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RESEARCH STATEMENT DAVID ANDERSON
 

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 questions--the polynomials have interesting combinatorial
structure, and due to positivity theorems, one can interpret the structure constants
enumeratively--and 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

  

Source: Anderson, Dave - Department of Mathematics, University of Washington at Seattle

 

Collections: Mathematics