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Topics in Arithmetic Geometry

算术几何专题

基本信息

批准号：
1601946
负责人：
Bjorn Poonen
金额：
$ 65万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-06-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601946&HistoricalAwards=false
关键词：
Topics Arithmetic Geometry

项目摘要

The project will explore various topics within algebraic number theory and algebraic geometry. The study of integer and rational number solutions to polynomial equations has been an ongoing area of research since the time of the ancient Greeks, and the project will develop a theory of "number field fragments" that may yield a new approach to answering some such questions, in particular those related to Fermat's last theorem and its generalizations. In algebraic geometry, the project aims, among other things, to show that one of the simplest parameter spaces already has bad singularities, and in fact contains every singularity imaginable. Many of the specific research topics contained in the project are at a level accessible to beginning graduate students, and one of the goals is to involve students in the research.More specifically, given a Galois extension K of Q with Galois group G, the project will extend geometry-of-numbers methods to study the "fragments" of K cut out by an idempotent of the group ring QG, in the hope of developing a new way to prove the nonexistence of certain Galois representations. It will also study correspondences from a curve to itself that are unramified on one side, from the point of view of arithmetic dynamics, since such setups yield new examples of almost-everywhere-unramified arboreal representations. In algebraic geometry proper, the project will study infinitesimal neighborhoods of the vertex of a cone to try to prove that the Hilbert scheme of points satisfies Murphy's law. Finally, the project will explore whether there is an idele class group variant of the Cohen-Lenstra heuristics.

该项目将探索代数数论和代数几何中的各种主题。自古希腊时代以来，对多项式方程的整数和有理数解的研究一直是一个正在进行的研究领域，该项目将发展一种“数域碎片”理论，该理论可能会产生一种新的方法来回答一些这样的问题，特别是那些与费马大定理及其推广有关的问题。在代数几何中，该项目的目的之一是证明最简单的参数空间之一已经有了糟糕的奇点，并且实际上包含了所有可以想象到的奇点。项目中包含的许多具体研究主题都是初学者可以接触到的，其中一个目标是让学生参与到研究中来。更具体地说，给定Q的Galois扩张K和Galois群G，该项目将扩展数字几何方法来研究K被群环QG的幂等元切割出的“片段”，希望开发一种新的方法来证明某些Galois表示的不存在。它还将从算术动力学的角度研究从一条曲线到其自身的对应关系，因为这种设置产生了几乎处处未分支的乔木表示的新例子。在代数几何本身中，该项目将研究圆锥体顶点的无穷小邻域，试图证明点的希尔伯特格式满足墨菲定律。最后，该项目将探索是否存在Cohen-Lenstra启发式的Idele类小组变体。