Quantum Algebras, Quiver Varieties, and Applications

表示论研究的是数学中的对称性，而代数几何则是研究可以用代数方程描述的空间。这两个领域之间的界面是一个丰富的学科，它最近在数学、理论物理和组合学的各个分支中得到了应用。在这个项目中，首席研究员将研究一类特殊的空间，称为箭形簇，它由某些图决定，它们从图中继承了许多有趣的对称性。通过抽象箭形变种的性质，数学家们可以发现许多令人着迷的公式，这些公式的应用范围从弦理论到纽结的研究等遥远的领域。主要研究人员的方法有两个：第一，通过使用一种称为Shuffle代数的技术工具来研究箭图簇(上同调，K-理论，派生范畴)的一般性质，第二，将这些技术应用于重要的特殊情况，以解决具体问题。例如，对FLAG-希尔伯特方案的研究导致了纽结不变量的几何实现。同样，通过对高阶层模空间的研究，我们可以从数学上理解规范理论和保形场理论之间的关系。这些申请和其他申请将由首席研究员和他的合著者继续进行。

Representation theory is the study of symmetries in mathematics, while algebraic geometry is the study of spaces that can be described by algebraic equations. The interface between these two fields is a rich subject, which has had recent applications to various branches of mathematics, theoretical physics, and combinatorics. In this project the principal investigator will study a particular class of spaces called quiver varieties, which are determined by certain graphs, from which they inherit many interesting symmetries. Abstracting the properties of quiver varieties allows mathematicians to discover many fascinating formulas, whose applications range between such distant fields as string theory and the study of knots. The approach of the principal investigator is two-fold: first, to study the general properties of quiver varieties (cohomology, K-theory, derived categories) by using a technical tool called the shuffle algebra, and second, to apply these techniques in important particular cases in order to solve concrete problems. For example, the study of flag Hilbert schemes leads to a geometric realization of knot invariants. Similarly, the study of moduli spaces of higher rank sheaves leads to a mathematical understanding of the relations between gauge theory and conformal field theory. These and other applications will be pursued by the principal investigator and his coauthors.