Abstract. My talk will have two parts:
The work is very new. I'm quite confident of the overall picture but the details are subject to change.
- In the first part I will describe the (tentative and speculative) "Projectivization Paradigm", which says, roughly speaking, that everything graded and interesting is the associated graded of something plain ("ungraded", "global") and even more interesting. The paradigm is absolutely general, encompassing practically every algebraic structure that might exist, and there is a diverse base of interesting examples and candidates for future examples.
- In the second part I will describe my latest example of an instance of the Projectivization Paradigm: I will show that the projectivization of "the circuit algebra of welded tangles" describes a good part (and maybe, in the future, all) of the recent work by Alekseev and Torossian on associators and the Kashiwara Vergne conjecture. This is cool: it leads to a nice conceptual construction of tree-level associators which might even be brought to a closed form, and it seems like a step towards a better understanding of quantum universal enveloping algebras and the work of Etingof and Kazhdan.
Video. At MSRI's VMath. Experimentally, also on my site.
Source Files. wHandout.zip.
See also. Finite Type Invariants of W-Knotted Objects: From Alexander to Kashiwara and Vergne (very preliminary).