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

二次型理论是数学研究中最古老的分支之一，起源可以追溯到巴比伦时代。在现代的外表下，它代表了当代代数的基石，最好的背景是作为波雷尔、切瓦利、塞雷、提斯、韦尔和其他人在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。最初被设想为李群理论的代数类比(在几何和物理中至关重要)，代数群理论已经成为统一代数、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大趋势一致，二次型理论(实际上，代数群)最近通过注入强有力的几何和拓扑学的新概念而经历了一次重大转变。从沃沃茨基著名的米尔纳猜想证明(菲尔兹奖，2002年)中涌现出的复杂代数几何工具被证明是特别有成效的，导致了对该主题的一种新的“动机”方法的诞生。这一发展不仅在长期存在的问题上取得了惊人的进展，这些问题似乎超出了经典技术的能力范围，而且还揭示了二次型和一些基本行为者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加强特征域上二次型理论的发展动机方法，而不是2，以便解决其许多核心问题的核心“指数缩减”问题。特别是，我的目标是研究与控制二次形式的各向同性行为的基本问题有关的长期悬而未决的问题，该问题是在二次曲线的函数场和与经典代数群相关的其他齐次变元(具体地用厄米特形式描述)的函数域下控制的。这里的主要挑战是解决这些变种上某些代数循环的合理性问题。此外，该项目的另外两个目标是(A)研究特征为2的域上的某些(所谓的拟线性)二次型的类似的指数约简问题，其中许多当代的动机工具是不可用的；(B)将域上的二次型理论的一些主要成功(与Milnor猜想有关)推广到关于半局部环上的二次型的陈述(从算术代数几何的观点来看)；后者还涉及到对Motivic Homoty理论(Milnor和Milnor-Witt K-群)中元素重要性的密切相关结构的研究。对年轻研究人员的培训将是该项目的一个关键组成部分，该项目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。

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.