Pre-MSRI Semester Seminar, Fall 2013:

Seminar Information

In the Fall 2013 semester, we ran a seminar at UC-Berkeley meant to introduce a variety of techniques from classical and modern homotopy theory, in preparation for the upcoming MSRI semester. On this page, you can find a variety of information about this seminar.

Contact information

Meeting time and location

Schedule of Talks

Future Talk Ideas

What follows is an inexhaustive list of topics that would fit well in this preparation seminar. I prepared a dependency graph of some of these topics so that participants can get a feel for what level of knowledge is required: [link].

This list is pretty intimidating! Sorry about that. It's mostly meant to point out key words, so that you can go looking and see what strikes your interest. The goal of this seminar is to give an introduction to ideas and methods in homotopy theory, so a light-hearted talk about any of these topics --- individually or in part, and using only a small fraction of the available references --- would serve perfectly well. Pick what looks coolest to you, learn a little about it, and then tell us. :)

Spectral sequences in general

Their construction, their use, convergence, cellular homology and the Atiyah-Hirzebruch spectral sequence.



Vector bundles, the splitting principle, the K-theory spectra, the complex orientation of KU, connective K-theories, the homology of K-theory, the J-homomorphism, Thom complexes, Adams operations via representation theory, Snaith's theorem.


The Hopf invariant one problem

Vector fields on spheres, Clifford algebras and K-theory, application of the Steenrod algebra, the cell structure of Stiefel manifolds and the cell structure of projective space, Spanier-Whitehead duals of projective spaces, James periodicity.


The Steenrod algebra

The algebra and its dual, multiplication and comultiplication, its action on cohomology groups, some examples of such actions, presentation as quadratic power operations.


The Serre spectral sequence

Construction of the spectral sequence (perhaps not with the identification of its E2-page), calculation of the cohomology of Eilenberg-Mac Lane spaces, examples of the cohomology of loopspaces, the cohomologies of some matrix groups and their classifying spaces, Serre's method for computing the homotopy groups of spheres, Kudo transgression theorem.


The (classical) Adams spectral sequence

Its construction, the identification of the E2-page, computations in a low range, the first differential and the 7-stem, comparison with Serre's method, applications to computations with K-theory, computations of cobordism rings.


The May spectral sequence

Its construction and philosophy, identification of the E1-term and its differential, Nakamura's formula, computing the 2-adic homotopy of real and complex K-theory, computations of the Adams E2-term through a large range.



The definition via chain complexes, the Pontryagin-Thom construction, cobordisms with structure (e.g., oriented, complex, framed), genera.


Bredon equivariance

Definition of G-spaces, reformulating basic tools for equivariant homotopy theory, examples of computing G-equivariant cohomology groups, the transfer map, stabilization and G-spectra, the fixed point and orbit spectral sequences, real Bott periodicity from complex Bott periodicity (more generally, geometric differentials), comparison with Borel equivariance.


Monoidal structures

A and E-structures, deloopings and the bar construction, Gamma-spaces, the smash product of spectra, structured ring spectra and Dyer-Lashof operations, recasting in ∞-categories.


Localization and completion

Arithmetic localizations, Bousfield localizations in general, chromatic localizations.


The Adams-Novikov spectral sequence

Generalized Chern classes for complex oriented cohomology theories, Quillen's theorem for complex bordism, formal groups and structure theorems for MU, low-range computations.


Formal groups

Formal group laws and isomorphisms, stacks, the p-local classification, Landweber's exact functor theorem, homology theories from the classification of formal groups, Morava's change of rings isomorphism, Lubin-Tate theory and the stabilizer group, the Serre-Tate theorem.


Chromatic homotopy theory

The image of J, nilpotence and periodicity, classification of field spectra, the chromatic spectral sequence and the Greek letter families, periodic homotopy groups and the homotopy of the E(1)-local sphere, chromatic convergence.


Abstract homotopy theory

Model categories, homological algebra as an example of a model category, homotopy limits and colimits, the connection to ∞-categories, co/simplicial objects and descent.


Unstable phenomena

The EHP spectral sequence and its connection to RP, low-dimensional computations, Mahowald's unstable v1-periodic calculations.



The spectrum of units, spherical fibrations, Thom spectra and their universal property, bordism orientations, orientations as manifold invariants, the Atiyah-Bott-Shapiro orientation of real K-theory.


Functor calculus

Excisive functors, homogeneous functors and the cross effect construction, the derivatives of the identity and mapping space functors, connections to algebraic K-theory through the cyclotomic trace, interactions with chromatic homotopy theory.


Algebraic K-theory


Connections to algebraic geometry

More on formal groups, the "algebraic geometry of spaces", elliptic cohomology and topological modular forms, the beta family and congruences of modular forms, absolutely any kind of broad overview.


Connections to representation theory

Equivariant vector bundles, the Atiyah-Segal completion theorem, the Chern character, and generalizations of the Chern character.


Connections to physics

The Atiyah-Singer index theorem, the Witten genus and the String orientation, the whole business with TQFTs, anomalies and algebraic topology.


Some references in bulk

Additionally, here is a short list of notes from a few very successful courses. I would be happy to hear talks about any subset of the contents of these notes, up to and including just re-running one of the courses in its entirety.

Also, here are yet more lists of talks / reference papers from other seminars which you might find useful. Many of these are more classical (and so often more basic) than those above.

This is a work in progress.