Abstract (for two lectures): My subject is Schubert varieties and their classes. I shall speak about  some combinatorial rules for multiplication of  Schubert classes in cohomology rings of flag varieties. I will also give a sketch of a recent proof by J. Huh of the positivity conjecture for Chern classes of Schubert cells and  varieties in Grassmannians. Here is a more detailed plan:

Lecture 1: Semisimple groups, Borel subgroups, Tori,  Weyl groups,
Root systems, Schubert varieties and classes, structure constants
of multiplication in the basis of Schubert classes, characteristic map, BGG-operators, the Chevalley formula. 

Lecture 2: Littlewood-Richardson rule, Knutson-Tao puzzles , hives,...
An outline of Huh's proof.