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Andrius Kulikauskas
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博特周期性定理
 Compare the related Lie groups (and their connections with spheres) to the six specifications of geometry, the six transformations of perspectives
 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 anticommutativity relate to the constraints on Clifford algebras?
 Relate Bott periodicity with the nspheres from the 0sphere to the the 7sphere (and 8sphere).
Videos
Statement
Expositions
 Allen Hatcher. Vector Bundles & KTheory. Welcome.
 Tammo. Representation Theory. Chapter 3 Wedderburn Theorem
 John Baez. Bott periodicity.
 John Baez. Octonions.
 John Baez. Octonion projective geometry.
 John Baez. Projective lines.
 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.
 John Baez. The Magic Square.
 John Baez about Clifford algebra periodicity
 Eschenbur. Geometry of Octonions
 Arkadii Slinko. 1, 2, 4, 8,... What comes next?
 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 operatordoes for complex (or real) periodicity.
 M.A. Aguilar, Carlos Prieto. Quasifibrations and Bott periodicity.
 Mark J. Behrens. A new proof of the Bott periodicity theorem. A simpler proof based on an appropriate restriction of the exponential map to construct an explicit quasifibration with base space U and contractible total space.
 Zachary Halladay. Bott periodicity and Ktheory
 Max Karoubi. Bott Periodicity in Topological, Algebraic and Hermitian KTheory
 Raoul Bott. The Periodicity Theorem For The Classical Groups And Some Of Its Applications Δ
 Shinsei Ryu. General classification of topological insulators and superconductors.
 Baez: 2) H. Blaine Lawson, Jr. and MarieLouise Michelson, "Spin Geometry", Princeton U. Press, Princeton, 1989. or 3) Dale Husemoller, "Fibre Bundles", SpringerVerlag, Berlin, 1994. These books also describe some of the amazing consequences of this periodicity phenomenon. The topology of ndimensional manifolds is very similar to the topology of (n+8)dimensional manifolds in some subtle but important ways!'' Physics of fermions.
Extensions
Proofs
Bottperiodicity also arises in string theory.
Related concepts
Math facts
 Two types of results
 Periodicity of the homotopy groups of the unitary groups.
 Periodic theorem of vector bundles.
 Twocycle
 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}$}
 Eightcycle
 The fourfold periodicity within the Bott periodicity is a repeating pattern (0,1,1,0 ?)
 W: Symmetric group Note that the reverse on n elements and perfect shuffle on 2n elements have the same sign; these are important to the classification of Clifford algebras, which are 8periodic.
 This pattern is based on the sign of the pseudovector.
 This pattern expresses whether the sum of 1+...+n is even or odd. Even + even = even; + odd = odd; + even = odd; + odd = even; and so on. This pattern is likewise the sum of the numbers mod 2. Thus 0+0=0; +1=1; +0=1;+1=0.
 This pattern also comes up in multiplying by i. We get 1, i, 1, i  two positives followed by two negatives.
 A similar pattern comes up in the volumes of nspheres. The volume of an nsphere is {$\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}$} where {$\Gamma(n+\frac{1}{2})=(n\frac{1}{2})(n\frac{3}{2})\cdots \frac{1}{2}\cdot\pi^{\frac{1}{2}}$}. Thus we have: {$\frac{2n}{2}=n, \frac{2n+1}{2}\frac{1}{2}=n, \frac{2n+2}{2}=n+1, \frac{2n+3}{2}\frac{1}{2}=n+1$}, and so on.
 The pattern also comes up by two recurrences for volume and hyperarea (surface area) in terms of each other. Starting with {$V_0(R)=1, A_0(R)=2$} and continuing {$V_{n+1}(R)=\frac{R}{n+1}A_{n}(R), A_{n+1}(R)=(2\pi R) V_{n}$} we get the pattern for the factor {$\pi$}.
Real numbers = slack, Complex numbers = null, Quaternions = perspective
Clifford algebra periodicity
 Readings
 W: Classification of Clifford algebras
 Baez: Clifford algebra periodicity
 José FigueroaO'Farrill. Spin Geometry
 H. Blaine Lawson, Jr. and MarieLouise Michelson, "Spin Geometry", Princeton U. Press, Princeton, 1989. (For proof)
 Dale Husemoller, "Fibre Bundles", SpringerVerlag, Berlin, 1994. (For proof)
 Jayme Vaz, Jr., Roldão da Rocha, Jr. An Introduction to Clifford Algebras and Spinors.
 The Wikipedia article explains that in the complex case the quadratic form there is no signature because {$+1 = i^2(–1)$}. Whereas in the real case the signature distinguishes between the number of {$+1$} and the number of {$–1$}.
 The pseudoscalar (the product of all the basis elements) is important because it is similar to the scalar. When the dimension is odd, then the pseudoscalar commutes with everything, and so it is in the center. When the dimension is even, then the pseudoscalar does not commute with everything and so the center only consists of the identity.
 In the real case, what characterizes the Clifford algebra of signature {$(p,q)$} is the number {$pq \text{mod} 8$}.
 There are the progressions, adding basis elements which square to {$1$}, but likewise in the {$+1$} direction:
{$$R, C, H$$}
{$$R, R \bigoplus R, \begin{pmatrix} R & R \\ R & R \end{pmatrix} $$}
{$$ H, \begin{pmatrix} C & C \\ C & C \end{pmatrix}, \begin{pmatrix} R & R & R & R \\ R & R & R & R \\ R & R & R & R \\ R & R & R & R \end{pmatrix} $$}
 Wedderburn's theorem states that The ArtinWedderburn theorem generalizes this result to Artinian rings.
 Frobenius's theorem for real division algebras states that every finitedimensional associative division algebra over the real numbers is isomorphic to the real numbers, the complex numbers or the quaternions.
 Hurwitz's theorem: The only real normed division algebras are R, C, H, and the (nonassociative) algebra O.
 A division algebra is an algebra over a field in which division, except by zero, is always possible.
 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 {$C_n$}.
 The positive and negative directions (for the signature) are related as {$R$} and {$M_8(R)$}.
 For any n=s+t (exploring how we generate the perspectives in a division of everything into n perspectives, going from 0 to n in n steps) there are two patterns:
 even n, {$X=2^{\frac{N}{2}}$}: {$\dots R(2X), R(2X), H(X), H(X), R(2X), R(2X), H(X), H(X) \dots$}
 odd n, {$X=2^{\frac{N1}{2}}$}: {$\dots H(X)\bigoplus H(X), C(2X), R(2X)\bigoplus R(2X), C(2X), H(X)\bigoplus H(X), C(2X), R(2X)\bigoplus R(2X) \dots$}
 Combined, these two patterns yield for n, along the edge:
 {$\dots R(2^N), C(2^N), H(2^N), H(2^N) \bigoplus H(2^N), H(2^{N+1}), C(2^{N+2}), R(2^{N+3}), R(2^{N+3})\bigoplus R(2^{N+3}) \dots$}
Generalized Clifford Algebra has clockshift operators.
 Some matrices describe the 8cycle clock (the trolley stops).
 Generalized Pauli matrices describe the 3 shifts (the trolley cars of different increments +1, +2, +3). See the "clock and shift matrices".
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 anticommute with fixed {$J_1,\dots ,J_{k1}$}.
 {$Ω_0 \cong Ω_8$}
 Consider how complex structures relate to divisions of everything. Apparently, each {$J_i$} is a perspective. Anticommutativity {$J_iJ_j = J_jJ_i$} 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 twocycle is: the undefined (the nullsome) yields the defined (the foursome), which yields the defined undefined (the eightsome), which is the same as the undefined (the nullsome).
 The eightcycle arises from the details: adding four perspectives yields the foursome, the defined. Then we can, dually, think of subtracting perspectives from that. But the subtraction (relaxing, removing perspectives) is defined by adding perspectives, yielding the fivesome which lets a sensor be free in time and space, the sixsome lets the conditional sensor relate to the unconditional, the sevensome lets the external unity (of the whole) match the internal unity (of slack) in a dualistic way, the eightsome has the system collapse.
 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: 2periodicity  divisions having 4 (nežinojimas) or 2 (žinojimas) representations. Real case: 8peridocity.
 Perspective arises because of base point  there is a fixed point for the isometries. We are that fixed point.
 Understand the dimensions of a Lie group as perspectives. And look at Lie groups as rotations of a sphere.
John Baez about Clifford algebra periodicity
Notes
Bott Periodicity
 John Baez's comment about 4 dimensions being most interesting because halfway between 0 and 8. The Son turns around and reverses the Father.
 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.
 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 eightcycle 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 threefold braid.
 R nullsome
 H twosome 1+i (different) j+ij (the same)
 H+H threesome splits the twosome
 (H2) foursome  internally doubles
 (C4) fivesome
 (R8) sixsome
 R+R sevensome (dividing the nullsome into two perspectives)
(S16) what would S mean? half of R? positive reals?
 The Elliptic Umbilic Diffraction Catastrophe. Optics, Bott periodicity?
 Study orthogonal groups and Bott periodicity.
 Allen Hatcher. Algebraic topology. Explanation Homotopy, homology, cohomology. We will show in Theorem 3.21 that a finitedimensional division algebra over R must have dimension a power of 2. The fact that the dimension can be at most 8 is a famous theorem of [Bott & Milnor 1958] and [Kervaire 1958]. Example 4.55: Bott Periodicity.
 Allen Hatcher. Vector Bundles and KTheory. (Halfwritten). Explanation Table of Contents Bottperiodicity.
 Bott periodicity exhibits selffolding. Note the duality with the pseudoscalar. Consider the formula n(n+1)/2 does that relate to the entries of a matrix?
 Bott periodicity is the basis for 8fold folding and unfolding.
 What is the connection between Bott periodicity and spinors? See John Baez, The Octonions.
 Orthogonal add a perspective (Father), symplectic subtract a perspective (Son).
 My dream: Sartre wrote a book "Space as World" where he has a formula that expresses Bott periodicity / my eightfold wheel of divisions.
 A question about the topological proofs of Bott periodicity Answer by Peter May.
 Matthew Magill. Topological Ktheory and Bott Periodicity
 Daisuke Kishimoto. Topological proof of Bott periodicity and characterization of BR.
 Dyer, Lashof. A topological proof of the bott periodicity theorems.
 Guo Chuan Thiang. Lecture notes on symmetries, topological phases and Ktheory. A broad and helpful overview of the math and physics, including the tenfold way, Clifford algebras.
 ketverybė: H, nulybė: R
 2periodicity: Dievas  gerumas  Dievas
 sukeisti buvimą ir nebuvimą, tapimą ir netapimą
 10 = 4+6 = 4+(4+2) = 8 + 2 (2 tai laisvumo atvaizdai)
 Z 0 Z 0 Z 0
 Z 0 (replace Z 0 with Z2 Z2 particle) Z 0 (replace Z 0 with 0 0 hole)
 8 external dimensions (adding 4 dimensions or +2 or +3)
 internally expressed dimensions  adding integrity
 same pattern reversed internally
 syllabus by Roberto Rubio relate to Bott periodicity.
 Langlands program related to BottAtiyah. The Yang Mills Equations Over Riemann Surfaces, Morse theory  calculus of variations, thus to Bott periodicity. And Langlands is related to conjugacy classes on GL(2), which are eignevalues. There are three families of double coverings.
 Bott periodicity 8=0 could be the basis for reproduction.
 https://sites.google.com/view/dmonaco/teaching/ktheoryincondensedmatterphysics
 https://nptel.ac.in/courses/111/106/111106100/
 Atiyah: Ktheory is related to quantum theory, cohomology is related to classical theory.
 Dirac operator is the basis for spinors and for Bott periodicity.
 Dirac operator is, conceptually, the square root of the Laplacian. In what sense is a tableaux the square root of a matrix?
 Atiyah: What does octonionification mean for the magic square?
Alain Connes about Bott periodicity and CTP. "Why Four Dimensions and the Standard Model Coupled to Gravity...
Attiyah, Bott, Shapiro. Clifford Modules. 1963.
 Clifford modules are representations of Clifford algebras.
 They describe the eightfold periodicity.
 Related to Adams's work on vector fields on spheres.
 The odd and even dimensional orthogonal groups are related to odd and even dimensional (crosspolytopehypercube) choice frameworks. How does the duality of center and totality that distinguishes these two choice frameworks add an additional distinction that makes for eightfold Bott periodicity?
 What is the significance (for Bott periodicity) of {$S^{n1}=SO(n)/SO(n1)$} ?
Shintaro FushidaHardy. Notes for MATH 282B Homotopy Theory.
 Chapter 2 on Fibre Sequences contains examples in Sections 2.5 and 2.6.
 Seems relevant for mastering Bott periodicity. Mentions the fibration for orthogonal groups.
Is Bott periodicity related to the triality of SO(8) ? and thus the unit octonions? See John Baez.
Theo JohnsonFreyd. The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions.
 We show that the Morita equivalences Cliff(4) ' H, Cliff(7) ' Cliff(−1), and Cliff(8) ' R arise from quantizing the Hamiltonian reductions R04//Spin(3), R07//G2,
and R08//Spin(7), respectively.
 the Morita equivalence {$\textrm{Cliff}(7) \simeq \textrm{Cliff}(−1)$} arises from the Hamiltonian reduction {$\mathbb{R}^{07}//G_2$}, where {$G_2 ⊆ SO(7)$} is the exceptional Lie group of automorphisms of the octonion algebra {$\mathbb{O}$}.
 Hamiltonian Reduction by Stages
 M. F. Atiyah, R. Bott, and A. Shapiro. “Clifford Modules”
 So(8) and Bott?
 Could Feynman diagrams for four fields express the foursome? How goes to What in one direction and Why goes to Whether in the opposite direction. And could Feynman diagrams for N fields express the division of everything into N perspectives? And does Bott periodicity apply?
 In my understanding of Bott periodicity, is it the + signs that are growing or the  signs?
 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.
 Krulewski bott proof
 J. F. Adams, Infinite Loop Spaces, Ann. of Math. Studies 90, 1978.
 4.66. If F→E→B is a fibration or fiber bundle with E contractible, then there is a weak homotopy equivalence F→ΩB .
 For each topological group G there is a fiber bundle G→EG→BG with EG contractible, hence by the proposition there is a weak equivalence G ≃ ΩBG. There is also a converse statement
 Combining this duality between Σ and Ω with the duality between fibers and cofibers, we see a duality relationship between the fibration and cofibration sequences of §4.3:
 One formulation of Bott periodicity describes the twofold loop space, {$\Omega ^{2}BU$} of {$BU$}. Here, {$\Omega$} is the loop space functor, right adjoint to suspension and left adjoint to the classifying space construction. Bott periodicity states that this double loop space is essentially {$BU$} again.
 In the corresponding theory for the infinite orthogonal group, O, the space BO is the classifying space for stable real vector bundles. In this case, Bott periodicity states that, for the 8fold loop space,
 https://en.m.wikipedia.org/wiki/Periodic_table_of_topological_invariants
 There are ten discrete symmetry classes of topological insulators and superconductors, corresponding to the ten Altland–Zirnbauer classes of random matrices.
 Cameron Krulewski. The KTheoretic Classification of Topological Materials
 https://en.m.wikipedia.org/wiki/Spin_representation
 Steven Lehar at PhilPeople
 Understand in what sense the Clifford algebra for the complex numbers is +1 and 1. We have {$e_ie_j=e_je_i$}. What is the orientation of {$e_ie_i$}?
 Understand the Bott periodicity for the Clifford algebras.
 Interpret Clifford algebras geometrically, what it means for an area to become zero, one or negative one.
 Representation of Lorentz group and Clifford algebra
Morita equivalence is an additive equivalence
 Presumably, an additive equivalence is one in which the equivalence is by way of an additive functor.
 Additive category An additive category is one such that all of its homsets are abelian groups, composition of morphisms is bilinear, and it admits all finitary biproducts.
 Additive functors A functor F: C → D between preadditive categories is additive if it is an abelian group homomorphism on each homset in C. If the categories are additive, then a functor is additive if and only if it preserves all biproduct diagrams.
 Additivity of adjoint functors All adjoint functors between additive categories must be additive functors.
 If R is Morita equivalent to S, then their centers C(R) and C(S) are isomorphic.
 Morita equivalence is not interesting in the case of commutative rings because then they are isomorphic.
 The ring {$M_n(R)$} of nbyn matrices in R is Moritaequivalent to R for any n>0. Work out the functors here.
 Morita equivalence could describe a noncontradictory model of contradiction.
Bott periodicity
Complex vector bundles
