BIT Numerical Mathematics

, Volume 44, Issue 3, pp 425-438

First online:

Trees, Renormalization and Differential Equations

  • Ch. BrouderAffiliated withLaboratoire de Minéralogie-Cristallographie, CNRS UMR7590, Universités Paris 6 et 7, IPGP, 4 place Jussieu

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The Butcher group and its underlying Hopf algebra of rooted trees were originally formulated to describe Runge–Kutta methods in numerical analysis. In the past few years, these concepts turned out to have far-reaching applications in several areas of mathematics and physics: they were rediscovered in noncommutative geometry, they describe the combinatorics of renormalization in quantum field theory. The concept of Hopf algebra is introduced using a familiar example and the Hopf algebra of rooted trees is defined. Its role in Runge–Kutta methods, renormalization theory and noncommutative geometry is described.

Hopf algebra Runge–Kutta methods noncommutative geometry renormalization