权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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填充空间。确定哪些不规则四面体可以填充空间是一个2300年前的未解决问题，研究人员将使用代数几何和数论的工具使用新方法进行研究。作为实现这一目标的垫脚石，研究人员将研究一个更大的四面体类别，这些四面体可以被切成许多块并重新组装成立方体;自1974年以来，没有发现新的四面体。此外，研究者还将研究与方程组解有关的其他问题;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，研究者将研究多变量高阶多项式判别式的几何意义。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。研究人员将分类四面体的德恩不变0（剪刀-平行于立方体），通过将这种消失与指数丢番图方程组的解相关联，将研究伽罗瓦理论和p-adic方法的组合。可以拼接空间的四面体就是其中之一，因此这些德恩不变0四面体将通过使用几何和组合方法测试拼接的能力，利用计算来构建拼接或排除它们。研究者将扩展上同调障碍理论，以理解栈上的积分点，而不仅仅是变种;这样的研究将对允许态射进入低维栈的变种上的有理点产生影响。他将涉及几何的各种P^n在有限领域的时刻点计数的随机超平面部分。最后，他将解释投影超曲面的判别式的几何估值。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。