Arithmetic and Quantum Intersection Theory on Homogeneous Spaces

这项研究是在代数几何中交集理论的不同方面进行的：高维Arakelov理论，算术代数几何的一部分，以及量子上同调，数学和量子物理之间的边界理论。作为将经典代数几何的结果推广到新环境的一般程序的一部分，研究者研究了齐次空间的算术和量子交环，例如旗形流形。以下是所考虑的一些问题：a)理解经典李群齐次空间的算术和量子Schubert演算；b)找到拉格朗日和正交退化轨迹的行列式公式，并进一步发展研究人员的双重Schubert多项式理论；c)获得Fulton关于退化轨迹和数值正多项式的结果的算术相似；d)计算Shimura变量上的算术交点和高度。在19世纪，数学家和物理学家对几何对象中观察到的对称性着迷。当时进行的许多实验和计算最终导致了20世纪的上同调理论，这些理论被应用于解决这两个领域长期悬而未决的问题。今天，我们在现代数论和量子场论中都处于类似的情况，重要的是要有具体例子的计算来支持和指导我们的直觉。研究人员在许多具体的例子中检验了两个新的理论，一个是由数论驱动的，另一个是由量子物理驱动的，这些例子都是出于这个目的的原型。潜在的应用是更好地理解丢番图方程和近似，用于编码理论和理论计算机科学，以及曲线的计数几何，与弦理论和物理学中的对称性有关。

This research is in different aspects of intersection theory in algebraic geometry: higher dimensional Arakelov theory, a part of arithmetic algebraic geometry, and quantum cohomology, a theory on the border between mathematics and quantum physics. The investigator studies the arithmetic and quantum intersection rings of homogeneous spaces, such as flag manifolds, as part of a general program of extending results of classical algebraic geometry to the new settings. The following are some of the problems considered: a) Understanding the arithmetic and quantum Schubert calculus for homogeneous spaces of classical Lie groups; b) finding determinantal formulas for Lagrangian and orthogonal degeneracy loci and developing further the investigator's theory of double Schubert polynomials; c) obtaining arithmetic analogues of Fulton's results on degeneracy loci and numerically positive polynomials; d) computing arithmetic intersections and heights on Shimura varieties.During the 19th century, mathematicians and physicists were fascinated by the symmetries observed in geometric objects. The many experiments and computations made at that time eventually led to the cohomology theories of the 20th century, which were applied to solve long-standing open problems in both fields. Today, we are in a similar situation in modern number theory and quantum field theory, and it is important to have calculations of specific examples to support and guide our intuition. The investigator examines two new theories, one motivated by number theory and the other by quantum physics, in many specific examples which are prototypes for this purpose. The potential applications are a better understanding of Diophantine equations and approximation, used in coding theory and theoretical computer science, and enumerative geometry of curves, related to string theory and symmetry in physics.