Research

Here are my past and current research projects.

Current

• A motivic approach to equivariant synthetic spectra (j/w Keita Allen)

  Classically, (even, MU-based) synthetic spectra can equivalently be described in terms of cellular motivic spectra over the complex numbers after completion at a prime. We show that this persists in the equivariant setting at finite cyclic groups by comparing (suitably cellular) equivariant motivic spectra over the complex numbers with a candidate for (even, MU-based) equivariant synthetic spectra that is built to categorify the even filtration. The key technical tool is good control over the equivariant algebraic cobordism spectrum, in particular a sort of global functoriality and a cellular description, that allow us to use the machinery of global group laws and establish vanishing results needed to obtain an analogue of the Chow weight structure and Chow t-structure. A poster with a broad discussion of this project 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 Here’s a pdf (contains outdated results, some redundant exposition, and some mistakes so beware!)