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Explore the mathematical foundations behind Hilbert's Sixth Problem in this 46-minute conference talk that examines the transition from microscopic interactions to macroscopic continuum laws. Delve into two fundamental approaches: the particle setting, which traces the path from Newtonian particle dynamics through Boltzmann's equation to fluid equations, and the wave setting, which derives wave kinetic equations from nonlinear dispersive equations. Learn about groundbreaking recent research that successfully resolves both problems, starting from hard sphere dynamics for particles and the cubic nonlinear Schrödinger (NLS) equation for waves. Discover how these two proof frameworks, while sharing similar structures, exhibit distinctly different mathematical features and techniques. Gain insights into the rigorous mathematical justification of how atomistic interactions give rise to the fundamental laws governing continuum motion, addressing one of the most significant challenges in mathematical physics posed by David Hilbert over a century ago.
