With roots in classical enumerative geometry of the 19th century and Hilbert 15th problem, Schubert Calculus is nowadays understood as the study of classical and quantum intersection rings of flag manifolds. Among the most basic questions in this area is how many lines intersect 4 lines in the three dimensional space. A rigorous answer requires the study of the intersection ring of the Grassmann manifold of projective lines in the projective space, and to build a combinatorial model encoding the geometry of intersection ring.
There are more general intersection rings such as the K theory ring, and quantum versions of all of them. I study all these and more, using a mix of methods from Algebraic Geometry, Combinatorics, and Geometric Representation Theory. Often, my task is to combinatorialize a geometric problem, i.e. to find a combinatorial formula for a calculation in an intersection ring. I am especially interested in applications of Schubert Calculus to other areas, such as integrable systems or representation theory.