Research

Here are my past and current research projects.

Current

• Synthetic equivariant spectra for finite abelian groups and motivic homotopy theory (j/w Keita Allen)

  We apply techniques from global homotopy theory in the setting of equivariant motivic homotopy theory to obtain a structural description of the cellular subcategory. After completion at a prime, we show that this category is entirely topological, in that it is equivalent to the completion of synthetic equivariant spectra. The latter are an appropriate generalisation of syntehtic spectra to the equivariant setting: they give rise to a perfect even filtration on equivariant spectra, are related to (and in fact determined by) the equivariant Adams–Novikov spectral sequence, and form a one-parameter deformation with special fibre living over the moduli stack of equivariant formal groups. arXiv. An early poster with a more broad-audience discussion can be found here

• An equivariant Adams spectral sequence based on MO

  The global spectrum MO for elementary abelian 2-groups represent the universal equivariant 2-torsion global group law. Further, at the trivial group its associated Hopf algebroid is equivalent to the classical dual Steenrod algebra. Their respective Adams spectral sequences are related by a series of Bockstein spectral sequences, which give us a systematic way to produce classes in an equivariant analogue of the Adams spectral sequence. A poster with a broad discussion of this project can be found here.

Past

• Deformations of stable homotopy theory

  This was my master’s thesis in which I looked at a filtered model for synthetic spectra, and its interpretation as a relative affineness result. The thesis further attempts to construct a partial universal property for synthetic spectra as a deformation of spectra with a given algebraic fibre, in terms of this filtered characterisation. Here’s a pdf (contains outdated results, some redundant exposition, and some mistakes so beware!)