Explore the interplay between notions of convexity in complex, symplectic, and contact geometries in this lecture by Yakov Eliashberg from Stanford University. Delve into classical concepts such as holomorphic, polynomial, rational convexity, and pseudo-convexity in complex geometry, and discover their counterparts in symplectic and contact geometries. Examine the importance of understanding the relationships between these notions across different fields. Learn about pseudo convexity, strict convexity, rational convexity, and geometric notions of convection. Investigate convex intellectic manifolds, domains of age principles, and rational contexts. Study theorems of polynomial and rational convexity, as well as contact convexity in contact manifolds. Explore concepts like dividing sets, vector fields, characteristic relations, and semisymmetry. Analyze examples, arbitrary decompositions, and contact boundaries. Conclude with soft and hard proofs of static artistic relations in higher dimensions. Read more