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Explore advanced mathematical concepts in interface evolution through this comprehensive lecture series from the Hausdorff Center for Mathematics' trimester program. Delve into foundational theories with Camillo De Lellis's four-part exploration of De Giorgi and Almgren's work in simplified settings, covering essential geometric measure theory concepts. Master Brakke's mean curvature flow through Yoshihiro Tonegawa's detailed four-part introduction, examining the mathematical framework for understanding how surfaces evolve over time. Investigate convex integration and mixing flows with Daniel Faraco's four-part series, exploring connections between partial differential equations and fluid dynamics. Examine specialized topics including free boundaries on lattices and their scaling limits, singular perturbed elliptic systems with asymptotic phase segregation, and regularity theory for nonlinear PDEs. Study advanced problems in mathematical analysis such as infinity-harmonic potentials in convex rings, borderline Sobolev inequalities on symmetric spaces, and regularity of free boundaries in obstacle problems. Explore cutting-edge research in fluid dynamics including the Muskat problem with different mobilities, conservation of energy and entropy in fluid dynamics, and thermodynamically consistent boundary conditions for Korteweg fluids. Investigate geometric problems including isometric embeddings, vorticity measures, and non-uniqueness results for transport equations. Conclude with advanced topics in interface modeling, including diffuse and sharp interface models for two-phase flows and renormalization techniques for the axisymmetric Euler equation.
