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A link between lambda calculus and maps

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In the talk, I will report on a surprising and still quite mysterious/tenuous connection between lambda calculus and the theory of maps on surfaces (described in an arxiv draft with Alain Giorgetti: arxiv.org/abs/1408.5028). A term of the lambda calculus is standardly said to be (beta-)normal if it is fully evaluated, and linear if every variable is used exactly once. Let us moreover call it “planar” if the order in which variables are used strictly follows a last-in, first-out discipline (in the sense that lambda corresponds to a “push”). I will begin by describing a simple type system that characterizes the normal planar lambda terms, as well as a planar diagrammatic syntax for such terms based on the machinery of string diagrams. Then I will review the definition of rooted planar maps, and Tutte’s (1968) procedure for decomposing rooted planar maps by number of edges. Finally, I will show how to build a bijection between rooted planar maps and normal planar lambda terms (with one free variable) by replaying Tutte’s analysis in lambda calculus.

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