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# wClips-120530

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Interpreting ${\mathcal A}^w(\uparrow_n)$ as a universal space of invariant tangential differential operators.

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### The wClips

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Jan 11, 2012 : Introduction.
: Section 2.1 - v-Braids.
Jan 18, 2012 : An introduction to this web site.
: Section 2.2 - w-Braids by generators and relations and as flying rings.
: Section 2.2 - w-Braids - other drawing conventions, "wens".
Jan 25, 2012 : Section 2.2.3 - basis conjugating automorphisms of $F_n$.
: A very quick introduction to finite type invariants in the "u" case.
Feb 1, 2012 : Section 2.3 - finite type invariants of v- and w-braids, arrow diagrams, 6T, TC and 4T relations, expansions / universal finite type invariants.
Feb 8, 2012 : Review of u,v, and w braids and of Section 2.3.
Feb 15, 2012 : Section 2.5 - mostly compatibilities of $Z^w$, also injectivity and uniqueness of $Z^w$.
Feb 22, 2012 : Section 2.5.5, $\alpha:{\mathcal A}^u\to{\mathcal A}^v$, and Section 3.1 (partially), the definition of v- and w-knots.
Feb 29, 2012 : Sections 3.1-3.4: v-Knots and w-Knots: Definitions, framings, finite type invariants, dimensions, and the expansion in the w case.
Mar 7, 2012 : Section 3.5: Jacobi diagrams and the bracket-rise theorem.
Mar 14, 2012 : Section 3.6 - the relation with Lie algebras.
Mar 21, 2012 : Section 4 - Algebraic Structures.
Mar 28, 2012 Out-of-sequence not-on-tape we watched the video of Talks: GWU-1203.
Apr 4, 2012 : Section 3.7 - The Alexander Theorem (statement).
Apr 18, 2012 : Aside on the Euler trick, the differential of $\exp$, and the BCH formula.
Apr 25, 2012 : Section 3.8, a disorganized lecture towards the proof of the Alexander theorem.
May 2, 2012 : Section 4: Algebraic structures (review), circuit algebras, v- and w-tangles.
May 10, 2012 : Sections 5.1 and 5.2: tangles, their projectivization and its relationship with Alekseev-Torossian spaces.
May 23, 2012 : Section 5.2: Proof of the relationship with A-T spaces.
May 30, 2012 : Interpreting ${\mathcal A}^w(\uparrow_n)$ as a universal space of invariant tangential differential operators.
Group photo on January 11, 2012: DBN, ZD, Stephen Morgan, Lucy Zhang, Iva Halacheva, David Li-Bland, Sam Selmani, Oleg Chterental, Peter Lee.
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0:00:07  Blackboard shots by Stephen Morgan.
0:12:18 [add] Review of wheels, trees, div, and j.
0:27:16 [add] The map into differential operators.
0:31:56 [add] The tangential vector fields.
0:40:37 [add] Verifying that the bracket goes to the commutator.
0:43:04 [add] Verifying that the bracket goes to the commutator (2).