Maxim Kontsevich - Stability conditions in Fukaya categories: old and new ideas

Maxim Kontsevich - Stability conditions in Fukaya categories: old and new ideas

by Maxim Kontsevich

Formal & Physical Sciences Mathematics PBMGeometryPBMPDifferential and Riemannian geometry
🎙 Maxim Kontsevich 👥 79K 📅 June 12, 2026 ⏱ 78 min 👁 1K 📄 research talk 🧭 2026-08-02
Available in: English (current) Français

Keywords

stability conditionsFukaya categoriessymplectic manifoldsBridgeland stabilityhomological mirror symmetry

Summary

In this research talk, Maxim Kontsevich discusses stability conditions in Fukaya categories, a central topic in homological mirror symmetry. He begins by recalling the abstract notion of Bridgeland stability conditions on triangulated categories, including the support property and the geometry of semistable objects in the plane. He then introduces a conjectural framework called ‘Kähler stability’ for Fukaya categories, where stability structures are expected to arise from closed complex-valued differential forms on a symplectic manifold. The talk covers the construction of line bundles on moduli spaces of semistable objects and the idea that these moduli spaces should be quasi-projective. Kontsevich emphasizes the non-Archimedean nature of the Fukaya category and draws parallels with representations of quivers. He presents ongoing joint work with collaborators, aiming to formulate precise conjectures. The talk is highly technical, aimed at researchers in the field, and concludes with open questions and future directions.

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Critical Evaluation

The talk by Maxim Kontsevich is a high-level research presentation that offers deep insights into the current state of stability conditions in Fukaya categories. As a leading expert, Kontsevich provides a comprehensive overview of Bridgeland stability conditions, including the support property and the geometric interpretation of semistable objects. The introduction of ‘Kähler stability’ is a novel and speculative framework that aims to unify various aspects of the theory. The talk is well-structured, starting with foundational concepts and gradually building up to new conjectures. The mathematical rigor is high, but the speculative nature of the new ideas is acknowledged by the speaker. The sources cited are minimal, but the talk is based on published and ongoing work with collaborators. The title accurately reflects the content. The talk is not suitable for a general audience due to its advanced level, but for researchers in the field, it provides valuable insights and open problems. The absence of a detailed bibliography is a minor weakness, but the talk is intended as a research seminar rather than a comprehensive review.

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Title / Content Match

The title accurately reflects the content, which focuses on stability conditions in Fukaya categories, discussing both established ideas and new conjectures.

Quality & Reliability

8/10

Talk by a leading mathematician (Fields medalist) at a prestigious institution (IHES), presenting advanced research ideas. The content is highly technical and based on ongoing collaborations, but as a research talk it is not peer-reviewed and contains speculative elements.

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

Contribution & Novelties

The talk presents a novel framework called ‘Kähler stability’ for Fukaya categories, which is a significant new idea in the field. It also provides a geometric interpretation of stability conditions and proposes concrete conjectures for future research.

Pour aller plus loin :

74 words

Radar Profile

The radar profile shows very high scores in technical level and information quality, reflecting the advanced and rigorous nature of the talk. The quantity of information is also high, but the reliability is slightly lower due to the speculative aspects.

Reliability 8/10