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数学

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Andrius Kulikauskas

  • ms@ms.lt
  • +370 607 27 665
  • My work is in the Public Domain for all to share freely.

Lietuvių kalba

Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software


See: Math, Divisions

Investigation: Relate Bott periodicity and the eight-cycle of divisions of everything.


博特周期性定理


  • palyginti susijusias Lie grupes (ir jų ryšį su gaubliu) su požiūrių permainomis
  • Max Karoubi savo video paskaitoje paminėjo loop lygtį žiedams kurioje R,C,H,H' ir epsilon = +/-1 gaunasi 10 homotopy equivalences. Kodėl 10? 8+2=10? ar 6+4=10, dešimt Dievo įsakymų?
  • How might the fourfold periodicity of the sign of the pseudovector be related to the fourfold periodicity of the differentiation of sine and cosine functions?
  • Does the constraint {$J^2=−I_n$} on complex structures and their anti-commutativity relate to the constraints on Clifford algebras?

Study and understand:

{$$\dots\rightarrow \pi_nF\rightarrow \pi_nE\rightarrow \pi_nB\rightarrow \pi_{n-1}F\rightarrow \dots$$}

  • Characteristic class
  • Clifford algebras, clock shifts
  • Topological K-theory, infinite unitary group, classifying spaces.

Videos

Statement

Expositions

Extensions

Proofs

Related concepts

Math facts

  • Bott periodicity is based on the fourfold periodicity of the sign of the pseudovector.
  • Start from C - as the basic duality - and end up at C inverted (perhaps the conjugate) and go back. This circle is used to list the quotients of Clifford modules {$GL_{n}(\mathbb{R})/GL_{n}(\mathbb{C})$} check? etc If you multiply all the quotients together than you get the identity (?)

{$\begin{pmatrix} & & \mathbb{C}_{n} & & \\ & \mathbb{H}_{n} & & \mathbb{R}_{n} & \\ \mathbb{H}_{n} \times \mathbb{H}_{n} & & & & \mathbb{R}_{n} \times \mathbb{R}_{n} \\ & \mathbb{H}_{n} & & \mathbb{R}_{n} & \\ & & \mathbb{C}_{n} & & \end{pmatrix}$}

Clifford algebra periodicity

  • C0 R
  • C1 C
  • C2 H
  • C3 H + H
  • C4 H(2)
  • C5 C(4)
  • C6 R(8)
  • C7 R(8) + R(8)
  • C8 R(16)

''C_{n+8} consists of 16 x 16 matrices with entries in Cn ! For a proof you might try

Generalized Clifford Algebra has clock-shift operators.

  • Some matrices describe the 8-cycle clock (the trolley stops).
  • Generalized Pauli matrices describe the 3 shifts (the trolley cars of different increments +1, +2, +3).

Complex structures

  • We call {$J$} a complex structure on {$R^n$} if {$J\in O(n)$} and {$J^2=−I_n$}. Denote the space of complex structures {$Ω_1(n)⊂O(n)$}.
  • Define {$Ω_k(n)$} to be the space of complex structures that anti-commute with fixed {$J_1,\dots ,J_{k-1}$}.
  • {$Ω_0 \cong Ω_8$}
  • Consider how complex structures relate to divisions of everything. Apparently, each {$J_i$} is a perspective. Anti-commutativity {$J_iJ_j = -J_jJ_i$} means that the composition of perspectives is inverted if the order is switched. So the matrix {$-I$} can be interpreted as an inversion of perspective, and thus, of chains of perspectives. A set of eight perspectives brings us back to no perspectives, which is to say, the default perspective at the origin.

Ideas

  • The relevant Lie groups are all rotations about a fixed origin. That fixed origin represents a universal, absolute perspective, God's perspective upon everything, God's knowledge of everything.
  • Divisions of everything are perhaps chopping up a sphere where the sphere is everything also circle folding
  • Bott periodicity should be related to the collapse of the eightsome into the nullsome, and thus the definition of contradiction
  • Complex case: 2-periodicity - divisions having 4 (nežinojimas) or 2 (žinojimas) representations. Real case: 8-peridocity.
  • Perspective arises because of base point - there is a fixed point for the isometries. We are that fixed point.

BottPeriodicity


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Puslapis paskutinį kartą pakeistas 2019 balandžio 06 d., 12:49
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