A solution to the telescope conjecture, June, 2023.
In June 2023, a disproof of the telescope conjecture for chromatic heights ≥ 2 was announced by Robert Burklund, Jeremy Hahn, Ishan Levy and Tomer Schlank at the Panorama of Homotopy Theory Conference at the Mathematical Institute, Oxford University.
Their preprint appeared on October 26, 2023. K-theoretic counterexamples to Ravenel's telescope conjecture All items in their bibliography can be found in my archive.
NEW They will lead an Arbeitsgemeinschaft (Workshop) at Oberwolfach, Algebraic K-Theory and the Telescope Conjecture, October 13-18, 2024. The application deadline is June 15, 2024. Each participant is expected to give a talk on one of several listed topics. As of today (April 10) this list is not yet available.
Jeremy, Tomer, myself, Ishan and Robert at Oxford University, June 9, 2023. Photo by Matteo Barucco.
Videos of their four talks in Oxford: Ishan Levy, June 6. The lecture begins at 17:00. Tomer Schlank, June 7. The lecture begins at 18:00. Jeremy Hahn, June 8. This video includes a lecture by Soren Galatius. Jeremy's talk starts at 1:27:00. Robert Burklund, June 9. The lecture begins at 18:30. In each case you should put the window on the upper left in full screen mode.
Some background material. The conjecture first appeared in Localization with respect to certain periodic homology theories, 1984, as 10.5. My Glasgow lecture of May 24, 2022, "What is the telescope conjecture? A walking tour of modern homotopy theory." Slides. Zoom video recording. This talk was given for a general audience. It includes an account of "Morava's vision," which was first described in his unpublished 1972 preprint, The moduli variety for formal groups, recently transcribed by John Rognes. Jack sent me this long lost copy shortly after I gave the talk. Slides for my introduction to the Princeton Special Algebraic Topology Seminar of October 28, 2023, which featured lectures by each of the four authors. Contact Tony Bahri for notes on their talks. Print friendly version of my slides. I gave an eCHT minicourse (4 lectures) in December 2023. It covered the background and motivation for the conjecture, but not the BHLS disproof of it. Lecture 1: An algebraic prelude to chromatic homotopy theory, December 5 Slides   Print friendly version   Video Lecture 2: Morava's vision and the chromatic spectral sequence, December 7 Slides   Print friendly version   Video Lecture 3: From algebra to geometry, December 12 Slides   Print friendly version   Video Lecture 4: vh-periodic families and telescopes, December 14 Slides   Print friendly version   Video For deeper background on the algebraic apparatus behind the conjecture, see my first book, Complex cobordism and stable homotopy groups of spheres, also known as the green book. For more information about the chromatic filtration of the stable homotopy category, see my second book, Nilpotence and periodicity in stable homotopy theory, also known as the orange book. Graduate course by Mark Behrens on Algebraic K-theory and the telescope conjecture, Fall 2023. This site includes handwritten lecture notes and videos.
Quanta story of August 22, 2023 Podcast of January 24, 2024

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Created October 24, 2023.
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