Project description
Moving prime 2 barriers out of the way to reveal the foundations of special K-theories
The ubiquitous quadratic function is a second-degree polynomial – the highest exponent of any term is 2, and we say that term is second degree or quadratic. Quadratic forms are special nonlinear polynomials having only second-order terms – in other words, they are homogeneous polynomials of second degree. They encode so-called quadric surfaces, a generalisation of conic sections that are created by 'slicing' a 2D plane through a cone, that consist of ellipses, parabolas and hyperbolas. The theory of quadratic forms is thus very sensitive to the prime 2. The EU-funded MRKT project will expand the K-theory of quadratic forms, building on their novel framework that removes barriers and addresses subtleties imposed by the prime 2.
Objective
Quadratic forms are ubiquitous throughout mathematics, playing a fundamental role in areas from arithmetic through algebra and geometry. In surgery theory, quadratic forms feature prominently in the classification of smooth manifolds in a given homotopy type, while in arithmetic geometry they can be used to encode Galois and motivic cohomology classes via Milnor's conjecture. The theory of quadratic forms is naturally very sensitive to the prime 2. While in surgery theory this effect is critical, in algebraic geometry it was often set aside by assuming 2 to be invertible in all ground rings. A recent joint work of the PI and collaborators on the foundations of Hermitian K-theory uses state-of-the-art tools from higher category theory to develop a new framework for the subject, bringing a bordism theoretical approach to the algebraic study of quadratic forms, all while accommodating for the subtleties posed by the prime 2.
Building on this recent success, the project MRKT aims to remove the theoretical barrier of the prime 2 from the study of Hermitian K-theory in the domain of algebraic geometry, and set up the foundations of motivic Hermitian K-theory and real algebraic K-theory over the integers.
Fields of science
Programme(s)
Topic(s)
Funding Scheme
ERC-STG - Starting GrantHost institution
75794 Paris
France