Higher Grothendieck-Witt groups and A1-homotopy theory

An inner product space over a commutative ring R is a finitely generated projective R-module equipped with a non-degenerate symmetric bilinear form. Inner product spaces are important everywhere in mathematics but also for instance in physics (e.g., Minkowski space), chemistry (e.g., crystallography) and computer science (e.g., design of codes for a band limited channel).

In general, the classification of inner product spaces is a very difficult problem. As an example, the classification of projective modules over the ring of integers Z is easy (there is, up to isomorphism, precisely one for every given rank) whereas the classification of inner product spaces over Z is unknown: for a given rank there are only finitely many isometry classes but we don't know how many (even positive definite) inner product spaces of rank 32 there are over Z.

Though still far from being trivial, the study of inner product spaces simplifies when one introduces stable equivalence: two inner product spaces X and Y are stably equivalent if there is a third such space Z and an isometry between the orthogonal sum of X and Z with the orthogonal sum of Y and Z. For instance, two inner product spaces over the ring of integers are stably equivalent if and only if they have the same rank and signature.

The set of stable equivalence classes becomes an abelian monoid under orthogonal sum and embeds into the Grothendieck-Witt group GW(R) of formal differences of stable equivalence classes. For many rings (such as fields and local rings in which 2 is a unit) two inner product spaces are isometric if and only if they have the same class in GW(R). For such rings, the classification of inner product spaces thus amounts to computing the group GW(R). The computation of these groups is greatly aided by the fact that they are part of a cohomology theory which allows us to compute GW(R) from "local data".

So far, most tools to compute the groups GW(R) only work when 2 is a unit in R which is a (hopefully unnecessary) restrictive assumption. The main objective of the proposal is to develop tools for computing GW(R) that don't need 2 to be a unit in R. A second objective is the study of GW(R) in the context of an algebraic analogue (A1-homotopy theory) of the continuous world around us which was used by Voevodsky in his work on the Bloch-Kato conjecture which won him the Fields medal.

The results of this pure mathematical research are expected to be of significant interest to the mathematical community. They will be published in peer-reviewed mathematical journals of highest standing and on websites such as arXiv and the PIs professional website at Warwick University. The results will also be presented at important international mathematical conferences worldwide, and at workshops and national seminars. Mathematicians will be able to use our progress and develop our results further in their subsequent work.

As is typical for pure mathematical research, there is often a significant delay between the development of a mathematical theory and its implementation in industry. Potential long-term beneficiaries might be in business and industry, but this will only occur well beyond the duration of the project. The proposal is part of an effort to develop a new cohomology theory and to understand the classification of inner product spaces and lattices.
Potentially, this could have numerous applications outside mathematics for instance in crystallography and in the construction of optimal codes.