Abstract. In the first half of my talk I will tell a cute and simple story - how given a knot in R3 one may count all possible "cosmic coincidences" associated with that knot, and how this count, appropriately packaged, becomes an invariant Z with values in some space A of linear combinations of certain trivalent graphs.
In the second half of my talk I will describe (rather sketchily, I'm afraid) a part of the story surrounding Z and A: How the same Z also comes from quantum field theory, Feynman diagrams, and configuration space integrals. How A is a space of universal formulas which make sense in every metrized Lie algebra and how specific choices for that Lie algebra correspond to various famed knot invariants. How Z solves a universal topological problem, and how solving for Z is solving some universal Lie-algebraic problem. All together, this is the u-story.
In the remaining time I will mention several other Z's and A's and the parallel (yet sometimes interwoven) stories surrounding them - the v-story, and w-story, and perhaps also the p-story. Each of these stories is clearly still missing some chapters.
Talk video. Handout. CC.html, CC.pdf, CC.png. Source files. CC.zip. Pensieve. 2011-03.