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Explore the mathematical foundations of anomalous diffusion in turbulent systems through this advanced lecture from the Centre International de Rencontres Mathématiques. Delve into the construction of incompressible vector fields that exhibit properties characteristic of fully turbulent velocity fields, where the associated scalar advection-diffusion equation generically displays anomalous diffusion behavior. Learn about a novel analytical framework for studying anomalous diffusion through a backward cascade of renormalized eddy viscosities, with proofs developed using "fractal" homogenization techniques that perform cascades of homogenizations across arbitrarily many length scales. Examine how anomalous diffusion serves as the fundamental ansatz of phenomenological theories in passive scalar turbulence, gaining insights into the mathematical structures underlying these complex physical phenomena. This presentation was recorded during the thematic meeting "Physics and Mathematics of hydrodynamic and wave turbulence" and represents cutting-edge research at the intersection of mathematical analysis and turbulence theory.
