Schubert varieties in flag manifolds are a remarkable family of complex projective varieties indexed by permutations or more generally by elements in Weyl groups. Many properties of these varieties and their cohomology rings can be deduced from the combinatorial data in the indexing set. In this course, we will give a very concrete introduction to these varieties, their cohomology rings, Schubert calculus, and the structure of their tangent spaces. In the past 5 years, there has been a lot of research activity in computations for higher cohomology theories for Schubert varieties such as equivariant cohomology and K-theory. We will cover the main tools being used which include the Knutson-Tao puzzles and the Littlemann paths. We will summarize the current open problems in the field and the most recent work.

The material included in this proposed course is closely related to the representation theory of GLn and Young tableaux that will be presented in Monty McGovern's course in the fall. His course would be very beneficial though not required.