Andrius Kulikauskas

  • m a t h 4 w i s d o m - g m a i l
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  • My work is in the Public Domain for all to share freely.


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Introduction E9F5FC

Questions FFFFC0



Modeling divisions of everything

  • In my understanding of Bott periodicity, is it the + signs that are growing or the - signs?
  • How is the Yates Index Set Theorem {$+3$} related to Bott periodicity?
  • Consider how Bott perioidicity relates to choice, the binomial theorem, Grassmannians and symmetric spaces.
  • Consider how Bott periodicity relates to God and the field with one element.

Building blocks

Matrix symmetry

  • Does the formula for the sign of the square of the pseudoscalar, based on {$n(n+1)/2$}, which sums the integers from {$1$} to {$n$}, relate to the diagonal plus upper triangular entries of a matrix? and to the number of independent equations for orthogonality of vectors?

Choice frameworks

  • How do the choice frameworks match up to Lie groups? and to Bott periodicity?
  • Is there a Clifford algebra {$Cl_{0,k}$} for which the identity and the pseudoscalar are symmetric in every way?


  • What is the significance (for Bott periodicity) of {$S^{n-1}=SO(n)/SO(n-1)$} ?

Fourfold periodicity

  • 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?
  • Where have I encountered the fourfold periodicity for the shuffle permutation?

Artin-Wedderburn theorem

Flavors of Bott periodicity

Representations of Clifford algebras

  • Compare John Baez's representations over Clifford algebras (the forgetful functor and symmetric spaces) with the Clifford modules described by Attiyah, Bott, Shapiro.


Del Pezzo surfaces

Related math

  • Relate Bott periodicity with the n-spheres from the 0-sphere to the the 7-sphere (and 8-sphere).
  • Does {$SO(8)$}, the unit octonions and triality relate to Bott periodicity?

Is Bott periodicity related to the triality of SO(8) ?

  • Atiyah: What does octonionification mean for Freudenthal's magic square?
  • How can the fact that {$\Gamma(\frac{1}{2})=\sqrt{\pi}$} relate {$\pi$} and {$e$}? Note that {$\Gamma(1)=0!$} and {$\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{2}$}. Consider the integral definition.



Homotopy groups

  • John Baez. Bott periodicity. Considering how {$\pi_0(O(\infty))$} relates to {$\mathbb{R}$}, {$\pi_1(O(\infty))$} relates to {$\mathbb{C}$}, {$\pi_3(O(\infty))$} relates to {$\mathbb{H}$} and {$\pi_7(O(\infty))=\pi_{-1}(O(\infty))$} relates to {$\mathbb{O}$} and the Grassmannian.
  • Caleb Ji. Various Statements of Bott Periodicity. A short overview of statements in homotopy theory and K theory.
  • Jonathan Block. The Bott Periodicity Theorem. First, periodicity allows one to deloop classifying spaces and thus define cohomology theories. Second, using periodicity, “wrong way” functoriality maps can be defined and these are of integral importance in the index theorem. Siebenmann periodicity states that for M a closed manifold {$S(M)∼=S(M×I4,M×∂I4)$}. A map {$S(M)→KO[12]∗(M)$} which intertwines the two periodicities. It should be noted that {$KO[1/2]$} is four periodic with the signature operator playing the same role as the inverse of the Bott element as the Dirac operator does for complex (or real) periodicity.

Loop spaces and suspension

Homology, cohomology

Symmetric spaces

  • John Milnor. Morse Theory. (AM-51), Volume 51 Contains a helpful proof of the Bott periodicity theorem, both real and complex, in terms of Morse theory, minimal geodesics and symmetric spaces. Relates the Lie groups to the symmetric spaces and loop spaces.

Lie group embeddings

Symplectic structures

Morse theory and geodesics

Real, Complex, Quaternions, Octonions

  • Eschenburg. Geometry of Octonions Relates the representations of Clifford algebras to the normed division algebras. Identifies the octonions with the representations for {$Cl_{0,4}$}, {$Cl_{0,4}$}, {$Cl_{0,4}$}, {$Cl_{0,4}$} and identifies {$M_2(\mathbb{O})$} with the representations for {$Cl_{0,8}$}. Gives a proof of Bott periodicity in terms of symmetric spaces, symmetric subspaces, poles and centrioles.
  • Arkadii Slinko. 1, 2, 4, 8,... What comes next? Parallelizability of spheres.

Octonionic line bundles

  • John Baez. OP1 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.

Clifford modules



Morita equivalences as quantum Hamiltonian reductions

CPT symmetry

Unit spheres




  • Max Karoubi mentioned in his video that loop equations for rings with R,C,H,H' and {$\epsilon = +/-1$} yields 10 homotopy equivalences.
  • Orthogonal add a perspective (Father), symplectic subtract a perspective (Son).
  • John Baez's comment about 4 dimensions being most interesting because half-way between 0 and 8. The Son turns around and reverses the Father.
  • In Bott periodicity, the going beyond oneself (by adding perspectives) and the going outside of oneself (by subtracting perspectives) is, in mathematics, extended to all possible integers, and thus the two directions are related in the four ways as given by the four classical Lie algebras, including gluing, fusing, folding.
  • In the eight-cycle of divisions, the going beyond oneself and the going back outside oneself are reminiscent of spinors like electrons.
  • The flip side of going beyond yourself - if you can add perspectives, then you can substract perspectives - on the flip side. Thus nullsome and foursome model the extremes: nullsome models the void of everything, foursome models the point of nothing. Thus adding a perspective takes us from the indefinite to the definite and back again. The three operations +1,+2,+3 make for a three-fold braid.
  • Goedel's incompleteness theorem. There is irrelevant, inaccessible knowledge, such as that which describe different implementation of equivalent rings (for Bott periodicity) where the equivalence means that they have the same (isomorphically, structurally) representations.
  • Bott periodicity exhibits self-folding. Note the duality of 1 with the pseudoscalar.

Raoul Bott

  • Raoul Bott mirė Carslbad, Kalifornijoje. Attiyah apie Bott gyvenimą. Nekrologas New York Times Dukra Jocelyn Scott gyveno Rancho Santa Fe, dirbo C.B.S. Scientific. Jisai gyveno: assisted living - Sunrise at La Costa, 7020 Manzanita St, Carlsbad, CA 92011-5123 Studied geodesics on {$SU(2)$}.

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