Seeing Geometry Through Combinatorics: Diagrams, Algorithms, and Symmetry

My work is about turning complicated algebraic and geometric objects into concrete combinatorial models that let us see structure, compute invariants, and connect to geometry and representation theory. In this talk, I will talk about several different but connected polynomials arising from Schubert calculus, which originated in enumerative geometry.  We will look at various combinatorial models, including diagrams (representing cells in a plane) and tableaux (arrangements of numbers in boxes), with many examples. These discrete objects, along with operations defined on them, exhibit rich combinatorial properties. They can be used to simplify computations, extract algebraic and geometric information, and uncover hidden symmetries within these polynomials.