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Explore the mathematical foundations of anomalous diffusion in passive scalar turbulence through this advanced lecture delivered at 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. This presentation forms part of the thematic meeting on "Physics and Mathematics of hydrodynamic and wave turbulence" and provides deep insights into the mathematical structures underlying turbulent phenomena, making it valuable for researchers and advanced students working in mathematical physics, fluid dynamics, and turbulence theory.
