A Simplicial Set Whose Homology is the Higher Chow Groups of Bloch

A Simplicial Set Whose Homology is the Higher Chow Groups of Bloch

by James Austin Myer

Formal & Physical Sciences Mathematics PBMGeometryPBMWAlgebraic geometry
🎙 James Austin Myer 👥 1K 📅 March 4, 2026 ⏱ 85 min 👁 91 📄 original study 🧭 2026-08-16
Available in: English (current) Français

Keywords

simplicial sethigher Chow groupsBlochresolution of singularitiesSteenrod problem

Summary

The talk, given by James Austin Myer at the New York City Category Theory Seminar, presents a construction of a simplicial set associated to any algebraic variety, whose homology recovers the higher Chow groups of Bloch. The motivation stems from the unresolved problem of resolution of singularities in positive characteristic. Myer explains that in characteristic zero, resolutions exist (Hironaka), but in positive characteristic the problem remains open. He draws a connection to the classical Steenrod problem in topology, which asks whether every homology cycle can be represented by a manifold. Sullivan suggested that obstructions to this problem could be adapted to attack resolution of singularities. Myer’s construction aims to provide a framework for defining such obstructions in the algebraic setting. He outlines the classical obstruction theory using Steenrod operations and discusses the role of blowups and normal bundles. The talk is technical and assumes familiarity with algebraic geometry and topology, but the speaker provides intuitive explanations and engages with audience questions.

161 words

Critical Evaluation

Value of the Information & Strength of the Argument

The talk presents a novel construction with clear motivation from two major open problems: resolution of singularities and the Steenrod problem. The argumentation is logical, starting from the motivation, explaining the classical theory, and then introducing the new simplicial set. The speaker is careful to distinguish known results from conjectures and acknowledges uncertainties. The value lies in providing a concrete tool that may lead to progress on these problems. However, the talk is more of a research announcement than a full proof, and the construction is not fully detailed in the presentation.

Scientific Rigor, Source Quality, Title Accuracy

The speaker references classical works: Hironaka’s resolution of singularities, Sullivan’s 2004 memorial article for Thom, and the Steenrod problem as formulated by Eilenberg. These are appropriate and well-known. The title accurately describes the main content, though the talk also covers broader context. The presentation is rigorous in its mathematical reasoning, but the informal style and lack of written references in the description limit the ability to verify all claims. No comments were provided for analysis.

182 words

Title / Content Match

The title accurately reflects the main construction presented, though the talk also covers broader motivation and context.

Quality & Reliability

7/10

The talk presents original research in algebraic geometry, with a clear logical structure and references to classical results (Hironaka, Sullivan, Thom). However, the presentation is informal and lacks detailed proofs, making it difficult to fully verify the claims without additional literature.

Key Moments

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

Cited Sources

  • Sullivan's memorial article for René Thom (2004) — Mentioned as the source of the suggestion to adapt Steenrod problem obstructions to resolution of singularities.
  • Hironaka's resolution of singularities — Referenced as the solution in characteristic zero.
  • Steenrod problem — Classical problem in topology, attributed to Steenrod and Eilenberg.

Concurring Sources

  • Hironaka's resolution of singularities — Supports the claim that resolution exists in characteristic zero.

Contribution & Novelties

The talk presents a new construction of a simplicial set associated to any variety, whose homology is the higher Chow groups of Bloch. This provides a bridge between algebraic geometry and algebraic topology, potentially enabling the use of topological obstruction theory in the study of resolution of singularities. The construction is original and could lead to new insights.

Pour aller plus loin :

95 words

Radar Profile

The radar profile shows high scores in quality and technical level, reflecting the advanced mathematical content. The quantity of information is moderate, as the talk focuses on a specific construction. The overall reliability is good, but the informal presentation and lack of published references slightly lower the score.

Reliability 7/10