Zygotop seminar

We are an informal pedagogical seminar concerning algebraic topology and homotopy theory, localized at Harvard and MIT. Unless otherwise specified, talks will proceed on Wednesdays at 4:30pm ET, located in SC 310, and following a topic chosen by the speaker, organized according to these principles.

This seminar is organized by Keita Allen, who you should email with questions or to be added to the signal group chat. Click here to add this seminar to your google calendar.

• • Dec 10 2025

  Synthetic spectra and equivariant lifts of tau

  Keita Allen

  One of the most salient features of \(\mathbb{C}\)-motivic homotopy theory is the existence of the map \(\tau\). The results of various authors come together to tell us that the Adams-Novikov spectral sequence can be recovered as the \(\tau\)-Bockstein spectral sequence, and the work of Pstrągowski and Gheorghe, Isaksen, Krause and Ricka tell us that in fact a sizable portion of \(p\)-complete stable \(\mathbb{C}\)-motivic homotopy theory admits a description in terms of a ``categorification" of the Adams-Novikov spectral sequence, constructed purely topologically. These categorifications have come to be known as synthetic spectra, and they have been closely related to many compelling recent developments in homotopy theory.

  In my talk, I'll give a slightly ahistorical review of these two approaches to synthetic spectra, and how they are related to each other. I'll then discuss joint work with Lucas Piessevaux providing a generalization of these results to the \(A\)-equivariant setting, when \(A\) is a finite abelian group, and some consequences in equivariant motivic homotopy theory.

  References:

  • Perfect even modules and the even filtration by Pstrągowski.
  • Synthetic equivariant spectra for finite abelian groups and motivic homotopy theory by Allen and Piessevaux.