HHH | Prof. Lars Hesselholt | Rothschild Lecture: Higher algebra and arithmetic

Item: HHH | Prof. Lars Hesselholt | Rothschild Lecture: Higher algebra and arithmetic
Rating: 4
Author: PostSingularity Institute

by Prof. Lars Hesselholt

Formal & Physical Sciences Mathematics PBMathematics PBFAlgebra

🎙 Prof. Lars Hesselholt 👥 2K 📅 December 15, 2025 ⏱ 65 min 👁 108 📄 original study 🧭 2026-08-15

Available in: English (current) Français

Keywords

higher algebraspectraTHHTCzeta functionDeningerWitt vectorsperfectoidFrobeniusregularized determinant

Summary

In this Rothschild Lecture, Professor Lars Hesselholt explores the deep connections between higher algebra and arithmetic, motivated by a 20,000-year-old mistake: the natural numbers record only the result of counting, not the process. He proposes that by working with the sphere spectrum instead of the integers as the ground ring, one obtains a more fundamental arithmetic that eliminates denominators. The lecture begins by recalling the zeta function of a scheme and its conjectural cohomological interpretation, particularly Deninger’s vision. Hesselholt then introduces topological Hochschild homology (THH) and topological periodic homology (TP) as the correct cohomology theories in this higher algebraic setting. He demonstrates that for schemes over a finite field, TP gives a cohomological interpretation of the zeta function via regularized determinants, as envisioned by Deninger. Key results include the computation of THH and TP for F_p, revealing the ring of Witt vectors, and the appearance of an internal Frobenius operator, a purely higher algebraic phenomenon. The lecture also touches on the Bhatt-Morrow-Scholze integral p-adic Hodge theory and the role of the sphere spectrum in eliminating denominators. The talk is highly technical, aimed at experts, and provides a coherent vision for a new arithmetic based on higher algebra.

197 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture presents a compelling and original perspective on arithmetic, arguing that working over the sphere spectrum rather than the integers resolves longstanding issues such as denominators in Chern characters and provides a natural cohomological interpretation of zeta functions. The argumentation is rigorous, building from classical definitions to modern higher algebra, and is supported by concrete computations (e.g., THH(F_p)) and references to known theorems. The value lies in unifying diverse areas: algebraic K-theory, homotopy theory, and arithmetic geometry. The speaker’s expertise is evident, and the logical flow is clear, though the advanced nature of the content limits accessibility.

Scientific Rigor, Source Quality, Title Accuracy

The lecture demonstrates high scientific rigor, with precise definitions and careful reasoning. The speaker cites key works by Bökstedt, Hsiang, Madsen, Bhatt, Morrow, Scholze, and Deninger, among others, grounding the presentation in established research. The title accurately reflects the content, which indeed explores higher algebra and its arithmetic applications. The talk is part of a research programme at the Isaac Newton Institute, ensuring institutional credibility. No public comments were provided, so no analysis of audience reception is possible.

191 words

Title / Content Match

The title accurately reflects the content: the lecture connects higher algebra (spectra, THH, TC) to arithmetic (zeta functions, p-adic Hodge theory).

Quality & Reliability

8/10

Lecture by a leading expert in algebraic K-theory and higher algebra, presenting original research and known results with mathematical rigor. The content is advanced and assumes expertise, but the arguments are coherent and based on established theories.

Key Moments

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

Cited Sources

Isaac Newton Institute for Mathematical Sciences — Institutional page for the lecture series and research programme.
Isaac Newton Institute LinkedIn — Social media profile of the institute hosting the lecture.

Concurring Sources

Bhatt, Morrow, Scholze - Integral p-adic Hodge theory — Related work on integral p-adic Hodge theory, which the lecture references.
Bökstedt, Hsiang, Madsen - Topological cyclic homology — Foundational work on TC, which is central to the lecture.

Dissenting Sources

No discordant sources found — The lecture is consistent with established literature in the field.

Contribution & Novelties

The lecture presents a novel synthesis of higher algebra and arithmetic, arguing that the sphere spectrum provides a more fundamental foundation that eliminates denominators and yields a natural cohomological interpretation of zeta functions. The key innovation is the use of topological periodic homology (TP) to realize Deninger’s vision, with concrete computations for F_p. The talk also highlights the internal Frobenius operator as a purely higher algebraic phenomenon, offering new insights into characteristic p phenomena.

Pour aller plus loin :

Topological Hochschild homology — Background on THH and its role in algebraic K-theory.
Deninger’s approach to zeta functions — Overview of Deninger’s conjectural cohomological interpretation.
Integral p-adic Hodge theory — Related work by Bhatt, Morrow, and Scholze.
Spectrum (topology) — Foundational concept in stable homotopy theory.
Witt vectors — Appearance in TP(F_p) and arithmetic geometry.

134 words

Radar Profile

The radar profile shows very high scores in technical level and information quality, reflecting the advanced and rigorous nature of the lecture. The quantity of information is also high, but the global reliability is slightly lower due to the speculative elements (e.g., Deninger's conjecture). Overall, the lecture is a strong contribution to mathematical research.

Reliability 8/10