Convexity: Convex Analysis, Geometry and Optimization

Convexity lies at the heart of modern mathematics, extending its influence into analysis, geometry, and optimization. This course will explore various topics related to convexity, drawn from a wide range of fields yet unified by common analytical and geometric techniques. As a senior seminar course, our aim is not to cover one specific subject exhaustively, but rather to introduce diverse concepts that provide a broader understanding of the methods involved in addressing both theoretical and applied mathematical questions related to convexity. The course will broadly cover: --Convex Sets: Separation properties, Supporting Hyperplanes, Volume, Surface area, Minkowski Problem, Isoperimetric Inequality, Steiner’s formula, Caratheodory’s theorem, Duality. --Convex Functions: Derivatives, Rademacher’s theorem, Helly’s Theorem, Minmax principles, Duality, and Cyclic monotonicity. --Convex Optimization: Optimality conditions, Duality, Gradient methods, Subgradients, Newton’s method, and relevant applications.