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.

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.

The preprint

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.

Recognition

NEWIn 2024 Ishan won a Clay Research Fellowship.

April 14, 2026, from the Clay Mathematics Institute:

A Clay Research Award is made to Robert Burklund (Copenhagen), Jeremy Hahn (MIT), Ishan Levy (IAS and CMI), and Tomer Schlank (Chicago) in recognition of their remarkable construction of counterexamples to Ravenel's Telescope Conjecture.

The Telescope Conjecture was the last open conjecture from Ravenel's visionary paper Localization with respect to certain periodic homology theories. That paper, and the body of work it inspired, form the bedrock of chromatic homotopy theory. In one version, the telescope conjecture postulates an upper bound on the growth rate of the chromatic layers of the stable homotopy groups of spheres. The work of Burklund, Hahn, Levy and Schlank is the crest of a revolutionary new wave in K-theoretic techniques, to which they have each, independently, contributed. Their counterexamples imply that the p-rank of the stable homotopy groups of spheres grows faster than expected, and contains a proliferation of elements that are unaccountable by any prior understanding of the subject. This is a milestone achievement.

In 2024 they led

• An Arbeitsgemeinschaft (study group) at Oberwolfach, Algebraic K-Theory and the Telescope Conjecture, October 13–18.
• A Hot Topics Workshop at SLMath, Berkeley, Life after the Telescope Conjecture, December 9–13.

Some background material

• NEWI have recently written two expository papers on the background of the BHLS proof, What are cyclotomic spectra and why do we need them? and Cyclotomic extensions in stable homotopy theory.
• 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. Jack sent me this long lost copy shortly after I gave the talk. It was later retypeset by John Rognes here.
• 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: v_h-periodic families and telescopes, December 14
    Slides Print friendly version
• 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.

In the press