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.

二次型理论是数学研究中最古老的分支之一，其起源可以追溯到巴比伦时代。在其现代的伪装下，它代表了当代代数的基石，最好的背景是作为由Borel， Chevalley, Serre, Tits， Weil和其他人在20世纪中期发展的代数群的一般理论的一个非常丰富的亲戚。代数群理论最初被认为是李群理论（在几何和物理中至关重要）的代数模拟，现在已经成为统一代数、算术和几何中看似不同的陈述的非常强大的工具。与席卷现代代数和数论的普遍趋势一致，二次型理论（实际上，代数群）最近经历了一个重大的转变，通过注入强有力的几何和拓扑的新思想。沃沃斯基著名的米尔诺猜想证明（菲尔兹奖，2002年）中出现的复杂的代数几何工具被证明特别富有成果，导致了研究该主题的新“动机”方法的诞生。这一发展不仅在经典技术无法解决的长期问题上取得了惊人的进展，而且揭示了二次型与“动机同伦理论”的一些基本行为体之间的深刻的新联系，从而开辟了全新的研究方向。该项目的长期愿景是在非2特征域上增强二次型理论的发展动机方法，以解决许多核心问题的核心“指数缩减”问题。特别是，我的目标是研究长期存在的开放问题，这些问题涉及到控制二次型在扩展到二次型函数场的各向同性行为的基本问题，以及与经典代数群相关的其他齐次变量（具体地用厄米特形式描述）。这里的主要挑战是解决这些变量上某些代数循环的合理性问题。此外，该项目的两个进一步目标是(a)研究某些（所谓的拟线性）二次型在特征为2的域上的类似指数约简问题，其中许多当代动机工具是不可用的；(b)将二次型在域上的一些主要成功理论（与米尔诺猜想有关）全球化到关于半局部环上的二次型的陈述（从算术代数几何的角度感兴趣）；后者还涉及动机同伦理论（Milnor和Milnor- witt k群）中密切相关的元素重要性结构的研究。培训年轻研究人员将是该项目的一个关键组成部分，这将加强加拿大在上述基本研究领域的存在，并提高其整体科学能力。