06-1350/Class Notes for Thursday November 16

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Today's Agenda

Formulas are a Chore (Bore?)

  • Sweeping clean a tree and {\mathcal A}(\Gamma)\equiv{\mathcal A}(\uparrow_{b_1(\Gamma)}).
  • In the coordinates above, write the TR\Phi B relations in various algebraic notations.
    • R4: (1230)^\star B\cdot(1213)^\star B\cdot(1023)^\star\Phi=(1123)^\star\Phi\cdot(1233)^\star B or (B_{1a}B_{2a}\Phi_{1a}; B_{1b}B_{2b}; B_{1c}B_{2a}\Phi_{1b}; B_{2c}\Phi_{1c}) = (\Phi_{2a}B_{3a}; \Phi_{2a}B_{3b}; \Phi_{2b}B_{3c}; \Phi_{2c}B_{3c}).
    • R3: (1230)^\star B\cdot(1213)^\star B\cdot(1023)^\star B = (1123)^\star B\cdot(1203)^\star B\cdot(1231)^\star B or (B_{1a}B_{2a}B_{3a}; B_{1b}B_{2b}; B_{1c}B_{2a}B_{3b}; B_{2c}B_{3c}) = (B_{4a}B_{5a}B_{6a}; B_{4a}B_{5b}B_{6b}; B_{4b}B_{6c}; B_{4c}B_{5c}B_{6a}).
    • R2: (123)^\star B^\pm\cdot(132)^\star B^\mp=1_3 or (B^\pm_{1a}B^\mp_{2a}; B^\pm_{2b}B^\mp_{3c}; B^\pm_{1c}B^\mp_{2b}) = (1;1;1).
    • R1: (B^\pm_a; B^\pm_bB^\pm_c)=(1;T^{\pm 2}).
  • But for now, skip the writing of the following relations:
    • Symmetry of \Phi and of B^{\pm}.
    • u, d and \#
    • Idempotence for T, R, \Phi and B^{\pm}.
    • B^{\pm} in terms of \Phi and R and R in terms of T.

Exponentiation is a Miracle

  • Description of the problem.
  • Beads and strands.
  • The perturbative approach, linearization.
  • The syzygies: relations between the errors.
  • The Hochschild complex and homology.