Talks on the Arf-Kervaire invariant problem

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Talks about our solution to the Arf-Kervaire invariant problem and related topics

Lectures at the Isaac Newton Institute, Fall 2018
These informal talks are part of the program Homotopy harnessing higher structures. They cover selected topics from the book (see below), from which all handouts are taken. Most of them were recorded. Available video links are given below.
- October 15, 2018. Orthogonal G-spectra I. Hand written notes.
- October 17, 2018. Orthogonal G-spectra II. Hand written notes. Frank Adams writing about the hare and the tortoise, and my letter to him.
- October 22, 2018. Orthogonal G-spectra III. (This lecture was not recorded.) Hand written notes. First handout, diagram in middle of page. Second handout, table at bottom of page.
- November 12, 2018. Outlining the proof of the Kervaire invariant theorem.
- November 19, 2018. The Kervaire invariant problem at odd primes.
- November 26, 2018. The slice filtration and the slice spectral sequence.
- November 30, 2018. More about the slice filtration and the slice spectral sequence.
Slides for my talk on G-spectra at Lehigh University, May, 2015.
Slides for my talk on the Gap Theorem at Princeton University, March, 2015.
Slides for my talk on the C₄ spectrum at Northwestern University, March, 2013.
Mike Hill's Kervaire invariant workshop talks at University of Haifa, Israel, May, 2012.
The Browder's work on the Kervaire invariant problem, May, 2012.
Slides Conference photos
The 3-primary Arf-Kervaire invariant problem, February, 2012.
Slides
Abel Conference talk, February, 2012. Slides
Mike Hopkins' course on Equivariant Stable Homotopy Theory at Harvard, Fall 2011.

MSRI Hot Topics Workshop October 25-29, 2010.

Monday
- John Jones: The Kervaire invariant problem
- John Jones: Constructing elements with Kervaire invariant one
- Doug Ravenel: Overview of the proof
- Peter May: Introduction to equivariant homotopy theory I
  See also EQUIVARIANT ORTHOGONAL SPECTRA by Mandell and May, a 39 page version (written for this workshop) of their 107 page memoir of 2002 on the same subject.
Tuesday
- John Greenlees: Introduction to equivariant homotopy theory II For more background, see EQUIVARIANT STABLE HOMOTOPY THEORY (1995) by Greenlees and May.
- Mike Hill: The slice filtration
- Teena Gerhardt: The gap theorem
- Igor Kriz: The slice spectral sequence This is a report on the relevant parts of Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence (2001) by Hu and Kriz.
Wednesday
- Haynes Miller: The norm functor
- Nitu Kitchloo: Special elements
Thursday
- Mark Behrens: The Adams-Novikov spectral sequence
- Po Hu: The periodicity theorem
- Doug Ravenel: The detection theorem
Friday
- Andrew Blumberg: The homotopy fixed point theorem
- Dan Dugger: The slice theorem from the reduction theorem
- Mike Hill: The reduction theorem I
- Mike Hopkins: The reduction theorem II

Namboodiri Lectures, University of Chicago, April 1, 4 and 5, 2011.

First talk, April 1, slides print version, contains time sensitive material.
Second talk, April 4, slides print version.
Third talk, April 5, slides print version.

Oberwolfach talks of September 20-23, 2010, one by me and three by Hill, followed by a 2 hour question and answer session by him that amounted to a talk on the slice filtration.

First talk, September 20, slides print version

Second talk, September 20, my handwritten notes

Third talk, September 21, my handwritten notes

Fourth talk, September 23, my handwritten notes

Question and answer session, September 23, my handwritten notes

Middle East Technical University, May 18, 2010
Photo by Turgut Onder

My Ankara talks of May 17 and 18, 2010.

First lecture: Slides Print version
Second lecture: Slides Print version

Proceedings of Gökova Geometry-Topology Conference 2010
Includes a 43 page writeup of the two talks above.

The Princeton University Workshop on the Kervaire invariant problem of February 11, 2010, see below.
Our result was first announced on April 21, 2009, in a lecture by Hopkins at the Geometry and Physics: Atiyah80 conference. (This link has some more information.) His title was Applications of algebra to a problem in topology. (This title was often used for theses on algebraic topology written at Oxford in previous decades. It was used on this occasion to conceal the nature of the talk for dramatic effect.) Here are the slides for that talk; the file is 44MB. There is also a video.
I gave four talks on the theorem in Lisbon on May 5-7, 2009. Special thanks to Peter Landweber for bringing numerous typos in them to my attention.
- Lecture 1, May 5, 2009. Printer friendly version.
  This lecture discusses the work of Kervaire-Milnor on the classification of exotic spheres. Andrew Ranicki's talk on Kervaire's work, given in Geneva in February, with this more recent addendum, is a good online reference for this material. My lecture also includes a brief introduction to the classical Adams spectral sequence.
  
  Frank Adams
  1930-1989 Michel Kervaire
  1927-2007 John Milnor Andrew Ranicki
- Lecture 2, May 6, 2009. Printer friendly version.
  This lecture covers MU, formal group laws with Quillen's theorem, the Hopkins-Miller theorem, equivariant stable homotopy theory and MU as a C₂ spectrum.
  
  Peter Landweber Peter May Haynes Miller and Frank Peterson Daniel Quillen
- Lecture 3, May 7, 2009. Printer friendly version.
  Covers the homotopy of MU⁽⁴⁾, Postnikov towers, equivariant Postnikov towers, the slice spectral sequence, the gap theorem and RO(G)-graded homotopy.
- Lecture 4, May 7, 2009. Printer friendly version.
  Covers geometric fixed points, the slice and reduction theorems, the periodicity and slice differentials theorems, and the homotopy fixed point theorem.
- Supplement on the Detection Theorem. Printer friendly version.
Hill's talk at the Georgia Topology Conference, May, 2009. Slides
Hill's talk at the Isle of Skye Conference, June 15, 2009. Slides Handout
Hopkins' talk at the MIT seminar, July 8, 2009. Notes by Angelica Osorno.
Hill's talk at the MIT seminar, July 15, 2009. Notes by Angelica Osorno.
My talk at the MIT seminar, July 29, 2009. Slides Print version
Hill's talk at the Loen Conference, August 7, 2009. Slides

My talks in the UR Topology Seminar of September 25 and October 2, 2009.
- The first lecture is a colloquium style talk designed to be accessible to nontopologists. It includes information about the "World Without End Hypothesis" thought to hold before we proved our theorem. Slides Print version
- The second lecture introduces equivariant spectra and outlines the proof of the Gap Theorem. Slides Print version
Hopkins' talk at the Mayday Conference, October 16, 2009. My handwritten notes
Hill's talk at the Mayday Conference, October 18, 2009. My handwritten notes
My talk at the AMS Special Session on Homotopy Theory at Penn State, October 25, 2009. Slides Print version This talk is very similar to the one I gave at UR on September 25.
I gave six talks at Tokyo City University on November 4-6, 2009. Here are some photographs taken by Atsushi Yamaguchi. I also have videos taken by Nori Minami.
- Lecture 1: History and background of the problem. Slides Print version
- Lecture 2: Formal group laws and the Hopkins-Miller theorem. Slides Print version
- Lecture 3: Equivariant stable homotopy theory. Slides Print version
- Lecture 4: The slice spectral sequence and the gap theorem Slides Print version
- Lecture 5: The periodicity theorem. Slides Print version
- Lecture 6: The detection theorem. Slides Print version
Hopkins' talk (20 MB, be patient) at the 2010 AMS meeting (January 15), part of the AMS Current Events Bulletin.

The Princeton University Workshop on the Kervaire invariant problem of February 11, 2010, organized by William Browder and Martin Bendersky. This was the first occasion in which Hill, Hopkins and I have all spoken at the same time.
- Hill's talk, Equivariant computations and the gap theorem, handwritten notes.
- Hopkins' lunch time talk, The Kervaire invariant of immersions, handwritten notes.
- Browder's talk, The history of the problem, handwritten notes.
- Hopkins' colloquium talk, The Kervaire invariant problem, handwritten notes.
- My talk, The periodicity theorem in the solution to the Arf-Kervaire invariant problem, Slides Print version
  The slides contain more material than I was able to cover in an hour. This account is better than my earlier talks on this topic.
My talk at the Eighth Annual Graduate Student Topology and Geometry Conference in Ann Arbor, April 10, 2010.
Slides Print version These files are 13MB each, so the download may take some time.

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Created October 28, 2009; last modified February 11, 2015.