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，其起源可以追溯到巴比伦时代。在其现代的幌子，它代表了一个基石，当代代数，最好的语境作为一个显着丰富的相对一般理论的代数群开发的博雷尔，Chevalley，塞尔，山雀，韦伊和其他人在中期的世纪。最初被认为是李群理论（在几何和物理中至关重要）的代数类比，代数群理论已经成为一个非常强大的工具，用于统一代数，算术和几何中看似不同的陈述。 与席卷现代代数和数论的一般趋势一致，二次型理论（实际上，代数群）最近通过注入几何和拓扑味道的强有力的新思想而经历了重大转变。Voevodsky（菲尔兹奖，2002年）著名的Milnor猜想证明中出现的复杂的代数几何工具已证明特别富有成效，导致了对该主题的新的“动机”方法的诞生。这一发展不仅看到了壮观的进展，长期存在的问题，似乎超出了经典技术的范围，但揭示了深刻的二次型和一些基本演员的“动机同伦理论”之间的新的联系，从而开辟了全新的研究方向。 这个项目的长期目标是加强对特征不为2的领域上的二次型理论的发展动机方法，以解决处于其许多中心问题核心的核心“指数减少”问题。特别是，我的目标是调查长期悬而未决的问题有关的基本问题，控制的各向同性行为的二次型下扩展到功能领域的二次和其他齐次品种相关的经典代数群（具体描述的埃尔米特形式）。这里的主要挑战是解决这些品种的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）研究类似的指数缩减问题（所谓的准线性）二次型的领域的特点2，其中许多当代motivic工具是不可用的，（B）将域上二次型理论的一些主要成果全球化（与Milnor猜想有关）关于半局部环上二次型的陈述（从算术代数几何的观点来看是有趣的）;后者还涉及到对动机同伦理论中重要元素的密切相关结构的研究（Milnor和Milnor-Witt K-群）。培训青年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，并提高其总体科学能力。

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.