Collaborative Research: Calculus beyond Schubert

该研究项目旨在解决枚举代数几何中的悬而未决的问题。一般来说，寻找多个方程的同时解的方法对物理学、计算机科学和工程学的进步有着重要的意义。这些解可以用某些几何空间的交集来表示。枚举几何中的激励问题是预测满足一系列条件的具有指定性质的几何图形的数量。例如，我们可以寻找包含一组点并且与一列直线相切的曲线的数量。令人惊讶的是，虽然很难确定满足条件的数字的精确列表，但通常可以预测这些数字的数量。在计数几何中寻找解的个数的精确公式是一个活跃的研究领域，与许多领域有关，包括几何学、组合学、表示论、计算机科学中的复杂性理论和理论物理学中的镜像对称。这笔赠款将支持研究人员及其研究生在这些领域的继续工作。 理解几何图形模空间的交理论是解决计数几何问题的有效途径。舒伯特微积分领域松散地适用于这些调查中的大类品种在齐性空间。各种上同调理论可以用来提取枚举信息。例如，奇异上同调适用于计算几何图形交叉点的点数，量子上同调适用于计算与其他图形相交的曲线，而等变上同调则产生依赖于群作用的更一般的几何不变量。具有群作用的空间的上同调理论通常具有自然定义的基，与这些基相关的几何不变量往往具有有趣的正性性质，通常与捕捉几何本质方面的美丽组合结构相关。研究人员将在几个几何空间的背景下研究这些现象，包括旗簇及其余切束，弓簇和Hessenberg簇。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research project aims to resolve outstanding questions in enumerative algebraic geometry. Broadly speaking, methods for finding simultaneous solutions to multiple equations have significant implications for progress in physics, computer science, and engineering. These solutions may be expressed in terms of the intersection of certain geometric spaces. The motivating question in enumerative geometry is to predict the number of geometric figures with specified properties that satisfy a list of conditions. For example, we may seek the number of curves that contain a set of points and are tangent to a list of lines. Surprisingly, while it can be very difficult to determine the precise list of figures that satisfy the conditions, it is often possible to predict the number of such figures. The search for exact formulas for the number of solutions in enumerative geometry is an active area of research with relations to numerous fields, including geometry, combinatorics, representation theory, complexity theory in computer science, and mirror symmetry in theoretical physics. This grant will support continued work in these areas by the investigators and their graduate students. An effective approach to solving enumerative geometric problems is to understand the intersection theory of moduli spaces of geometric figures. The field of Schubert calculus loosely applies to these investigations among large classes of varieties in homogeneous spaces. Various cohomology theories can be used to extract enumerative information. For example, singular cohomology is useful for counting the number of points in intersections of geometric figures, quantum cohomology is designed for counting curves meeting other figures, and equivariant cohomology produces more general geometric invariants that depend on a group action. Cohomology theories of spaces with group actions often have naturally defined bases, and geometric invariants related to such bases tend to possess intriguing positivity properties, often related to beautiful combinatorial structures that capture the essential aspects of the geometry. The investigators will study these phenomena in the context of several geometric spaces, including flag varieties and their cotangent bundles, bow varieties, and Hessenberg varieties. Techniques include intersection theory, equivariant localization, symmetric functions, Hecke algebra actions, and geometric representation theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.