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Here are my current notes (last updated May 4th).

- John

Due to a conference in Regensburg, the Easter holidays and Ib Madsen's retirement conference, I will replace the April 7th and April 21st classes with lectures at the following times:

• Tuesday April 11th at 10.15-12.00 in room B738, and
• Wednesday April 19th at 14.15-16.00 in room B738.

We will resume the regular schedule on April 28th and May 5th. The exam will be on May 12th.

- John

Some references on the stable homotopy category:

• Adams: Stable homotopy and generalised homology (1974)
• Margolis: Spectra and the Steenrod algebra (1983)
• May: The additivity of traces in triangulated categories (2001)
• Whitehead: Generalized homology theories (1962)

The student representative this term is Alice Hedenlund.

The lectures have been moved to Fridays 10.15-14.00 in room B738.

- John

• Mandell-May: Equivariant orthogonal spectra and S-modules (2002)
• Stolz: Equivariant structure on smash powers of commutative ring spectra (2011)
• Hausmann: G-symmetric spectra, semistability and the multiplicative norm (preprint, 2014)
• Hill-Hopkins-Ravenel: On the nonexistence of elements of Kervaire invariant one (2016)
• Schwede: Lectures on equivariant stable homotopy theory (in preparation, 2016)

• Elmendorf-Kriz-Mandell-May: Rings, modules, and algebras in stable homotopy theory (1997)
• Hovey-Shipley-Smith: Symmetric spectra (1999)
• Mandell-May-Schwede-Shipley: Model categories of diagram spectra (2000)
• Schwede: Symmetric spectra ( in preparation, 2012)

Some references on point set topology:

• May-Ponto: More concise algebraic topology (2012)
• McCord: Classifying spaces and infinite symmetric products (1969)
• Steenrod: A convenient category of topological spaces (1967)
• Strickland: The category of CGWH spaces (notes, 2009)
• Str?m: The homotopy category is a homotopy category (1972)

Some references on topological K-theory:

• Atiyah-Hirzebruch: Vector bundles and homogeneous spaces (1961)
• Milnor: Morse theory (1963)
• Atiyah: K-theory (1967)
• Hatcher: Vector Bundles & K-Theory https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html

Some resources on (co-)bordism:

• Thom: Quelques propri¨¦t¨¦s globales des vari¨¦t¨¦s diff¨¦rentiables (1954)
• Stong: Notes on cobordism theory (1968)
• Milnor-Stasheff: Characteristic classes (1974)
• Hopkins: The Kervaire invariant problem (2016)

I plan to lecture about foundations and applications of stable homotopy theory.

As motivation, we may study examples of generalized (co)homology theories, such as bordism [Pontryagin, Thom] and $K$-theory [Bott, Atiyah-Hirzebruch].  Such functors become representable in the stable homotopy category [Boardman, Adams], which is triangulated and closed symmetric monoidal.  In the 1990s, this structure was found to arise as the homotopy category of a model category, in several ways, including $S$-modules, symmetric spectra and orthogonal spectra [Elmendorf-Kriz-Mandell-May, Hovey-Shipley-Smith, Lydakis, May-Mandell-Schwede-Shipley, ...].  We will follow a book project by Stefan Schwede, titled "Symmetric Spectra", as an introduction to these results.  Thereafter we will turn to the equivariant theory [Adams, Lewis-May-Steinberger, Mandell-May, Stolz, Hausmann, ...], with possible applications to the Kervaire invariant one problem [Hill-Hopkins-Ravenel].