This award supports the research of Maclagan to develop multigraded analogues of several fundamental properties of graded polynomial rings. This is motivated by the many naturally multigraded objects arising in commutative algebra. An important example is the Cox homogeneous coordinate ring of a simplicial toric variety, which allows algebro-geometric questions about sheaves on the toric variety to be studied using commutative algebra techniques. Applications outside commutative algebra of Maclagan's research include those to algebraic geometry (explicit construction of multigraded Hilbert schemes, and projective normality of smooth toric varieties) and combinatorics (properties of matroids). One of her projects allows concrete calculations of moduli spaces arising in the study of the McKay correspondence, which has been an active area of research for the last two decades.
This research lies in the intersection of commutative algebra and computational and combinatorial algebraic geometry, in the area of toric geometry. The most classical version of these fields studies the geometry of solutions to polynomial equations in several variables. Such questions have been studied for hundreds of years, but continue to arise in applications such as robotics, statistics, and coding theory. The particular aspects to be studied here have applications in optimization (integer programming through toric ideals) and theoretical physics (through the McKay correspondence).
