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Deformation Theory Seminar

Monday, April 18, 2022 - 2:00pm

Arthemy Kiselev

Bernoulli Inst


University of Pennsylvania

Part III essentially self-contained

   Kontsevich's directed graphs are a way to represent non-commutative associative star-products using pictures: every such directed graph encodes an expression which is a differential-polynomial with respect to the coefficients of Poisson brackets (that are placed in the internal
vertices) and which is a bi-differential operator with respect to the content of the graph sinks.
   We design and implement an algorithm which, given a graph encoding, offers several nice drawings in the LaTeX picture environment, the most economical way to draw pictures in scientific texts. We thusobtain the drawings of those 247 Kontsevich's graphs with 4 internal vertices which do show up at order h^4 in the star-product.
   The talk builds around this visualisation problem for a very
special class of directed graphs (and particularly the LaTeX encoding format of their drawings). By learning how to draw hundreds and thousands of Kontsevich's weighted graphs we gain a better understanding of these objects --in fact, Feynman diagrams !-- in deformation quantisation; e.g., the graphs can be tame, neutral, or wild (and then, vacuum, gauge, or zero). We explore why Kontsevich's Formality theorem itself suggests how the graphs in star-products want to be drawn in the upper half-plane and what we can learn about the(linear constraints upon their) largely unknown (ir?)rational weightsand Riemann zeta values.
   The mathematical rendering of `beauty' is the hardest challenge. (The talk is based on recent joint work with S.Kerkhove (Utrecht).)