Topics in Arithmetic Geometry

该项目将探讨代数数论和代数几何中的各种主题。 自古希腊时代以来，多项式方程的整数和有理数解的研究一直是一个持续的研究领域，该项目将开发一种“数域碎片”理论，这可能会产生一种新的方法来回答一些这样的问题，特别是那些与费马最后定理及其推广有关的问题。 在代数几何中，该项目的目标之一是证明最简单的参数空间之一已经具有坏奇点，并且实际上包含所有可以想象的奇点。 该项目中包含的许多具体研究课题都处于研究生入门阶段，其中一个目标是让学生参与研究。更具体地说，给定具有伽罗瓦群G的Q的伽罗瓦扩张K，该项目将扩展数字几何方法来研究K被群环QG的幂等元切割的“片段”，希望开发一种新的方法来证明某些伽罗瓦表示的不存在。 它还将研究从一条曲线到它本身的对应关系，从算术动力学的角度来看，这是因为这样的设置产生了几乎无处不在的非分歧树表示的新例子。 在代数几何中，该项目将研究圆锥顶点的无穷小邻域，试图证明希尔伯特点的方案满足墨菲定律。 最后，这个项目将探讨是否有一个Idele类群的Cohen-Lenstra算法的变体。

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.