RUI: Hurwitz Number Fields

数论和几何是数学中最古老的两个领域。数场在数论中扮演着核心角色，而曲面在几何中扮演着核心角色。该研究项目涉及从表面几何中自然产生的数场，通过一个可以追溯到1900年左右数学家Adolph Hurwitz工作的结构。从纯粹的数论观点来看，这些赫维茨数域具有一些非常不寻常的性质：它们的度和伽罗瓦群通常非常大，而它们的分枝素数集非常小。事实上，Hurwitz数域在这些不变量方面是如此极端，以至于它们的存在与最近成功指导其他数域研究的启发式相矛盾。本研究项目的目标是建立这些外围数字域都以统一和高度结构化的方式表现。该项目的结果预计将支持围绕数论的几个学科的未来发展。该项目包括与三名本科生合作，专门为他们的科学事业做好准备。这些合作将成为未来学生参与高水平数学研究的典范。赫尔维茨数域作为黎曼曲面到黎曼球映射的定义域出现。研究项目的主要焦点将是对任意赫维茨数域中分支的明确描述。从定义场的几何数据中，我们可以读出一个有限的素数集，在这个素数集之外的分支是已知的驯服的。该项目将在这些温和素数处寻找分枝的精确公式，并在其余素数处寻找野生分枝的尖锐上限。除了研究分枝外，研究者的目标是建立一个关于给定分枝素数集的度无界性的猜想的几何类比。在一个不同的方向上，在允许衍生素数集增长的地方，研究者的目标是通过找到一次定义无限多个Hurwitz数域的多元多项式的多索引序列来进一步证明行为的一致性。

Two of the oldest areas of mathematics are number theory and geometry. Number fields play a central role in number theory, while surfaces play a central role in geometry. The research project concerns number fields that arise naturally from the geometry of surfaces, via a construction dating back to work of the mathematician Adolph Hurwitz around 1900. These Hurwitz number fields have a combination of properties that is very unusual from a purely number-theoretic point of view: their degrees and Galois groups are typically very large, while their sets of ramifying primes are very small. In fact, Hurwitz number fields are so extreme with respect to these invariants that their existence contradicts a heuristic that has successfully guided other recent research into number fields. The goal of this research project is to establish that these outlying number fields all behave in a uniform and highly structured way. The results of the project are expected to support future developments interrelating several disciplines surrounding number theory. The project includes collaborations with three undergraduates, designed specifically to better prepare them for scientific careers. These collaborations will serve as models for future involvement of students in high-level mathematics research.Hurwitz number fields arise as fields of definition of maps for Riemann surfaces to the Riemann sphere. The main focus of the research project will be on the explicit description of ramification in an arbitrary Hurwitz number field. From the geometric data defining the field, one can read off a finite set of primes outside of which ramification is known to be tame. The project will pursue exact formulas for ramification at these tame primes and sharp upper bounds for the wild ramification at the remaining primes. Besides studying ramification, the investigator will aim to establish a geometric analog of a conjecture on the unboundedness of degrees for a given set of ramifying primes. In a different direction, where the set of ramifying primes is allowed to grow, the investigator aims to demonstrate further uniformity of behavior by finding multi-indexed sequences of multivariate polynomials that define infinitely many Hurwitz number fields at once.