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

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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.