Links to my publications can be found on the "My Website" part of this website.
My research interests lie mainly in the area of quadratic and bilinear form theory over fields of characteristic two. Due to the lack of correspondence between quadratic and bilinear forms over fields of characteristic 2, problems here often require slightly different or more subtle arguments than those in other fields.
There is a strong connection between bilinear forms, and indeed quadratic forms, and differential forms when working over fields of characteristic 2. Discovered by Kato in 1982, this connection means that many problems concerning bilinear forms and quadratic forms can be translated into the language of differential forms, where they are often easier to handle.
During my doctoral work I used this correspondence to find the Witt kernel of bilinear forms when passing to any simple extension. This was then extended to calculate the Witt kernel when passing to any function field in joint work between Deltlev Hoffmann and myself. Our calculations on differential forms were held for any field of characteristic different from 0.
Algebras with involution
Much work has been done recently on generalising concepts from quadratic form theory to the closely related, but more general, setting of algebras with involution. Of particular interest in characteristic 2 is the concept metabolicity. I have been studying the connection between metabolic hermitian forms and algebras with involution in characteristic 2, and have determined which involutions become metabolic over a given quadratic extension.
I have also been working on an extension of the usual Witt decomposition theorem which occurs only over fields of characteristic 2. This theorem has allowed me to obtain strong results on the effect of generic splitting of the algebra and the effect of passing to the separable closure on the anisotropy its involutions. It can also be used to show that one can associate a bilinear Pfister form to any decomposable algebras with involution which captures the isotropy behavior of the involution.
Algebras with quadratic pair
Quadratic pairs are associated with quadratic forms in a similar way to how we associate an involution to a bilinear form. In characteristic different from 2, a quadratic form is the same thing as a symmetric bilinear form, and analogously quadratic pairs are simply orthogonal involutions over these fields. In characteristic 2, however, these objects are distinct.
I have been working on generalising known results on involutions over fields of characteristic different from 2, to quadratic pairs in characteristic 2.