This is a research-level graduate class geared towards advanced graduate students. The topics change from year to year, to reflect latest developments in the field, exciting topics, and topics of most interest. In the past, the first half of the course has focused on Convex Analysis, developing the geometry of convexity and proving the basic duality results of convex sets and then convex functions. The goal is to impart an appreciation of the questions, tools, and techniques central to convex analysis, to enable the student to go on to read the foundational texts on the topic. After this, the course then considers advanced special topics in optimization. In the past, this list has included: Moment problems and sum of squares polynomials, applications of Semidefinite optimization, Robust optimization and applications, combinatorial optimization.