## Research

jump to publicationsMy 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 20^{th} 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 A^{1}-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:

- Jens Hornbostel, Matthias Wendt, Heng Xie, MZ.

*The real cycle class map*(Preprint 2019)

- Manuel Amann, David González-Álvaro, MZ.

*Vector bundles of non-negative curvature over cohomogeneity one manifolds*

(Preprint 2019)

- Jens Hornbostel, Heng Xie, MZ.

*Chow-Witt rings of split quadrics*

To appear in the proceedings

*Motivic homotopy theory and refined enumerative geometry*in the AMS series*Contemporary Mathematics*. - David González-Álvaro, MZ.

*Open manifolds with non-homeomorphic positively curved souls*

Accepted for publication in Math. Proc. Cambridge Philos. Soc. (2019)

The accompanying C++ code for generating Eschenburg spaces and computing their invariants is available from zenodo.

- David González-Álvaro, MZ.

*The stable converse soul question for positively curved homogeneous spaces*

Accepted for publication in J. Differential Geom. (2019)

*The γ-filtration on the Witt ring of a scheme*

*Nilpotence in Milnor-Witt K-Theory*

Appendix to: Hornbostel, Jens:

*Some comments on motivic nilpotence*

Trans. Amer. Math. Soc. 370 (2018), 3001–3015*Symmetric representation rings are λ-rings*

*KO-Rings of Full Flag Varieties*

*Witt Groups of Curves and Surfaces*

*Twisted Witt Groups of Flag Varieties*

*Witt Groups of Complex Cellular Varieties*

Documenta Math. 16 (2011), 465–511.

*Witt Groups of Complex Varieties*

PhD thesis (2011). Also available from DSpace@Cambridge.

*Comparing Grothendieck-Witt Groups of a Complex Variety to its Real Topological K-Groups*

Smith-Knight/Rayleigh-Knight Prize Essay (2009). Also available from DSpace@Cambridge .