Dualities in Enumerative Geometry and Representation Theory

枚举几何是数学的一个分支，它试图确定满足某些条件的几何物体的数量，例如，一个空间可以嵌入另一个空间的方式的数量。表征理论是研究对称性的线性代数的一部分。这个项目的重点是这些被称为三维镜像对称的区域的深层相互作用。这种相互作用导致了这些领域中不同对象之间的强大同一性，并发现了应用于量子场论、代数几何和组合学的新公式。这个项目包括学生研究的机会。更确切地说，这是一个研究等变k理论和辛变簇的椭圆上同调中的顶点函数和稳定包络类的性质的课题。通过三维镜像对称建立连接这些对象的显式公式。这项研究的主要技术工具包括稳定包络的阿贝尔化和量子差分方程。由此产生的稳定包络类之间的恒等式自然导致了量子群表示理论中新的对偶性。PI还将研究重要的特殊情况，包括表面上点的希尔伯特格式和瞬时模空间，以发展在组合学和理论物理中的应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Enumerative geometry is a branch of mathematics which seeks to determine the number of geometric objects that satisfy certain conditions, for instance, the number of ways in which one space can be embedded into another. Representation theory is a part of linear algebra studying symmetries. This project is focused on deep interaction of these areas known as 3-dimensional mirror symmetry. This interaction leads to powerful identities between various objects in these fields and to discovery of new formulas which find applications in quantum field theory, algebraic geometry and combinatorics. This project includes opportunities for student research. More precisely, this is a project to investigate properties of the vertex functions and the stable envelope classes in equivariant K-theory and elliptic cohomology of symplectic varieties. Explicit formulas connecting these objects for pairs of varieties related by 3-dimensional mirror symmetry will be established. The main technical tools for this investigation include the abelianization of stable envelopes and the quantum difference equations. The resulting identities between the stable envelope classes lead naturally to new dualities in representation theory of quantum groups. The PI will also investigate important special cases, including Hilbert schemes of points on surfaces and instanton moduli spaces, in order to develop applications in combinatorics and theoretical physics.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.