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

亚里士多德错误地宣称，人们可以用正四面体的复制品来填充空间。确定哪些非正四面体可以填充空间是一个2300年未解决的问题，研究者将使用代数几何和数论工具的新方法来研究。作为实现这一目标的垫脚石，研究者将研究更大类的四面体，这些四面体可以被切成有限多块并重新组装成一个立方体；自1974年以来，没有发现新的四面体。此外，研究者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次多项式的判别式的几何意义。获得该奖项的研究生和本科生将接受培训，为这些项目做出贡献。研究者将通过将这种消失与指数丢番图方程系统的解联系起来，对Dehn不变量0（剪刀-同立方体）的四面体进行分类，该系统将结合伽罗瓦理论和p进方法进行研究。可以平铺空间的四面体就在其中，所以这些Dehn不变的四面体将通过使用几何和组合方法来测试平铺的能力，利用计算来构建平铺或排除它们。研究者将扩展上同调障碍理论来理解堆栈上的积分点，而不仅仅是变量；这样的研究可能会对承认低维堆栈形态的变种上的有理点产生影响。他将把有限域中P^n的变化几何与随机超平面截面的点计数矩联系起来。最后，他将从几何角度解释射影超曲面的判别式。该奖项反映了美国国家科学基金会的法定使命，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。

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.