Epistemology m a t h 4 w i s d o m - g m a i l +370 607 27 665 My work is in the Public Domain for all to share freely. 读物 书 影片 维基百科 Introduction E9F5FC Questions FFFFC0 Software Investigation: Relate Bott periodicity to the eight-cycle of divisions of everything. 博特周期性定理 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? Spheres 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 Understand the Artin-Wedderburn theorem, starting with Wedderburn's original result. 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. Spinors What is the connection between Bott periodicity and spinors? See John Baez, The Octonions. Del Pezzo surfaces Do Del Pezzo surfaces from degree 2 to 9 manifest an eightfold phenomenon? Is that related to M-theory? How is it related to the Veronese surface and the conics? 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. Overviews Wikipedia: Bott periodicity theorem Raoul Bott. The Periodicity Theorem For The Classical Groups And Some Of Its Applications In 1970, Bott looked back at his original proof, reconsidered it in terms of functors, vector bundles and K-theory, and considered applications, such as parallelizability. Math Overflow. Overview of proofs of Bott-periodicity Emphasizes the role of Thom isomorphisms. Asking for the gist of why this should be periodic. Videos Ravi Vakil slides Symmetric spaces and loop space iteration. Complex and real. Focuses on double loop space iteration, using short exact sequences to show that mapping spheres into one space is the same as mapping points into another space. 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 Tai-Danae Bradley. "One-Line" Proof: Fundamental Group of the Circle. Tai-Danae Bradley, Bryson, Terilla. Chapter 6. Paths, Loops, Cylinders, Suspensions... Suspension-loop adjunction. Algebraic Topology July 2016 University of Chicago Peter May. Introduction to Algebraic Topology 10:30 Freudenthal Suspension theorem. {$X\wedge Y=X\times Y/X\cup Y$}, {$\Sigma X = X\wedge S^1$}, {$S^n\wedge S^1=S^{n+1}$} Peter May. An excellent overview of Bott periodicity. Text He goes through Bott's approach. Given symmetric space {$M$} and a triple {$ν = (P, Q; h)$} where {$h$} is the homotopy class of curves joining point {$P$} to {$Q$} in {$M$}. Define {$M^{\nu}$} as the set of all geodesics of minimal length which join {$P$} to {$Q$} and are in the homotopy class {$h$}. Bott shows that {$M^{\nu}$} is a symmetric space. {$\pi_k(M^{\nu})=\pi_{k+1}(M)$} for {\$0
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