Topics in Arithmetic Geometry and K-Theory

吉列教授将研究涉及k理论和算术几何相互作用的问题。他将研究离散值域上的变量与Arakelov理论的代数类比。他希望建立算术黎曼-洛克定理。希望这些研究将导致对算术变量上的有理点集的结构有更好的认识。除了这些代数问题外，Gillet教授还将从纯解析的角度研究解析循环的交叉点。分析、几何和代数是研究现代数学问题的三种主要方法。这三种方法都被反复地用于解决仅用分数解方程的古老问题。从几何上讲，这个问题相当于在曲线（被视为图形的方程）上寻找有理点（用分数描述的点）。给这样的解决方案一个几何解释使它们更加具体。但是这样做之后，现代的几何方法构建了抽象的代数对象，它们的行为就像熟悉的几何对象一样。同时，使用微积分的分析技术可以帮助以不同的方式理解相同的几何对象。在这个项目中，Gillet教授将使用几何、代数和分析来研究方程、变量的有理解的某些集合。几种现代思想的结合将导致对理性方程的理性解的结构的新见解。

Henri GilletDMS-98 01219Professor Gillet will work on problems involving the interaction between K-theory and arithmetic geometry. He will study algebraic analogs to Arakelov theory for varieties over discretely valued fields. He hopes to establish an arithmetic Riemann-Roch theorem. It is hoped that these researches will lead to improved insights into the structure of sets of rational points on arithmetic varieties. In addition to these algebraic problems, Professor Gillet will study intersections of analytic cycles from a purely analytic perspective.Analysis, geometry and algebra are three of the major attacks used to investigate modern mathematical problems. All three have been used repeatedly on the ancient problem of solving equations using fractions alone. Geometrically this problem amounts to finding rational points (points described by fractions) on curves (equations viewed as graphs.) Giving such solutions a geometric interpretation makes them more concrete. But having done this, the modern approach to geometry constructs abstract algebraic objects that behave like familiar geometric objects. At the same time analytic techniques that use Calculus can help understand the same geometric objects in a different way. In this project Professor Gillet will study certain collections of rational solutions of equations, varieties, using geometry, algebra, and analysis in tandem. The combination of several modern ideas will lead to new insights into the structure of the rational solutions of rational equations.