Arakelov theory and quantum cohomology of homogeneous varieties

点击翻译按钮获取中文摘要

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.