Research

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 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 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:

• The stable converse soul question for positively curved homogeneous spaces
  

  with David González-Álvaro (Preprint 2017)
  Also available from arXiv.org.

• Chow-Witt rings of split quadrics
  

  with Jens Hornbostel and Heng Xie
  To appear in the proceedings Motivic homotopy theory and refined enumerative geometry in the AMS series Contemporary Mathematics.

• Open manifolds with non-homeomorphic positively curved souls
  

  with David González-Álvaro
  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.

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

  Q. J. Math. 69 (2) (2018), 549–583.

• 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
  

  New York J. Math. 21 (2015), 1055–1092.

• KO-Rings of Full Flag Varieties
  

  Trans. Amer. Math. Soc. 367 (2015), 2997–3016.

• Witt Groups of Curves and Surfaces
  

  Math. Zeits. 278 (2014), no. 1–2, 191–227.

• Twisted Witt Groups of Flag Varieties
  

  J. K-Theory 14 (2014), no. 1, 139–184.

• 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 .