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

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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.