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Explore advanced topics in harmonic analysis and partial differential equations through this comprehensive collection of lectures delivered during a trimester program at the Hausdorff Research Institute for Mathematics. Delve into cutting-edge research presented by leading mathematicians including Terence Tao on equidistribution estimates for primes, Ben Green on covering congruences, James Maynard on gaps between primes, and many other distinguished speakers. Master fundamental concepts in nonlinear dispersive equations, singular integral operators, spectral multipliers, and boundary value problems for elliptic systems through detailed presentations by experts like Sebastian Herr, Pascal Auscher, and Michael Christ. Investigate the dynamics of critical generalized KdV equations with Yvan Martel, examine decoupling theorems and applications with Ciprian Demeter, and study extremal functions and bounds for the Riemann zeta function with Emanuel Carneiro. Discover recent progress on Onsager's conjecture through Tristan Buckmaster's work, explore maximal functions along curves with James Wright, and understand global well-posedness results for various nonlinear equations including the cubic Dirac equation and Kadomtsev-Petviashvili problems. Learn about applications ranging from fiber optics and water wave models to vortex filament dynamics and free boundary problems, while gaining insights into uniform distribution theory, lacunary Fourier series, and multi-frequency NLS limits from world-renowned researchers in mathematical analysis.
