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.
References:
- The discussion of exact couples in Hatcher's Spectral Sequences in Algebraic Topology,
- Mosher-Tangora Chapter 7,
- Boardman's Conditionally Convergent Spectral Sequences,
- this.
K-theory
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.
References:
- Atiyah's K-theory,
- Units 3-5 and 10-11 of the VFoS notes,
- Switzer's Algebraic topology: Homotopy and homology around Theorem 16.2,
- Stong's Determination of H*(BO(k, ..., ∞); Z/2) and H*(BU(k, ..., ∞); Z/2),
- Snaith's Localised stable homotopy of some classifying spaces,
- Gepner-Snaith's On the motivic spectra representing algebraic cobordism and algebraic K-theory,
- Section 7 of the COCTaLoS notes.
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.
References:
- The first half of the VFoS notes,
- Section 6 of Atiyah's Thom Complexes,
- Atiyah-Bott-Shapiro's Clifford Modules,
- James's Cross-sections of Stiefel manifolds,
- Mosher-Tangora's section 4.
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.
References:
- Milnor's On the Steenrod algebra and its dual,
- Sections 2-3, 6 of Mosher and Tangora,
- Sections 6-9 of the VFoS notes,
- Section 7 of the COCTaLoS notes,
- Mike Hill's course notes Steenrod Operations: Axioms and Properties and Construction of the Squares.
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.
References:
- Mosher and Tangora's sections 8-9, 11-12,
- Sections 1-3 of Neil Strickland's Spectral Sequences notes,
- Switzer's chapter 16.
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.
References:
- Mosher and Tangora's section 18,
- Switzer's section 19.14,
- Chapters 2 and 3 of Ravenel's Complex cobordism and the stable homotopy groups of spheres,
- The first third of Haynes Miller's course notes The Adams Spectral Sequence,
- Nerses Aramian's notes The Adams Spectral Sequence,
- Sections II.3 and II.7 of Mark Behrens' Infinite families notes,
- Mark Behrens's videotaped lectures at a summer MSRI conference.
- Mike Hill's notes titled More Ext & Adams Differentials and ko Homology.
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.
References:
Cobordism
The definition via chain complexes, the Pontryagin-Thom construction, cobordisms with structure (e.g., oriented, complex, framed), genera.
References:
- Sections II.7 and II.8 of Buchstaber's notes,
- Chapter 12 of Switzer,
- Section IV.7 of Rudyak's Thom spectra, orientability, and cobordism,
- Chapter 2 of Hirzebruch's Topological methods in algebraic geometry,
- Neil Strickland's notes Cobordism and formal power series.
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.
References:
- Bredon's original paper Equivariant cohomology theories,
- Adams's notes Prerequisites (on equivariant stable homotopy) for Carlsson's lecture,
- Any subset of Paolo Masulli's masters thesis Equivariant homotopy: KR-theory,
- Any subset of Peter May's Equivariant homotopy and cohomology theory ("the Alaska notes"),
- Becker-Gottlieb's The transfer map and fiber bundles,
- Mike Hill's videotaped lectures at a summer MSRI conference,
- this MathOverflow question,
- Hill-Hopkins-Ravenel's The homotopy of EO2p-2.
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.
References:
- Segal's Categories and cohomology theories,
- May's The geometry of iterated loopspaces,
- Hovey-Shipley-Smith's Symmetric spectra,
- Elmendorf-Kriz-Mandell-May's Rings, modules, and algebras in stable homotopy theory (especially chapters I and II),
- Cohen-Lada-May's The homology of iterated loopspaces (mostly section 1, I think),
- Chapter 3 of Bruner-May-McClure-Steinberg's H∞ Ring Spectra and their Applications,
- Lurie's Higher Algebra. (This is asking a lot. An abridged, less polished version of Lurie's thesis is available [here], see sections 2.4-2.6.)
Localization and completion
Arithmetic localizations, Bousfield localizations in general, chromatic localizations.
References:
- Section III.14 of Adams's Stable homotopy and generalised homology,
- Adams's Localization and Completion (UChicago lecture notes),
- David White's pre-Talbot talk notes (there's also a video here),
- Sullivan's course notes Geometry Topology: Localization, Periodicity, and Galois Symmetry (especially sections 1-3),
- Margolis's Spectra and the Steenrod algebra (sections 7-9),
- Hirschhorn's Model categories and Their Localizations,
- Ravenel's Localization with respect to certain periodic cohomology theories,
- Hovey's Bousfield Localization Functions and Hopkins' Chromatic Splitting Conjecture.
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.
References:
- Ravenel's A Novice's Guide to the Adams-Novikov Spectral Sequence,
- Section II.9 of Mark Behrens's Infinite Families notes (these are a bit abbreviated),
- Sections 4-5 of the COCTaLoS notes,
- Kyle Ormsby's pre-Talbot talk [video],
- Section 1.3 and chapter 4 of Ravenel's Complex cobordism and the stable homotopy groups of spheres (the "green book").
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.
References:
- Sections 6-20 of the COCTaLoS notes,
- Sections 4-5 of Charles Rezk's 512 Notes,
- Hazewinkel's Formal groups and applications,
- Neil Strickland's course notes Formal groups,
- Neil Strickland's Formal schemes and formal groups,
- Various pieces of Neil Strickland's Functorial Philosophy for Formal Phenomena,
- Michel Lazard's Sur les groups de Lie formels a un parametre,
- The first section of Nicholas Katz's Serre-Tate local moduli,
- Devinatz-Hopkins's The action of the Morava stabilizer group on the Lubin-Tate moduli of lifts.
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.
References:
- Adams's On the groups J(X),
- Ravenel's Localization with respect to certain periodic cohomology theories,
- Vitaly Lorman's lecture notes Image of J,
- Hopkins's lecture notes Global methods in homotopy theory,
- Ravenel's lecture notes Localization and periodicity in homotopy theory,
- Ravenel's lecture notes Nilpotence and periodicity theorems in stable homotopy,
- Aaron Mazel-Gee's notes The homotopy of the E(1)-local sphere at an odd prime,
- Jack Morava's Stable homotopy theory and local number theory,
- Morava's notes Complex cobordism and algebraic topology,
- Miller-Ravenel-Wilson's Periodic phenomena in the Adams-Novikov spectral sequence,
- this Mathoverflow question and answer,
- Wurgler's notes Morava K-theories: A survey,
- Hopkins-Smith's Nilpotence and stable homotopy theory II,
- Hopkins's talk The work of Jack Morava [audio only].
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.
References:
- Quillen's Homotopical algebra,
- Quillen's Homology of commutative rings and On the (co)homology of commutative rings (these are different articles with slightly different perspectives on the same material),
- Bousfield's Cosimplicial resolutions and homotopy spectral sequences in model categories,
- Hirschhorn's Model categories and their localizations,
- Aaron Mazel-Gee's lecture notes Model categories for algebraists,
- Aaron Mazel-Gee's lecture notes Homotopy (co)limits,
- Sections 1, 3-4 of Lurie's Higher Topos Theory.
Unstable phenomena
The EHP spectral sequence and its connection to RP∞, low-dimensional computations, Mahowald's unstable v1-periodic calculations.
References:
- Sections 14-16 and 21 in Haynes Miller's Vector Fields on Spheres notes,
- Section 1.5 of Ravenel's Complex cobordism and the stable homotopy groups of spheres,
- Day 3 of last semester's seminar notes,
- Mahowald's The Image of J in the EHP Sequence.
Orientations
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.
References:
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.
References:
- Hal Sadofsky's notes Introduction to Goodwillie calculus,
- Goodwillie's original papers Calculus I, Calculus II, and Calculus III,
- Chapter 7 ("The Calculus of Functors") of Lurie's Higher Algebra,
- Arone's A generalization of Snaith-type filtration,
- Ching's A chain rule for Goodwillie derivatives of functors from spectra to spectra,
- Arone-Ching's Operads and chain rules for the calculus of functors,
- Kuhn's Goodwillie towers and chromatic homotopy: An overview,
- Kuhn's Localization of Andre-Quillen-Goodwillie towers and the periodic homology of infinite loopspaces,
- Arone and Mahowald's The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres,
- Notes from the Goodwillie Calculus Arbeitsgemeinschaft at Uni-Bonn,
Algebraic K-theory
References:
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.
References:
- Morava's Forms of K-theory (I would really like someone to speak about this),
Connections to representation theory
Equivariant vector bundles, the Atiyah-Segal completion theorem, the Chern character, and generalizations of the Chern character.
References:
Connections to physics
The Atiyah-Singer index theorem, the Witten genus and the String orientation, the whole business with TQFTs, anomalies and algebraic topology.
References:
- Atiyah's Topological quantum field theory,
- Freed's lecture notes Lectures on twisted K-theory and orientifolds,
- Anything from the middle of Nakahara's book Geometry, Topology, and Physics,
- Witten's Global anomalies in string theory,
- Freed-Witten's Anomalies in string theory with D-branes,
- Witten's Global gravitational anomalies,
- Lurie's preprint On the classification of topological field theories.
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.
- List of talks for the summer 2013 MSRI workshop in algebraic topology as run by Andrew Blumberg, Teena Gerhardt, and Mike Hill,
- List of papers available to be talked about in the famous Dan Kan seminar.