ICM 2026 Plenary Lecture - Jacob Lurie

ICM 2026 Plenary Lecture - Jacob Lurie

by Jacob Lurie

Formal & Physical Sciences Mathematics PBMathematics
🎙 Jacob Lurie 👥 58K 📅 August 17, 2026 ⏱ 48 min 👁 12 📄 expert opinion 🧭 2026-08-17
Available in: English (current) Français

Keywords

prismatic cohomologystable homotopy theoryalgebraic K-theorytopological cyclic homologysyntomic cohomology

Summary

In this ICM 2026 plenary lecture, Jacob Lurie presents a series of conjectures inspired by recent developments in prismatic cohomology. He begins by reviewing complex K-theory as a generalized cohomology theory in topology, highlighting its utility in solving problems like the Hopf invariant one problem. He then introduces algebraic K-theory in algebraic geometry, noting its computational difficulty, and discusses topological cyclic homology (TC) as a computable approximation. Lurie explains that TC can be expressed as Ext-groups in a category of cyclotomic spectra, and that THH satisfies a Künneth formula. He draws parallels with the story of motivic cohomology and syntomic cohomology, which can be described via prismatic F-gauges. The central conjecture is the existence of a ‘prismatic stable homotopy category’ that unifies these stories, with objects representing cohomology theories on p-adic formal schemes, satisfying a Künneth formula. Lurie outlines the desired properties of this category and its relation to known invariants, suggesting a broad generalization of the phenomena observed in K-theory and cohomology.

163 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a high-level overview of recent developments in prismatic cohomology and its connections to algebraic topology. Lurie’s argumentation is clear and logical, building from classical examples to more abstract conjectures. He effectively motivates the need for a unifying framework and presents the conjectural prismatic stable homotopy category as a natural extension of existing theories. The value lies in its synthesis of diverse areas and the proposal of a new research program.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is scientifically rigorous, with Lurie referencing key works by Bhatt, Morrow, Scholze, and others. The sources are not explicitly cited in the video, but the content is based on well-established mathematical literature. The title accurately reflects the content, and the lecture is appropriate for a specialist audience. No external sources are provided in the description, so the analysis relies on the speaker’s expertise.

153 words

Title / Content Match

The title accurately reflects the content: a plenary lecture by Jacob Lurie on prismatic stable homotopy theory.

Quality & Reliability

9/10

Lecture by a leading mathematician (Jacob Lurie) at a prestigious venue (ICM 2026), presenting recent conjectures in prismatic stable homotopy theory. The content is highly technical and based on established mathematical frameworks, though it is a research-level presentation rather than a peer-reviewed publication.

Key Moments

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

Contribution & Novelties

The lecture presents a novel unifying conjecture for prismatic stable homotopy theory, extending the analogy between algebraic K-theory and topological cyclic homology to a broader framework. It synthesizes recent developments and proposes a new categorical structure that could have far-reaching implications.

Pour aller plus loin :

  • Prismatic cohomology — Overview of the theory introduced by Bhatt and Scholze.
  • Topological cyclic homology — Background on TC and its role in algebraic K-theory.
  • Stable homotopy theory — Foundational concepts in algebraic topology.

80 words

Radar Profile

The radar profile shows very high scores across all dimensions, reflecting the lecture's exceptional technical depth, clarity, and scientific rigor. The content is highly specialized and assumes advanced knowledge, making it most valuable for researchers in the field.

Reliability 9/10