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wClips-120321

Speaker: Zsuzsanna Dancso

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Videography by Karene Chu troubleshooting

Notes on wClips-120321:    [edit, refresh]

Section 4 - Algebraic Structures.


[edit]

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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Date Links
Jan 11, 2012 dbnvp 120111-1: Introduction.
dbnvp 120111-2: Section 2.1 - v-Braids.
Jan 18, 2012 dbnvp 120118-1: An introduction to this web site.
dbnvp 120118-2: Section 2.2 - w-Braids by generators and relations and as flying rings.
dbnvp 120118-3: Section 2.2 - w-Braids - other drawing conventions, "wens".
Jan 25, 2012 dbnvp 120125-1: Section 2.2.3 - basis conjugating automorphisms of F_n.
dbnvp 120125-2: A very quick introduction to finite type invariants in the "u" case.
Feb 1, 2012 dbnvp 120201: 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 dbnvp 120208: Review of u,v, and w braids and of Section 2.3.
Feb 15, 2012 dbnvp 120215: Section 2.5 - mostly compatibilities of Z^w, also injectivity and uniqueness of Z^w.
Feb 22, 2012 dbnvp 120222: 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 dbnvp 120229: 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 dbnvp 120307: Section 3.5: Jacobi diagrams and the bracket-rise theorem.
Mar 14, 2012 dbnvp 120314: Section 3.6 - the relation with Lie algebras.
Mar 21, 2012 dbnvp 120321: Section 4 - Algebraic Structures.
Mar 28, 2012 Out-of-sequence not-on-tape we watched the video of Talks: GWU-1203.
Apr 4, 2012 dbnvp 120404: Section 3.7 - The Alexander Theorem (statement).
Apr 18, 2012 dbnvp 120418: Aside on the Euler trick, the differential of \exp, and the BCH formula.
Apr 25, 2012 dbnvp 120425: Section 3.8, a disorganized lecture towards the proof of the Alexander theorem.
May 2, 2012 dbnvp 120502: Section 4: Algebraic structures (review), circuit algebras, v- and w-tangles.
May 10, 2012 dbnvp 120510: Sections 5.1 and 5.2: tangles, their projectivization and its relationship with Alekseev-Torossian spaces.
May 23, 2012 dbnvp 120523: Section 5.2: Proof of the relationship with A-T spaces.
May 30, 2012 dbnvp 120530: Interpreting {\mathcal A}^w(\uparrow_n) as a universal space of invariant tangential differential operators.
wClips Seminar Group Photo
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 [edit] Blackboard shots by Stephen Morgan.
0:03:41 [edit] "0-nary" operations (constants) are often called "nullary operation". --Drorbn 15:24, 25 March 2012 (EDT)
0:05:00 [add] Algebraic Structures.
0:07:54 [edit] "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)
0:09:49 [add] Examples - groups and group homomorphisms.
0:10:50 [edit] See http://www.math.toronto.edu/zsuzsi/research/thesis/.
0:14:03 [add] Examples - Actions, Quandles, QTGs, fibered situations.
0:19:06 [add] The augmentation ideal.
0:22:35 [edit] "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)

0:28:45 [add] ${\mathcal I}^m$ and the projectivization.
0:32:33 [add] Operations on proj.
0:39:32 [add] Groups and associative algebras.
0:43:34 [add] Quandles defined.
0:51:16 [add] Quandles defined (2).
0:56:20 [add] Quandles and Leibniz algebras.
1:01:41 [add] Quandles and Leibniz algebras (2).
1:05:50 [add] Quandles and Leibniz algebras (3).
1:14:45 [add] Quandles and Leibniz algebras (4).
1:21:05 [add] Quandles and Leibniz algebras (5).
1:26:07 [add] Expansions.
1:27:45 [add] Homomorphic expansions.