Arakelov theory and quantum cohomology of homogeneous varieties

研究者将研究代数几何中的两种不同的交理论：整数上定义的变异的算术周环和复流形的量子上同调环。所考虑的变异是李群的齐次空间，目标是理解算术和量子舒伯特微积分。这使得人们一方面可以有效地计算数值不变量，比如高度，另一方面可以计算Gromov-Witteninvariants。进一步的目的是回答有关现代舒伯特微积分的组合学和表示向量束的退化轨迹的多项式的相关问题。长期以来，对几何物体对称性的研究一直是数学和物理的交汇点。19世纪进行的许多实验和计算最终导致了20世纪数学的上同调和交理论，这些理论被应用于解决这两个领域的重要开放问题。目前，我们在现代数论和弦论中面临着类似的情况，具体例子的计算对于指导我们的直觉和更好地理解一般理论至关重要。研究者研究了两个新的交叉理论，Arakelov理论和量子肿瘤同源，并在许多例子中为这两个理论提供了很好的检验基础。潜在的应用是更好地理解丢芬图方程和近似，用于编码理论和理论计算机科学，以及曲线的枚举几何，与经典代数几何和物理中的量子场论有关。

The investigator will study two different intersection theories inalgebraic geometry: the arithmetic Chow rings of varieties definedover the integers and the quantum cohomology rings of complexmanifolds. The varieties considered are homogeneous spaces ofLie groups, and the goal is to understand the arithmetic and quantumSchubert calculus. This allows one to effectively compute numericalinvariants such as heights, on the one hand, and Gromov-Witteninvariants, on the other. A further aim is to answer related questionsabout the combinatorics of modern Schubert calculus and thepolynomials which represent degeneracy loci of vector bundles.The study of the symmetries observed in geometric objects has been ameeting point of mathematics and physics for a long time. The manyexperiments and computations made in the 19th century eventually ledto the cohomology and intersection theories of 20th century mathematics,which were applied to solve important open problems in both fields. Atpresent, we face a similar situation in modern number theory and stringtheory, and computations of concrete examples are crucial to guide ourintuition and to better understand the general theories. The investigatorstudies two new intersection theories, Arakelov theory and quantumcohomology, in many examples which are good testing grounds forboth of them. The potential applications are a better understanding ofDiophantine equations and approximation, used in coding theory andtheoretical computer science, and the enumerative geometry of curves,related to classical algebraic geometry and quantum field theory in physics.