Multigraded Commutative Algebra

该奖项支持Maclagan的研究，以开发分次多项式环的几个基本性质的多级类似物。这是由交换代数中出现的许多自然多分次对象所激发的。一个重要的例子是单纯环面簇的cox齐次坐标环，它解决了关于环面簇上的层的代数几何问题，需要用交换代数技巧来研究。Maclagan研究的交换代数之外的应用包括代数几何(多分次Hilbert格式的显式构造，以及光滑Toric簇的投影正规性)和组合学(拟阵的性质)。她的一个项目允许对McKay对应研究中产生的模空间进行具体计算，McKay对应在过去20年一直是一个活跃的研究领域。这项研究位于交换代数与计算和组合代数几何的交集，在环面几何领域。这些领域最经典的版本研究多元多项式方程的解的几何。这样的问题已经被研究了几百年，但在诸如机器人学、统计学和编码理论的应用中继续出现。这里要研究的具体方面在最优化(通过环理想的整数规划)和理论物理(通过McKay对应)中都有应用。

This award supports the research of Maclagan to develop multigradedanalogues of several fundamental properties of graded polynomialrings. This is motivated by the many naturally multigraded objectsarising in commutative algebra. An important example is the Coxhomogeneous coordinate ring of a simplicial toric variety, whichallows algebro-geometric questions about sheaves on the toric varietyto be studied using commutative algebra techniques. Applicationsoutside commutative algebra of Maclagan's research include those toalgebraic geometry (explicit construction of multigraded Hilbertschemes, and projective normality of smooth toric varieties) andcombinatorics (properties of matroids). One of her projects allowsconcrete calculations of moduli spaces arising in the study of theMcKay correspondence, which has been an active area of research forthe last two decades.This research lies in the intersection of commutative algebra andcomputational and combinatorial algebraic geometry, in the area oftoric geometry. The most classical version of these fields studies thegeometry of solutions to polynomial equations in several variables.Such questions have been studied for hundreds of years, but continueto arise in applications such as robotics, statistics, and codingtheory. The particular aspects to be studied here have applicationsin optimization (integer programming through toric ideals) andtheoretical physics (through the McKay correspondence).