Infinities and Flows, Renormalization  and the Renormalization Group

Physical theories of matter are plagued with infinities. At the classical level these seem difficult to avoid, even at the conceptual level. In quantum physics the problem is attenuated since quantum objects are described as probability amplitudes only. The inifinities reappear in relativistic quantum field theory, due to the interplay of locality and the uncertainty principle. Renormalization theory permits to absorb the infinities in the physical parameters of the theory. Embedding renormalization theory into the renormalization group opens new perspectives on the problem, conceptually and mathematically.