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There are some MO posts about the uses of topological modular forms:

I am interested in learning about topological modular forms. But from an outsider's perspective (mine), the learning curve looks steep. I will explain my background below, but here is my question.

To either (1) understand the definition of topological modular forms, or (2) begin reading into some of the computations/applications of topological modular forms, what topics are important to know, and what references are useful for learning about them?

It seems like an answer to this could be useful to many people. Topological modular forms remain an active subject of research, while resources to learn the relevant tools (homotopy theory et al.) are not easy to navigate.

I tried to separate 'understanding the foundations' and 'understanding the applications/computations' in my question. My impression is that neither fully requires the other. (Maybe this is wrong.)

My own relevant background is tiny. I have finished my university's algebraic topology coursework (very geometric, used Hatcher, little homotopy theory), and I know some complex analysis. I have no experience with algebraic geometry, except some exposure to sheaves and sheaf cohomology. I am somewhat comfortable with category theory. In any case, I'm not necessarily asking for a roadmap for me at this very moment. That might be too ambitious.

Descartes Before the Horse
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  • The very first thing you need to do is to learn about spectra (in the sense of algebraic topology). Unfortunately there aren't many good references for that. After that, learning some chromatic homotopy theory (say, from Lurie's notes) and you should be ready to face the Tmf book. Some background on elliptic curves from a more algebro-geometric perspective would be nice but not required. – Denis Nardin Sep 28 '21 at 21:21
  • For basics on spectra, there is now a good reference: Foundations of Stable Homotopy Theory by David Barnes and Constanze Roitzheim. – Dmitri Pavlov Sep 28 '21 at 23:15
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    Lennart Meier has an absolutely amazing set of lecture notes on tmf. Maybe one strategy would be to first learn about spectra and then start reading them, pausing once in a while to fill any gaps you find in your background as you go on. – Emily Sep 29 '21 at 07:49
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    @DmitriPavlov I'll be honest - I'm not a huge fan of that book because it spends too much time on foundational issues and not enough on examples. – Denis Nardin Sep 29 '21 at 07:58
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    Part of your question is about learning about computations involving tmf. The AMS's release date for the new 690-page book of Bruner and Rognes, "The Adams Spectral Sequence for Topological Modular Forms", is in two weeks: https://bookstore.ams.org/surv-253 For detailed and careful computations involving tmf, this ought to be a very valuable reference. –  Sep 29 '21 at 12:44

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