Investigations in the algebraic and geometric theory of quadratic and hermitian forms

二次和埃尔米特形式的代数和几何理论研究

• 批准号: RGPIN-2019-05607
• 负责人: Scully, Stephen
• 金额: $ 1.6万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759121
• 关键词: Investigations; algebraic; geometric; theory; quadratic

The theory of quadratic forms is among the oldest branches of mathematical research, with origins that can be traced back to Babylonian times. In its modern guise, it represents a cornerstone of contemporary algebra, best contextualized as a remarkably rich relative of the general theory of algebraic groups developed by Borel, Chevalley, Serre, Tits, Weil and others in the mid-20th century. Initially conceived as an algebraic analogue of the theory of Lie groups (crucial in geometry and physics), the theory of algebraic groups has emerged as a very powerful tool for unifying seemingly disparate statements in algebra, arithmetic and geometry. Consistent with a general trend sweeping modern algebra and number theory, the theory of quadratic forms (and indeed, algebraic groups) has recently undergone a major transformation via an infusion of potent new ideas of geometric and topological flavour. Sophisticated algebro-geometric tools emerging from the celebrated proof of the Milnor Conjecture by Voevodsky (Fields Medal, 2002) have proved particularly fruitful, leading to the birth of a new `motivic' approach to the subject. This development has not only seen spectacular progress on long-standing problems that seemed beyond the reach of classical techniques, but has revealed deep new links between quadratic forms and some of the basic actors of `motivic homotopy theory', thereby opening up entirely new directions of research. The long-term vision of this project is to enhance the developing motivic approach to the theory of quadratic forms over fields of characteristic not 2 in order to attack core `index-reduction' problems that lie at the heart of many its central questions. In particular, I aim to investigate long-standing open problems related to the fundamental issue of controlling the isotropy behaviour of quadratic forms under extension to function fields of quadrics and other homogeneous varieties associated to classical algebraic groups (described concretely in terms of hermitian forms). The main challenge here is to address rationality questions for certain algebraic cycles on these varieties. In addition, two further aims of the project are (a) to study similar index-reduction problems for certain (so-called quasilinear) quadratic forms over fields of characteristic 2, where many of the contemporary motivic tools are unavailable, and (b) to globalize some of the major successes of the theory of quadratic forms over fields (related to the Milnor Conjecture) to statements about quadratic forms over semilocal rings (of interest from the viewpoint of arithmetic algebraic geometry); the latter also involves the study of closely related structures of elemental importance in motivic homotopy theory (Milnor and Milnor-Witt K-groups). The training of young researchers will be a key component of the project, which will strengthen Canada's presence in the fundamental areas of research outlined above, and enhance its overall scientific capacity.

二次形式的理论是数学研究的最古老的分支之一，起源可以追溯到巴比伦时代。它以现代的幌子代表了当代代数的基石，最佳背景是由Borel，Chevalley，Chevalley，Chevalley，Serre，Titt，Weil，Weil，Weil，Weil，Weil，Weil和其他人在20世纪中叶开发的代数群体一般理论的最丰富的亲戚。最初被认为是谎言群体理论的代数类似物（在几何学和物理学中至关重要），代数群的理论已成为一种非常有力的工具，用于统一代数，算术和几何学的看似截然不同的陈述。 与一般趋势的现代代数和数字理论一致，二次形式的理论（甚至代数群体）最近通过注入有效的几何学和拓扑风味的新思想，进行了重大转变。事实证明，Voevodsky（Fields Medal，2002）的著名米尔诺（Milnor）猜想的著名证明已经出现了复杂的代数几何工具，已被证明特别富有成果，导致了对该主题的新“动机”方法的诞生。这一发展不仅在长期以来的问题上看到了超越古典技术的长期问题的壮观进步，而且还揭示了二次形式与“动机同义理论”的一些基本参与者之间的深刻新联系，从而开辟了全新的研究方向。 该项目的长期愿景是增强对二次形式理论的发展动机方法，而不是特征的领域而不是2个领域，以攻击许多核心问题的核心“减少索引”问题。特别是，我的目的是研究与控制二次形式在扩展到四边形功能领域的二次形式和其他与经典代数群体相关的统一品种的基本问题相关的长期开放问题（用遗传学形式进行了核心描述）。这里的主要挑战是解决这些品种某些代数周期的合理性问题。此外，该项目的两个进一步目标是（a）研究特征领域的某些（所谓的准二词）二次形式的类似（所谓的准二一个）形式2，其中许多现代动机工具不可用，（b）全球化，以全球化的某些主要形式（相关范围）（相关的竞争）（与米兰德的竞争）相关的某些主要形式（相关），而超过了对米兰的竞争，则构造了偶然的统一。 （从算术代数几何学的角度来看）；后者还涉及对动机同型理论（Milnor和Milnor-Witt K-groups）紧密相关的元素重要性结构的研究。对年轻研究人员的培训将是该项目的关键组成部分，该项目将加强加拿大在上面概述的研究领域的存在，并增强其整体科学能力。