Degeneracy loci, toric degenerations, and equivariant algebraic geometry

从根本上说，代数几何是研究多项式方程组的解。 感兴趣的系统通常涉及许多方程和许多变量，并且必须寻找简化它们的方法。 一种方法是将某些参数调整为零;如果仔细操作，所得到的系统保留了原始系统的许多属性，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系统拥有额外的对称性。 本计画将利用这些方法来研究枚举几何中的几个有趣问题，目的是利用等变与退化技巧来研究具有群作用的代数簇的几何。 PI将使用等变上同调的方法以及牛顿-奥昆科夫体理论来研究旗簇、舒伯特簇和复曲面簇。 激励问题是确定标准的品种承认一个单位退化的复曲面品种，并产生一个更广泛的类的例子，牛顿-Okounkov体可以明确计算。 PI还将继续他的研究退化轨迹，并调查其更精细的几何性质，通过开发组合数学的图表签署的排列。 最后，PI将探索最近发展的运算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.