Research

Here are my past and current research projects.

Current

• Synthetic isotropy separation

  We lift the theory of isotropy separation to the level of synthetic equivariant spectra, allowing us to reduce questions about the equivariant Adams–Novikov spectral sequence to nonequivariant information by means of iterated recollements. As a first application, we express the equivariant Adams–Novikov spectral sequence (for a finite abelian group) in terms of motivic stable homotopy groups of étale classifying spaces using a synthetic tom Dieck splitting and confirm the existence of a vanishing line of slope one and intercept zero, resolving a conjecture of Carrick. Second, we revisit the theory of the filtered circle (à la Raksit, Antieau–Riggenbach, etc.) as a Borel completion of the theory of genuine synthetic circle-equivariant spectra. This is part of my thesis work and will be uploaded soon.

Past

• Synthetic equivariant spectra and complex bordism (PhD thesis).

  My thesis, which is essentially a pushout of the motivic thing ↓ and the isotropy separation thing ↑. pdf.

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

  We analyse an appropriate cellular subcategory of equivariant motivic spectra over the complex numbers and show that (after completion at an arbitrary prime) this is equivalent to a suitable category of synthetic equivariant spectra, which categorifies an equivariant version of the perfect even filtration which we further relate to the Adams–Novikov spectral sequence. The key input in this motivic comparison argument is a description of the homotopy groups of torus-equivariant algebraic cobordism which we deduce using global methods. arXiv. An early research poster with a more accessible discussion can be found here

• 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. Available upon request.