CAREER: Models of curves and non-archimedean geometry

“特征p世界”（对于某个素数p）是这样一个世界，每次你向前走p步，你就会回到开始的地方。这个世界远非深奥，它描述了一周中每一天的过程（p = 7），计算机体系结构的基本原理（p = 2），也是重要密码系统的设置。特征p代数几何是研究这个世界上多项式方程的解给出的几何对象（“变种”）（例如，上述密码系统的基本椭圆曲线），而混合特征代数几何是研究“混合”世界中的变种，作为从特征p世界到我们更熟悉的世界的桥梁，在这个世界中，所有方向都永远持续下去。奇点是一个有交叉点、角落或其他不光滑的地方。该项目引入了新的代数技术来处理混合特征代数几何中的奇点问题，其主要目标是明确地分析需要哪些过程来“解决”奇点，即使奇点平滑。该项目的教育计划提供了从高中到研究生院的各级机会。具体来说，PI的研究生将在主要研究项目上直接与PI合作，PI将为数论相关领域的本科生研究提供建议，PI将帮助规划巴鲁克学院妇女数学协会分会的活动，PI将继续致力于利比里亚的数学教育，为利比里亚高中学生举办数学竞赛，并为他们的教师举办一系列远程讲习班。该项目的第一部分致力于Mac Lane估值的应用和推广，它给出了值域上投影线的正规模型的明确的，计算上有用的描述。Mac Lane估值可以追溯到80多年前，但直到最近十年，它们才被成功地用于解决涉及曲线模型的问题，例如计算某些算术曲面奇点的分辨率和验证导体判别不等式。PI将应用Mac Lane估值来(1)完善他之前关于超椭圆曲线的导体判别不等式的结果，并将其扩展到超椭圆曲线；(2)阐明正则模型、半稳定模型和Berkovich空间“纯粹算术”覆盖的分支理论之间的关系。此外，他将致力于建立更高属曲线的Mac Lane估值理论。项目的第二部分致力于理解曲线的分支伽罗瓦覆盖何时从特征p提升到特征0，并获得提升空间的几何信息。PI将使用模理论技术来解决循环盖塔的提升问题，他将继续他的长期工作，对所谓的局部奥尔特群和弱局部奥尔特群进行分类。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。

The "characteristic p world" (for p some prime number) is one where, every time you take p steps forward, you wind up back where you start. Far from being esoteric, this world describes the procession of the days of the week (p = 7), the fundamentals of computer architecture (p = 2), and is also the setting for important cryptosystems. Characteristic p algebraic geometry is the study of geometric objects ("varieties") given by solutions to polynomial equations in this world (e.g., the elliptic curves fundamental to the aforementioned cryptosystems), whereas mixed characteristic algebraic geometry is the study of varieties in a "hybrid" world serving as a bridge from the characteristic p world to our more familiar world where all directions go on forever. A singularity is a place where a variety has crossings, corners, or otherwise fails to be smooth. This project brings new algebraic techniques to bear on the topic of singularities in mixed characteristic algebraic geometry, with one main goal to explicitly analyze what processes are necessary to "resolve" them, i.e, smooth them out. The educational plan for the project provides opportunities at levels from high school to graduate school. Specifically, the PI's graduate students will work directly with the PI on the main research project, the PI will advise undergraduate research in related areas of number theory, the PI will help plan events for Baruch College's chapter of the Association for Women in Mathematics, and the PI will continue his commitment to mathematics education in Liberia by running a mathematics competition for Liberian high school students, as well as a series of remote workshops for their teachers.The first part of the project is dedicated to applications and generalizations of Mac Lane valuations, which give explicit, computationally useful descriptions of normal models of the projective line over a valued field. Mac Lane valuations date back over 80 years, but it is only this past decade that they have been successfully used to attack problems involving models of curves, such as computing resolutions of certain arithmetic surface singularities and verifying conductor-discriminant inequalities. The PI will apply Mac Lane valuations to (1) refine his previous results on conductor-discriminant inequalities for hyperelliptic curves and extend them to superelliptic curves and (2) shed light on the relationship between regular models, semistable models, and the ramification theory of ``purely arithmetic'' covers of Berkovich spaces. Furthermore, he will work to build a theory of Mac Lane valuations for higher genus curves. The second part of the project is devoted to understanding when branched Galois covers of curves lift from characteristic p to characteristic zero and to obtaining information about the geometry of the space of lifts. The PI will use moduli-theoretic techniques to attack the lifting problem in towers for cyclic covers, and he will continue his long-term work toward a classification of the so-called local Oort groups and the weak local Oort groups.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.