Integral points on stacks, hyperplane sections over finite fields, and vectors forming rational angles

堆栈上的积分点、有限域上的超平面截面以及形成有理角的向量

基本信息

批准号：
2101040
负责人：
Bjorn Poonen
金额：
$ 36万
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：
2021
资助国家：
美国
起止时间：
2021-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101040&HistoricalAwards=false
关键词：
Integral points stacks hyperplane sections

项目摘要

Aristotle incorrectly claimed that one could fill space with copies of a regular tetrahedron. Determining which nonregular tetrahedra can fill space is a 2300-year-old unsolved problem that the investigator will study using a new approach using tools from algebraic geometry and number theory. As a stepping stone towards this, the investigator will study a larger class of tetrahedra, those that can be sliced into finitely many pieces and reassembled into a cube; no new such tetrahedra have been found since 1974. In addition, the investigator will study other questions concerning the solutions to systems of equations; for example, generalizing the fact that the discriminant b^2-4ac of a quadratic polynomial ax^2+bx+c determines whether its roots coincide, the investigator will study the geometric meaning of the discriminant of a higher degree polynomial in many variables. Graduate and undergraduate students supported by the award will receive training to contribute towards these projects. The investigator will classify tetrahedra of Dehn invariant 0 (scissors-congrent to a cube) by relating this vanishing to the solutions of a system of exponential Diophantine equations, which will be studied a combination of Galois-theoretic and p-adic methods. The tetrahedra that can tile space are among these, so these Dehn invariant 0 tetrahedra will be tested for the ability to tile by using geometric and combinatorial methods, making use of computation to construct tilings or to rule them out. The investigator will extend the theory of cohomological obstructions to understand integral points on stacks instead of just varieties; such a study would have implications for rational points on varieties that admit morphisms to lower-dimensional stacks. He will relate geometry of a variety in P^n over a finite field to the moments of point counts of random hyperplane sections. Finally, he will explain the valuation of the discriminant of a projective hypersurface in terms of its geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

亚里士多德错误地声称人们可以用正四面体的副本来填充空间。确定哪种不规则四面体可以填充空间是一个 2300 年前未解决的问题，研究人员将使用代数几何和数论工具的新方法来研究这个问题。作为实现这一目标的踏脚石，研究人员将研究更大一类的四面体，即那些可以被切成有限多块并重新组装成立方体的四面体；自 1974 年以来，没有发现新的此类四面体。此外，研究者还将研究有关方程组解的其他问题；例如，推广二次多项式ax^2+bx+c的判别式b^2-4ac决定其根是否重合的事实，研究者将研究高次多项式的判别式在许多变量中的几何意义。受该奖项支持的研究生和本科生将接受培训，为这些项目做出贡献。研究人员将通过将此消失与指数丢番图方程组的解联系起来，对 Dehn 不变量 0（与立方体剪刀一致）的四面体进行分类，该方程组将结合伽罗瓦理论和 p-adic 方法进行研究。能够平铺空间的四面体就属于其中，因此我们将使用几何和组合方法来测试这些Dehn不变0四面体的平铺能力，利用计算来构造平铺或排除它们。研究者将扩展上同调障碍理论来理解堆栈上的积分点，而不仅仅是簇；这样的研究将对允许态射进入低维堆栈的簇上的理性点产生影响。他将有限域上 P^n 中的各种几何形状与随机超平面截面的点计数矩联系起来。最后，他将解释射影超曲面判别式在几何形状方面的估值。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。