Within the greater than a hundred years because the basic staff used to be first brought via Henri Poincaré it has developed to play an enormous function in numerous parts of arithmetic. initially conceived as a part of algebraic topology, this crucial idea and its analogies have came upon various functions in arithmetic which are nonetheless being investigated this present day, and that are explored during this quantity, the results of a gathering at Heidelberg college that introduced jointly mathematicians who use or research primary teams of their paintings with an eye fixed in the direction of purposes in mathematics. The e-book recognizes the numerous incarnations of the elemental workforce: pro-finite, ℓ-adic, p-adic, pro-algebraic and motivic. It explores a wealth of issues that variety from anabelian geometry (in specific the part conjecture), the ℓ-adic polylogarithm, gonality questions of modular curves, vector bundles in reference to monodromy, and relative pro-algebraic completions, to a motivic model of Minhyong Kim's non-abelian Chabauty process and p-adic integration after Coleman. The editor has additionally integrated the abstracts of all of the talks given on the Heidelberg assembly, in addition to the notes on Coleman integration and on Grothendieck's primary team with a view in the direction of anabelian geometry taken from a chain of introductory lectures given by means of Amnon Besser and Tamás Szamuely, respectively.
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