Quadratic Forms and Algebraic Cobordism

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The subject area of this proposal is the modern theory of quadratic forms using new geometric methods suggested by the principal investigator. These methods are based on and further extend the new strategically important developments in algebra and algebraic geometry related to the names of V. Voevodsky (Fields Medal 2002), M. Rost, A.Merkurjev, A. Suslin, M.Levine, F. Morel.The research of the proposal lies at the boundary between algebra, algebraic geometry and topology. Each of these fields has a world of its own, governed by laws specific to it but interacting in a peculiar way; with algebra catching the most basic essence of mathematical objects, algebraic geometry translating this into the language where one can use geometric reasoning, and topology providing the rough shade of the geometric object, where exact shape is lost, but the most important ``topological'' invariants are kept. In a sense, topology is just a ``toy model'' of the much richer field algebro-geometric topology. All the phenomena one encounters in topology appear also in the algebro-geometric world, but in incomparably more variable shapes. The simplest object of topology is a point, out of which with the help of suspension operation one produces its derivatives - the spheres. The problem of computing morphisms between these objects, the so-called homotopy groups of spheres, is the central question of topology, and one of the main problems of mathematics as a whole. Despite many attempts, this problem resists breaking. In algebraic geometry one also has a point, but now it depends on a base-field, and one has spheres, but now they have two dimensions - the ``round'' and the ``square'' one - because there are two different suspension operations. In particular, the homotopy groups are now parametrized not by one natural number, but by two. These groups give the topological counterparts as a certain degeneration, but otherwise they are much richer. These groups store in a compressed form the information about the difference between topological and algebro-geometric worlds. And the central one of them can be described in terms of quadratic forms (by the result of F.Morel). Thus, studying quadratic forms we, in reality, study homotopy groups of spheres,and the results obtained here can be applied to the central question of topology, since both (quadratic forms and topological homotopy groups) interact as parts of the same object. And the properties of quadratic forms are well seen through their invariants.The principal aim of this proposal is the study of invariants of quadratic forms (both, new ones introduced by the PI, and the classical ones as well). The second aim is the study of the new cohomological operations in algebraic cobordism (in connection with the first aim, and independently). The third aim is the development of the new homotopy-theoretic viewpoint on quadratic form theory (thus, extending the theory of Morel-Voevodsky).The research will be undertaken at the School of Mathematical Sciences, University of Nottingham.