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Abstracts for MPIM Topology Seminar: Miniseries on topological modular forms and synthetic spectra

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From topological to synthetic modular forms

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Speaker: 
Sven van Nigtevecht
Affiliation: 
Universität Bonn
Date: 
Mon, 26/01/2026 - 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
MPIM Topology Seminar: Miniseries on topological modular forms and synthetic spectra

This talk serves as a gentle introduction to both tmf and synthetic ideas. We will explain the need for and intricacies of the computation of the homotopy groups of tmf, and highlight the core principle of what makes our synthetic approach to this computation tick. This can even be demonstrated in the setting of chain complexes; in particular, familiarity with spectra is not required for this talk.

Recording

Techniques in filtered and synthetic homotopy theory

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Speaker: 
Christian Carrick
Affiliation: 
Universität Bonn
Date: 
Mon, 02/02/2026 - 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
MPIM Topology Seminar: Miniseries on topological modular forms and synthetic spectra

In this talk, we will survey a broad toolkit of techniques in filtered and synthetic homotopy theory that facilitate spectral sequence computations. There are many such techniques, and we will focus on the ones used crucially in our computation of the descent spectral sequence (DSS) of Tmf. We refine the descent spectral sequence to an E_infty-algebra in MU-synthetic spectra, and this endows the DSS with structure ranging from generalized forms of the Leibniz rule to temporary Toda brackets.

The actual computation

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Speaker: 
Jack Davies
Affiliation: 
Bergische Universität Wuppertal
Date: 
Mon, 09/02/2026 - 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
MPIM Topology Seminar: Miniseries on topological modular forms and synthetic spectra
The previous two talks have covered why we want to compute the descent spectral sequence for Tmf, the definition of the synthetic spectrum Smf, and seen an array of tools to work with synthetic spectra. In this talk, the rubber hits the road and we use detection methods, synthetic generalisations of the Leibniz rule and Moss' convergence theorem, and a truncated version of the Burklund--Hahn--Senger omnibus theorem to fully calculate this spectral sequence. This finally gives all the details to this computation, which was started around 30 years ago by Hopkins and Mahowald.

 

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