ICM 2026 Invited Section Lecture - Jeremy Hahn

ICM 2026 Invited Section Lecture - Jeremy Hahn

by Jeremy Hahn · View the channel on PSI

Formal & Physical Sciences Mathematics PBMathematics
🎙 Jeremy Hahn 👥 60K 📅 September 14, 2026 ⏱ 39 min 👁 1 📄 original study 🧭 2026-09-14
Available in: English (current) Français

Keywords

homotopy theoryring spectratopological cyclic homologychromatic homotopytelescope conjecture

Summary

The lecture begins by contrasting singular cohomology and topological K-theory for topological spaces, then introduces their algebraic analogues for schemes: étale cohomology and topological cyclic homology (TC). The speaker explains that for many problems, algebraic K-theory and TC carry equivalent information, so he focuses on TC. He then introduces E-infinity rings as a generalization of commutative rings, where addition and multiplication are commutative and associative only up to coherent isomorphism. Key examples include the sphere spectrum S and the complex K-theory spectrum KU. The homotopy groups of S, known as stable homotopy groups of spheres, are connected to counting exotic spheres via known results. The chromatic approach organizes these groups through a sequence of operations v1, v2, … after killing powers of earlier operations. The telescope conjecture of Ravenel predicted a specific finite-rank module structure after inverting v2 under certain conditions. The speaker presents a counterexample: for a carefully chosen height-one E-infinity ring R, the map from S to TC(R) yields an infinite image in degree one after inverting v2, contradicting the conjecture. The computation relies on the fact that TC raises chromatic height by one (redshift), and that killing p, v1, and v2 gives finite-dimensional homotopy groups. This counterexample, computed via syntomic cohomology and the motivic spectral sequence, provides new insight into the structure of stable homotopy groups.

219 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides substantial value by connecting classical algebraic topology with modern chromatic homotopy theory and demonstrating a concrete recent result that settles a long-standing conjecture. The argument is carefully structured: the speaker motivates each step, from classical invariants to E-infinity rings, then to TC and the height-raising property, culminating in the counterexample to Ravenel’s telescope conjecture. The exposition is mathematically rigorous, with clear explanations of the underlying concepts and their relevance. The speaker also contextualizes the result within the broader program of understanding homotopy groups of spheres via cohomological invariants.

Scientific Rigor, Source Quality, Title Accuracy

The talk is scientifically rigorous: it relies on established theorems (e.g., Atiyah-Hirzebruch spectral sequence, comparison theorem, work of Hopkins-Smith, Miller, etc.) and clearly distinguishes proven results from conjectures. The title accurately describes the content, as the lecture indeed focuses on algebraic cohomology of ring spectra. The speaker does not cite specific publications with URLs, but references key names and theorems, which is appropriate for a lecture. The content aligns with the latest research, and the source is a reputable foundation (Simons Foundation) hosting an ICM invited lecture.

193 words

Title / Content Match

The title accurately reflects the content: a lecture on algebraic cohomology of ring spectra, delivered as part of an ICM invited section.

Quality & Reliability

8/10

Rigorous mathematical exposition by a recognized expert, presenting recent research results with clear logical structure. The content is advanced and technical, but the speaker provides appropriate context and examples. As an invited lecture, it reflects peer-approved research but is not a formal publication.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Cited Sources

  • Atiyah-Hirzebruch spectral sequence — Relates singular cohomology and topological K-theory for spaces.
  • Artin comparison theorem — States that singular cohomology of complex points equals étale cohomology with profinite coefficients.
  • Hopkins-Smith periodicity theorem — Provides basis for chromatic structure and the existence of v_n operations.
  • Miller's computation of v1-localized homotopy — Gives explicit structure of homotopy groups after inverting v1.
  • Ravenel's telescope conjecture — A conjecture about the structure of idempotent E-infinity rings; disproved in this lecture.
  • Ausoni-Rognes redshift conjecture — Predicts that TC increases chromatic height by one; proved in many cases recently.
  • Kervaire-Milnor exotic spheres — Connects stable homotopy groups of spheres to classification of exotic smooth structures.

Contribution & Novelties

This lecture presents a significant new counterexample to Ravenel’s telescope conjecture, a central problem in chromatic homotopy theory. The key insight is that topological cyclic homology raises the chromatic height of a ring spectrum by one, and this can be used to detect families of homotopy classes that contradict the conjecture’s finiteness predictions. The approach uses a careful computation of the TC of a height-one ring, leveraging the fact that after killing p, v1, and v2, the homotopy becomes concentrated in finitely many degrees, allowing complete determination. This work provides a new tool for understanding stable homotopy groups and opens avenues for further computation using syntomic cohomology.

Pour aller plus loin :

  • Chromatic homotopy theory — A broad overview of the field that contextualizes operations v_n and the telescope conjecture.
  • Topological cyclic homology — A technical definition and properties of TC, the central object of the talk.
  • Exotic sphere — Connection between stable homotopy groups and exotic smooth structures, as mentioned in the lecture.
  • Redshift conjecture — Notes on the conjecture that TC increases chromatic height, with references to Ausoni–Rognes.

180 words

Radar Profile

The profile shows a very high level of technical rigor and content quality, with a slightly lower score for quantity of information due to the concise 40-minute format. The extremely high niveau technique and fiabilité indicate an advanced, trustworthy presentation, appropriate for a specialist audience.

Reliability 9/10