Derived categories in arithmetic and algebraic geometry

关于多项式方程组有许多未解决的问题，它们被称为变种：我们如何确定它们的解的坐标是否都是整数？我们如何确定两个变量是否有相同的解集合？直接计算这些问题的答案通常是不可能的，或者非常耗时，因此数学家将多项式转换为更容易分析的其他对象。一个这样的对象，称为衍生类别，在回答这些问题方面显示出有希望的初步结果，但它只是在过去十年中才在这方面进行了研究。为了充分利用这项新技术，需要更多的工具来分析派生类别，并进一步了解它可以检测到什么。多项式也可以转化为上同调理论。格罗腾迪克（Grothendieck）于1960年为了证明Weil猜想而提出了l-adic上同理论，被认为是不可或缺的。派生范畴是一个更精细的度量，因为它可以区分这个理论不能区分的品种，但一般来说，上同调和派生范畴之间的关系是未知的。为了充分利用派生范畴，重要的是要揭示它与这一既定理论的关系。关于上述问题，最有趣的变种是Kodaira维度0，即凹与凸之间的临界点。其中包括椭圆曲线，它以理论应用而闻名，如费马大定理的证明，以及密码学中的实际应用。我的长期目标是使用和发展衍生类别，以便对Kodaira维数0的品种进行分类。短期目标：理解hyperkähler 4-fold Kummer型的l-adic上同调的一部分，使用与派生范畴有连接的结构。2. 开发工具来比较堆栈在正特征域上的派生类别。3. 证明导出的torelli型定理，特别是特征为2的域上的Enriques曲面。4. 探索导出范畴是否检测Calabi-Yau 3-fold上整数坐标点的存在性。该计划将为衍生范畴的研究提供新的工具和见解，更重要的是，将把衍生几何的成果带给更大的代数几何学者和数论学者。推导范畴已经被证明是研究代数几何中许多重要课题的有用设置，包括变形、模、托雷利定理、有理点和镜像对称，它们在物理学中也有应用。这项建议还将支持培养代数几何方面的高素质人员，加强加拿大数学界。

There are many unresolved questions about systems of polynomial equations, which are called varieties: How do we decide if they have solutions whose coordinates are all integers? How do we decide if two varieties have the same set of solutions? Directly computing answers to these questions is often impossible or prohibitively time-consuming, so mathematicians convert polynomials into other objects that are easier to analyze. One such object, called the derived category, has shown promising initial results in answering these questions, but it has only been studied in this regard in the last decade. More tools for analyzing the derived category and further knowledge of what it can detect are needed to take full advantage of this new technique. Polynomials may also be converted into cohomology theories. The l-adic étale cohomology theory was introduced by Grothendieck in 1960 in order to prove the Weil conjectures, and is considered indispensible. The derived category is a more refined measure since it can distinguish varieties that this theory cannot, but in general the relationship between cohomology and derived categories is unknown. To access the full utility of the derived category, it is important to uncover its relationship to this well-established theory. The most interesting varieties in regard to the questions above are those of Kodaira dimension 0 -- a tipping point between concave and convex. These include elliptic curves, which are well-known for theoretical applications like the proof of Fermat's last theorem as well as practical applications in cryptography. My long-term goal is to use and develop the derived category in order to classify varieties of Kodaira dimension 0. Short-term objectives: 1. Understand a portion of the l-adic étale cohomology of hyperkähler 4-folds of Kummer type, using a construction with connections to the derived category. 2. Develop tools for comparing derived categories of stacks over fields of positive characteristic. 3. Prove derived Torelli-type theorems, particularly for Enriques surfaces over fields of characteristic 2. 4. Explore whether the derived category detects the existence of points with integer coordinates on Calabi-Yau 3-folds. This program will provide new tools and insight to the study of derived categories, and more significantly, will bring the fruits of derived geometry to the larger community of algebraic geometers and number theorists. The derived category has already proven a useful setting for studying many major topics in algebraic geometry, including deformations, moduli, Torelli theorems, rational points, and mirror symmetry, which has applications in physics. This proposal will also support the training of highly qualified personnel in algebraic geometry, enhancing the mathematical community in Canada.