators that square to +1 or -1. Recursion relations tensor with 2x2 real matrices or with quaternions.
1:05:00 Morita equivalence. Categorically, 16x16 matrices in R are indistinguishable from R itself. 
1:06:10 4 generators that square to -1 yields the same 2x2 matrices of quaternions as do 4 generators that square to +1.
1:06:35 Groups A_k in terms of the inclusion map for Clifford algebras, and the reverse map for their even graded representations.
1:10:15 Brings to mind the twosome for existence by which opposites coexist or all is the same.
1:11:20 Explicit representations (matrices) and implicit structure (complex, quaternion) are separately expressed and related by the pattern.

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