Quadratic forms are ubiquitous throughout mathematics, playing a fundamental role in areas ranging from arithmetic through algebra and geometry. In surgery theory, quadratic forms feature prominently in the classification of smooth manifolds. In algebraic geometry, quadratic forms are used to construct rich and interesting invariants of commutative rings (such as the rational numbers, or the integers) by considering forms with coefficients in these rings. This topic in algebraic geometry is called hermitian K-theory, and gives rise to invariants of rings (and more generally algebraic varieties) known as Grothendieck-Witt groups and L-groups. While these invariants obey a highly structured theory, they remain difficult to compute in practice.
The theory of quadratic forms is very sensitive to the prime 2. For example, if one is working with coefficients where one can divide by 2, then there is essentially no difference between using quadratic forms and using symmetric bilinear forms. On the other hand, if one cannot divide by 2 (for example, if one is working over the ring of integers), then one needs to specify the precise type of forms one is interested in, and the theory is highly sensitive to this choice. As a result, a significant portion of the theory had previously been developed under this divisibility hypothesis, and many aspects of hermitian K-theory remained well understood only when one can divide by 2. This includes, for example, the precise relationship between Grothendieck-Witt theory and L-theory, the periodicity phenomenon witnessed by Karoubi's fundamental theorem, the Nisnevich descent property and homotopy invariance of Grothendieck-Witt theory, and the computations of Grothendieck-Witt groups of explicit rings of interest.
The overall goal of the project MRKT is to pursue hermitian K-theory using a new point of view, which allows for the dependence on the prime 2 to be overcome. This new approach is built on a series of papers by the PI and eight collaborators, where new theoretical foundations to hermitian K-theory are proposed using the modern apparatus of higher category theory, leading to a proof of Karoubi's conjecture and computations of the Grothendieck-Witt groups of the integers. In the MRKT project we take this approach further and use it to remove the dependence on the divisibility by 2 from hermitian K-theory in the context of motivic homotopy theory, to extend trace methods to the hermitian setting, and to tackle problems arising from quadratic enumerative geometry.
