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对象的数量，例如，一个空间可以嵌入另一个空间的方式的数量。表示论是研究对称性的线性代数的一部分。该项目的重点是这些领域的深层相互作用，称为三维镜像对称。 这种相互作用导致强大的身份之间的各种对象在这些领域和发现新的公式，发现应用在量子场论，代数几何和组合。该项目包括学生研究的机会。更准确地说，这是一个项目，研究的性质的顶点功能和稳定的包络类等变K-理论和椭圆上同调的辛簇。明确的公式连接这些对象对有关的3维镜像对称的品种将被建立。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。由此产生的身份之间的稳定包络类自然导致新的对偶表示理论的量子群。PI还将研究重要的特殊情况，包括表面上的点和瞬子模空间的希尔伯特方案，以开发组合数学和理论物理中的应用。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。

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.