Topological modular forms literature list

Links to other people's reference lists

Doug Ravenel's

John Rognes'

Andrew Baker's

Audio files and slides from Jack Morava's birthday conference

Paul Goerss' preprints

The Newton Institute proceedings

This page compiles a list of suggested reading for the upcoming Talbot workshop on the construction of tmf. Many of the documents here are preliminary versions of papers, which I have posted with the kind permission of the authors. For subjects where I did not know any printed references, there are scanned notes from seminars and workshops, whose contents should be viewed with even more caution.

Suggested reading

On the Classical Constructions of Elliptic Cohomology

Segal: Bourbaki and the long version of the Bourbaki article

Jens Franke: On the construction of elliptic cohomology, Math. Nachr. 158, 1992, pp.43-65.

P.S. Landweber, D.C. Ravenel, R.E. Stong, Periodic cohomology theories defined by elliptic curves, Contemporary Mathematics 181, 1995, pp.317-337.

P.S. Landweber (Editor), Elliptic curves and Modular Forms in Algebraic Topology: Proceedings, Priceton 1986, LNM 1326, Springer-Verlag, Berlin, 1986.

Matthew Greenberg's Master's thesis: Constructing elliptic cohomology

Survey talks and articles

Jacob Lurie: A Survey of Elliptic Cohomology

Mike Hopkins (ICM 2002): Algebraic Topology and Modular Forms

Haynes Miller (slides from a talk in Barcelona, 2002) Elliptic Cohomology: A Perspective and Some Prospects

Mike Hopkins (Notes from a talk at Santa Barbara): Algebraic Topology and Differential Forms

General papers and the construction of topological modular forms

Stefan Schwede's notes from the Muenster Conference: Mike Hopkins I, Mike Hopkins II, Matthew Ando, Charles Rezk, (some Charles Rezk and) Haynes Miller, Haynes Miller II, Paul Goerss I, Paul Goerss II.

Lars Hesselholt's notes from Mike Hopkins' 2004 class on tmf: Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10, Part 11, Part 12, Part 13, Part 14, Part 15.

John Rognes: Topologiske Modulaere Former

Hopkins, Miller: Elliptic curves and stable homotopy theory

Hopkins, Miller: On the Hopkins Miller theorem

Mike Hopkins (lecture notes): Course Notes for Elliptic Cohomology

Paul Goerss, Charles Rezk: Bonn lectures

Charles Rezk: Supplementary Notes for Math 152

M. Hopkins and M. Mahowald: From Elliptic Curves to Homotopy Theory, preprint, MIT and Northwestern University, June 1998

The book by Thomas: Ellliptic cohomology

E.S. Devinatz and M.H. Hopkins Homotopy fixed point spectra for closed subgroups of the Morava Stabilizer groups

Charles Rezk Notes on the Hopkins-Miller Theorem

Mark Mahowald and Charles Rezk: topological modular forms at level 3.

Stacks and their role in homotopy theory

Paul Goerss: (Pre-)sheaves of ring spectra over the moduli stack of formal group laws

Mike Hopkins: Complex Oriented Cohomology Theories and the Language of Stacks

Bertrand Toen, Gabriele Vezzosi: "Brave New" algebraic geometry and global derived moduli spaces of ring spectra

Niko Naumann: Comodule categories and the geometry of the stack of formal groups

Niko Naumann: Quasi-isogenies and Morava stabilizer groups

Haynes Miller: Sheaves and the exact functor theorem

A seminar talk by Jacob Lurie on the role of stacks in homotopy theory and the Landweber exact functor theorem: Part 1 Part 2

Goerss-Hopkins obstruction theory

Paul Goerss and Mike Hopkins: Moduli spaces of commutative ring spectra

Paul Goerss and Mike Hopkins: Moduli Problems for structured ring spectra

K(1)-local topological modular forms

Gerd Laures: K(1)-local Topological Modular Forms

Mike Hopkins: K(1)-local E_oo Ring Spectra

K(2)-local topological modular forms

Some of this can be found in Hovey Strickland: Morava K-theories and localizations. This paper also is the only one with a phantom discussion which is strong enough to deal with the construction of the Morava E-theories.

Another reference that speaks about the K(2)-local picture are the Bonn notes by Goerss-Rezk (above) and the string orientation paper by Ando-Hopkins-Rezk below.

Here are a few handwritten notes from a seminar talk on the topic:

Talk on the construction of the Hopkins-Miller spectra and an overview over what happened so far in the seminar and where we are going.

Talk on the construction of tmf, the K(2)-localization of tmf ...

... notes from a conversation with Mike ... and ... a second conversation with Mike ... (both on K(2)-local topological modular forms)

... and a third one ... on K(2) local topological modular forms and the construction of tmf.

Notes from conversations with Johan ... and ... Haynes.


Tilman Bauer: Computation of the homotopy of the spectrum tmf

Tilman Bauer: Elliptic cohomology and projective spaces - a computation -


Ando, Hopkins, Rezk: orientations

Ando, Hopkins, Rezk: MString --> tmf

Ando, Hopkins and Strickland: Elliptic Spectra, the Witten Genus and the Theorem of the Cube

Equivariant Elliptic Cohomology

Ginzburg, Kapranov, Vasserot: Elliptic Algebras and Equivariant Elliptic Cohomology

Ioanid Rosu: Equivariant elliptic cohomology and rigidity

David Gepner's thesis: Homotopy topoi and equivariant elliptic cohomology

Greenlees Hopkins Rosu: Rational circle equivariant elliptic cohomology

Jorge Devoto: Equivariant elliptic cohomology and finite groups

Matthew Ando: Equivariant Elliptic Cohomology and Rigidity

Ando, Basterra: The Witten Genus and Equivariant Elliptic Cohomology

Ando: The sigma orientation for analytic circle-equivariant elliptic cohomology

John Greenlees:

Power Operations and Hecke Operators

Many of Andy Baker's papers

Charles Rezk (MIT notes): Notes on power operations

Matthew Ando: Thesis

Matthew Ando: Power Operations in Elliptic Cohomology and Representations of Loop Groups

Ando, Hopkins, Strickland: The Sigma Orientation is an H_oo-map

The K(2)-local sphere

Mark Behrens: A modular description of the K(2)-local sphere at the prime 3

Goerss, Henn, Mahowald, Rezk: A resolution of the K(2)-local sphere at the prime 3

Geometric Approaches

Teichner, Stolz: What is an elliptic object?

Po Hu, Igor Kriz: Conformal Field Theory and Elliptic Cohomology

Baas, Dundas, Rognes: Two-Vector bundles and forms of elliptic cohomology

On Elliptic genera


Haynes Miller:

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