Topics in Arithmetic Geometry and K-theory

这一建议涉及到高维Arakelov理论和代数k理论中的问题。PI打算将前人与soule关于代数变量的动机权复形的工作成果推广到算术情况。在与J. Hu的合作中，他打算将Arakelov理论（由J. Hu在他的论文中发展）的变形到正锥体技术用于堆栈的情况。他还打算用微分代数和阿拉克洛夫理论的方法研究算术几何中与莫德尔猜想有关的问题。本提案涉及算术几何和代数k理论。算术几何是研究系数为整数的方程的性质，使用数论（研究整数的性质）和代数几何的方法，代数几何研究可以用最简单的方程，即多项式来定义的几何图形。代数k理论用几何学的方法研究线性方程的性质。数论与几何相结合所产生的问题和现象是当代许多数学研究的推动力。此外，算术几何还为许多应用做出了贡献，包括密码学和理论计算机科学。

This proposal concerns questions in higher dimensional Arakelov theory andalgebraic K-theory. The PI intends to extend results of previous work withSoule on motivic weight complexes of algebraic varities to the arithmetic case.In collaboration with J. Hu, he intends to use the deformation to the normalcone techniquess for Arakelov theory (developed by J. Hu in his thesis) to thecase of stacks. He also intends to study questions in arithmetic geometryrelated to the Mordell conjecture using methods from differential algebra andArakelov theory. This proposal deals with arithmetic geometry and algebraic K-theory.Arithmetic geometry is the study of the properties of equations withcoefficients that are whole numbers, and using methods both from numbertheory (the study of properties of whole numbers) and algebraic geometry, whichstudies geometric figures that can be defined by the simplest of equations,namely polynomials. Algebraic K-theory studies properties of linear equationsusing methods from geometry. The questions and phenomena which arise fromcombining number theory and geometry serve as driving forces in much ofcontemporary mathematics research. Moreover, arithemetic geometry hascontributed to many applications including cryptography and theoreticalcomputer science.