J. Francis Baer | Searcharxiv

arXiv2025

Stable comodule deformations and the synthetic Adams-Novikov spectral sequence

We study the Adams-Novikov spectral sequence in $\mathbb{F}_p$-synthetic spectra, computing the synthetic analogs of $\mathrm{BP}$ and its cooperations to identify the synthetic Adams-Novikov $\mathrm{E}_2$-page, computed in a range with a synthetic algebraic Novikov spectral sequence. We then identify deformations associated to the Cartan-Eilenberg and algebraic Novikov spectral sequences in terms of stable comodule categories, categorifying an algebraic Novikov spectral sequence result of Gheorghe-Wang-Xu. We then apply Isaksen-Wang-Xu methods in $\mathbb{F}_2$-synthetic spectra to deduce differentials in the $p=2$ synthetic Adams-Novikov for the sphere, producing almost entirely algebraic computations through the 45-stem.

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arXiv2024

The algebraic Novikov spectral sequence for topological modular forms

We compute the $\mathbb{C}$-motivic Adams spectral sequence for $\mathit{mmf}/τ$. Up to reindexing, this spectral sequence is isomorphic to the algebraic Novikov spectral sequence for topological modular forms. We give a full analysis of the inclusion and projection maps which occur in the long exact sequence induced by multiplication by $τ$ on Adams $E_2$-pages. Our computation is totally algebraic and can be used to obtain the $E_2$-page of the Adams-Novikov spectral sequence for topological modular forms in a way that is independent from previous computations using the elliptic curves Hopf algebroid.

math.AT↗