Research

jump to publications

My interests lie in the areas of topology, homotopy theory and algebraic geometry. Within this broad context, a reoccuring theme of my research is the theory of symmetric forms on vector bundles. Such “symmetric bundles” appear in different guises in both topology and geometry, and the flavour of the methods available in their study depend heavily on the respective context.

In topology, symmetric bundles are a very classical theme: the theory of symmetric complex vector bundles over a topological space is equivalent to the theory of real vector bundles. In many cases, these can be studied effectively using real topological K-theory. This is a well-established generalized cohomology theory, formally similar to its complex counterpart in many ways, but intrinsically more complicated. For example, while complex K-theory is 2-periodic, real K-theory is 8-periodic.

In algebraic geometry, interest in true symmetric bundles is more recent. There are, however, classical roots: a symmetric bundle over a point (a field) is simply a non-degenerate symmetric bilinear form (over the field), the study of which was pioneered by Ernst Witt in the early 20th century. The definitions were first extended to a geometric context in the 1970’s by Knebusch, but it was not until around 2000 that Balmer lifted the theory of symmetric bundles and Witt groups into the realm of cohomology theories, by introduced Witt groups of triangulated categories.

In my thesis, I used advances in A1-homotopy theory to compare the topological and algebro-geometric approaches for certain “cellular” spaces. My further publications pay tribute to the different flavours of the subject, meandering between purely topological, algebra-geometric, and purely algebraic perspectives: