Schur Positivity and Recent Trends in Algebraic Combinatorics

Schur positivity of symmetric functions plays a crucial role in many outstanding problems of algebraic combinatorics, with natural ties with several other areas of mathematics and theoretical physics. After recalling the necessary background for symmetric function calculations, I recall how one may derive interesting enumerative combinatorics identities from symmetric function identities involving positive Schur function expressions. Next, I explain why such expressions are very rare in general, but frequent in the right context. I then go on with a survey of recent history and of some of the important current open problems regarding Schur positivity, for instance in rectangular Catalan combinatorics. If time allows several new conjectures are presented, as well as a general framework that unifies them.

Lecture 1. Appetizers
Lecture 2. Main Courses
Lecture 3. Desserts