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# wClips-120321

#### Speaker: Zsuzsanna Dancso

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Section 4 - Algebraic Structures.

Announcements. small circle, UofT, LDT Blog (also here). Email Dror to join our mailing list!

Resources. How to use this site, Dror's notebook, blackboard shots.

### 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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0:00:00  Blackboard shots by Stephen Morgan.
0:03:41  "0-nary" operations (constants) are often called "nullary operation". --Drorbn 15:24, 25 March 2012 (EDT)
0:07:54  "Objects" in our context are just "objects" in the literal sense of the word. Mathematically it is the same as saying "elements". They have nothing to do with "objects" in a category. --Drorbn 15:24, 25 March 2012 (EDT)
0:22:35  "Addition" is never one of the operations in our structures - it is added later by allowing formal linear combinations of objects. I guess if you started with a structure that already has an addition operation - say, "connect sum" of knots - you'll have to rename it "original addition" so as to distinguish it from the "addition" we add later when we allow formal linear combinations.

When talking about powers of the augmentation ideal, we only use the original operations of our structure.

So in the example where the structure is a group $G$, its group-ring ${\mathbb Q}G$ is again a structure with just one binary operation (multiplication) plus an artificially-added auxiliary operation "addition" which does not participate in taking powers of the augmentation ideal.

An alternative to all that is to start with a structure whose ${\mathcal O}_\alpha$'s are linear spaces (or at least, ${\mathbb Z}$-modules) and all of whose operations are multi-linear. Here again "addition" will have a special role and will not participate in forming powers of the augmentation ideal. --Drorbn 15:46, 25 March 2012 (EDT)