Misha Gromov - Novikov Conjecture and Scalar Curvature with Corners

Item: Misha Gromov - Novikov Conjecture and Scalar Curvature with Corners
Rating: 4
Author: PostSingularity Institute

by Misha Gromov

Formal & Physical Sciences Mathematics PBPTopology PBPDAlgebraic topology

🎙 Misha Gromov 👥 79K 📅 July 20, 2026 ⏱ 65 min 👁 596 📄 research talk 🧭 2026-08-02

Available in: English (current) Français

Keywords

Novikov conjecturescalar curvaturemanifolds with cornerssignaturepolyhedra

Summary

In this research talk, Misha Gromov discusses the Novikov conjecture and scalar curvature in the context of manifolds with corners. He begins by recalling the classical Novikov conjecture, which states that the higher signatures of a manifold are homotopy invariants. He then introduces a more elementary formulation based on the signature of pullbacks of generic points under maps to a background manifold. Gromov proposes extending these ideas to manifolds with corners, which are stratified spaces locally modeled on convex polyhedra. He defines a category of manifolds with corners and maps that preserve the corner structure, and conjectures that a version of the Novikov conjecture holds in this setting. He also relates this to scalar curvature, suggesting that positive scalar curvature conditions may be linked to the topology of the corner structure. Gromov discusses specific examples, such as convex polyhedra and reflection groups, and poses open questions about the interplay between combinatorial data and topological invariants. The talk is exploratory and informal, aimed at researchers in geometry and topology.

168 words

Critical Evaluation

The talk presents original and thought-provoking ideas at the intersection of topology and geometry. Gromov’s approach to the Novikov conjecture via manifolds with corners is innovative, offering a combinatorial perspective that may simplify or refine the classical conjecture. The argumentation is based on intuition and examples rather than formal proofs, which is typical for a research talk. The mathematical content is rigorous in its use of established concepts, but the lack of detailed derivations or references to specific literature limits its immediate verifiability. The speaker’s authority and the context of IHES lend credibility, but the informal style and exploratory nature mean that the claims should be treated as conjectures rather than established results. The talk does not provide a comprehensive review of existing work, but rather a personal perspective on potential research directions. The adéquation between title and content is good, as the talk indeed focuses on the Novikov conjecture and scalar curvature in the context of corners. Overall, the talk is valuable for researchers seeking new perspectives, but it requires a strong background in differential geometry and topology to fully appreciate.

182 words

Title / Content Match

The title accurately reflects the content, which focuses on the Novikov conjecture and scalar curvature in the context of manifolds with corners.

Quality & Reliability

8/10

Talk by a leading mathematician, presenting original research ideas and conjectures. The content is advanced and relies on established mathematical concepts, but the presentation is informal and exploratory, with no formal proofs or peer-reviewed references provided.

Key Moments

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

Cited Sources

Carmin.tv — Platform hosting the video and related scientific content

Concurring Sources

Novikov conjecture — General reference for the Novikov conjecture.
Scalar curvature — General reference for scalar curvature.

Contribution & Novelties

The talk proposes a novel framework for studying the Novikov conjecture and scalar curvature using manifolds with corners, potentially simplifying the conjecture and revealing new combinatorial invariants. It suggests that the topology of polyhedra can be probed through corner structures, and poses specific open questions.

Pour aller plus loin :

Novikov conjecture — Background on the classical conjecture.
Scalar curvature — Definition and properties.
Manifold with corners — General concept.
Reflection group — Relevant to the discussion of reflection groups.
Signature (topology) — Definition and properties.

86 words

Radar Profile

The radar profile shows high scores in technical level and information quality, reflecting the advanced mathematical content and the speaker's expertise. The lower score in information quantity is due to the exploratory nature of the talk, which focuses on ideas rather than comprehensive coverage.

Reliability 8/10