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Explore advanced analytic techniques in dynamics and geometry through this comprehensive lecture on renormalization for nilflows and Weyl sums. Delve into topics such as motivation, Weyl sums, proof of liquid distribution, effective implementation, renormalization, inflows, Eisenberg inflows, Eisenberg organization, and scaling. Gain insights from Professor G. Forni of the University of Maryland as he presents a thorough examination of these complex mathematical concepts, providing a deeper understanding of their applications in the field.