This is a book about the theory of topological modular forms. It is also a
record of the efforts of a group of graduate students to learn that theory at the
2007 Talbot Workshop, and so a book born of and steeped in the Talbot vision.
In the fall of 2003, Mike Hopkins taught a course at MIT about tmf . Our gen-
eration of Cambridge algebraic topologists, having survived and thrived in Haynes
Miller’s Kan seminar, found in Mike’s class our next, and really our last common,
mathematical crucible. The course hacked through the theory of algebraic modu-
lar forms, formal groups, multiplicative stable homotopy theory, stacks, even more
stacks, moduli stacks of elliptic curves, Bousfield localization, Morava K- and E-
theory, the arithmetic and Hasse squares, Andr´ e–Quillen cohomology, obstruction
theory for moduli of associative and commutative ring spectra—by this point we
were having dreams, or maybe nightmares, about the spiral exact sequence.
In the middle of the course, we all flew over to M¨ unster for a week-long workshop
on tmf with lectures by Mike, Haynes, Matt Ando, Charles Rezk, and Paul Goerss.
A transatlantic mix of students spent the late afternoons coaxing and cramming the
knowledge in at a cafe off Steinfurter Strasse; there we devised a plan to reconvene
and sketched a vision of what would become the Talbot Workshops: a gathering
for graduate students, focused on a single topic of contemporary research interest,
lectured by graduate students and guided by a single faculty mentor, having talks in
the morning and in the evening and every afternoon free for discussion and outdoor
activities, with participants sleeping and lecturing and cooking together under the
same roof. We pitched it to Mike and Haynes and they agreed to back a ragtag
summit. Talbot was born.
Three years later, in 2007, we decided to bring Talbot home with a workshop
on tmf , mentored by Mike Hopkins. Mike stopped by Staples on his way to the
workshop and picked up a big red “That was easy” button. Throughout the work-
shop, whenever he or anyone else completed a particularly epic spectral sequence
computation or stacky decomposition, he’d hit the button and a scratchy electronic
voice would remind us, “That was easy!” It became the workshop joke (for much
of it was evidently not easy) and mantra (for shifting perspective, whether to mul-
tiplicative stable homotopy or to stacky language or to a suitable localization, did
make the intractable seem possible).
This book is a record and expansion of the material covered in the Talbot
2007 workshop. Though the authors of the various chapters have brought their
own expositional perspectives to bear (particularly heroically in the case of Mark
Behrens), the contemporary material in this book is due to Mike Hopkins, Haynes
Miller, and Paul Goerss, with contributions by Mark Mahowald, Matt Ando, and