My research interests lie within the algebraic theory of quadratic forms. In addition, I am interested in the overlap between quadratic form theory and the theory of algebras with involution, cohomology theories and algebraic geometry. 

A quadratic form, or homogeneous polynomial of degree two, is said to be isotropic if it has a non-trivial representation of zero, and is anisotropic otherwise. The hyperbolic plane H is the unique 2-dimensional isotropic (quadratic) form. By a theorem of Witt, every form q has a decomposition q = q_an + i(q) x H, where the integer i(q) and the anisotropic form q_an are uniquely determined. 

To every quadratic form q over a field F, one can associate a field extension called the function field of q, denoted F(q) (this is the function field of the quadric q(X)=0 over F). By construction, the extension of q to F(q), denoted q_F(q), is isotropic. Crucially, however, function fields of quadratic forms satisfy the following important property:

Letting q and p be two anisotropic forms over F, one has that i(p_F(q)) is less than or equal to i(p_K) for any field extension K of F such that q_K is isotropic.

In other words, the function field of an anisotropic form can be thought of as the most general field extension over which the form becomes isotropic. In particular, they represent the best hope for discriminating between the isotropy of one form and the anisotropy of another. As field invariants such as the u-invariant, the Hasse number, the level, the Pythagoras number, the length, etc., can be defined as the supremum of the dimensions of anisotropic forms of an associated type, the problems of determining their possible values represent one class of questions to which function fields (of quadratic forms) are ideally suited. Indeed, ever since their introduction, function fields have played a central role in the development of the theory of quadratic forms, underpinning a range of spectacular breakthroughs (for example, the resolution of the Milnor conjecture).

My current research has two central goals:

(i) furthering the understanding of the isotropy/anisotropy of forms with respect to function field extensions,

(ii) applying this understanding to resolve open problems in quadratic form theory

(for example, invariant determination problems, as motivated above).