 
Summary: RESEARCH STATEMENT
DAVID ANDERSON
My research interests lie in algebraic geometry, combinatorics, and representa
tion theory. The mathematics I find most exciting involves interactions among these
subjects. My work aims to understand, as concretely as possible, the cohomology
rings of certain algebraic varieties. Specific problems include interpreting the coho
mology classes of subvarieties as polynomials, and computing structure constants
for the cohomology ring with respect to geometrically defined bases. Combina
torics permeates these questionsthe polynomials have interesting combinatorial
structure, and due to positivity theorems, one can interpret the structure constants
enumerativelyand the varieties in question are often closely related to the repre
sentation theory of reductive groups.
1. EQUIVARIANT GEOMETRY
1.1. Positivity in Schubert calculus. Let X be a Grassmannian, flag variety, or
more generally any projective homogeneous space. The Schubert varieties in X
are defined via incidence conditions with respect to a fixed flag (or as orbit closures
for the action of a Borel group); their classes w form an additive basis for the co
homology of X, as w runs through certain elements of a Weyl group. In Schubert
calculus, one seeks to understand the structure constants cw
u,v for multiplication in
