Séminaire de Mathématique

Has anyone seen Kontsevich's star-product? or: How to draw Formality graphs nicely in LaTeX

by Arthemy Kiselev (Univ. of Groningen & IHES)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette

How can mathematical results be represented visually in such a way that much information can instantly be read off the images in minimal time and easily?
   How can visualization problems contribute to our understanding and progress in the theory from which these tasks arose?

   Built of wedges, like "V" or "Lambda", Kontsevich's directed graphs are a pictorial realization of non-commutative associative star-products: every such directed graph encodes an expression which is 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. More general Formality graphs can contain tridents or higher out-degree vertices; Kontsevich's Formality theorem itself suggests how, by the defition of their weights using integral formulas, the graphs in star-products want to be drawn in the upper half-plane with hyperbolic metric.

   The visualization problem which we solve is how the graphs in star-product theory can be drawn -- nicely! -- in large quantities by using the LaTeX {picture} environment, i.e. the most economical way to draw pictures in scientific texts. In a joint work with S.Kerkhove (Utrecht) we design and implement an algorithm which, given a graph encoding, offers its several drawings in the LaTeX picture environment.
   For graphs which do show up in star-products, we obtain the drawings up to order 4 in the deformation parameter. For similar graphs which never show up in the star-product (such as the vacuum diagrams) we explore their visual representations and properties.
   Finally, we examine how neural networks can be deployed in two classes of problems about Kontsevich's graphs and their weights in star-products.
   We shall discuss what the Mathematics of "beauty" is in graph visualization: `nice' is informative, `nice' is simple, `nice' is balanced, `nice' is flexible, `nice' is whole, `nice' is unexpected.


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Organized by

Maxim Kontsevich