Degeneracy loci, toric degenerations, and equivariant algebraic geometry

从根本上说，代数几何是对多项式方程组解的研究。我们感兴趣的系统通常涉及许多方程和许多变量，寻找简化它们的方法是很重要的。一种方法是将某些参数调整为零；如果做得很仔细，生成的系统保留了原始系统的许多特性，并且可能更易于分析。另一种简化的方法是在解集中寻找对称性，并利用这一点来减少问题并从较小的集合中提取有关原始解集的信息。这些技术，分别被称为“退化”和“等变定位”，可以协同使用：例如，一个可以退化到一个具有额外对称性的系统。这个项目将使用这些方法来研究枚举几何中几个有趣的问题。本课题的目的是利用等变和退化技术研究具有群作用的代数变异的几何问题。PI将使用等变上同调的方法以及牛顿-奥库科夫体理论研究旗型品种、舒伯特品种和环型品种。激励问题是确定一个变量的标准，以允许平面退化为环面变量，并产生一个更广泛的例子类别，其中牛顿-奥孔科夫体可以明确计算。PI还将继续他的退化轨迹的研究，并通过发展符号置换图的组合学来研究它们更精细的几何性质。最后，PI将探讨最近发展的运算k理论的应用，重点是在这种情况下等变局部化和黎曼-洛克定理之间的关系。

Fundamentally, algebraic geometry is the study of solutions to systems of polynomial equations. The systems of interest typically involve many equations and many variables, and it is essential to look for ways to simplify them. One way to do this is by tuning certain parameters to zero; if done carefully, the resulting system retains many properties of the original one, and may be more amenable to analysis. Another way to simplify is to look for symmetry in the set of solutions, and take advantage of this to reduce the problem and extract information about the original solution set from a smaller set. These techniques, respectively known as "degeneration" and "equivariant localization", can be used in concert: for example, one may degenerate to a system possessing extra symmetry. This project will use these methods to study several problems of interest in enumerative geometry.The aim of this project is to study the geometry of algebraic varieties with group actions using equivariant and degeneration techniques. The PI will study flag varieties, Schubert varieties, and toric varieties using methods from equivariant cohomology, as well as the theory of Newton-Okounkov bodies. Motivating problems are to determine criteria for a variety to admit a flat degeneration to a toric variety, and to produce a wider class of examples where the Newton-Okounkov body can be computed explicitly. The PI will also continue his study of degeneracy loci, and investigate their finer geometric properties by developing the combinatorics of diagrams for signed permutations. Finally, the PI will explore applications of the recently-developed operational K-theory, focusing on the relationship between equivariant localization and Riemann-Roch theorems in this context.