ICM 2026 Special Section Lecture - Andrew Snowden

ICM 2026 Special Section Lecture - Andrew Snowden

by Andrew Snowden · View the channel on PSI

Formal & Physical Sciences Mathematics PBMathematics
🎙 Andrew Snowden 👥 60K 📅 September 10, 2026 ⏱ 41 min 👁 10 📄 original study 🧭 2026-09-10
Available in: English (current) Français

Keywords

tensor categoriespre-TannakianRep(S_t)oligomorphic groupsDeligne

Summary

This is an ICM 2026 Special Section Lecture by Andrew Snowden on oligomorphic groups and tensor categories. The talk begins with a background on tensor categories, emphasizing pre‑Tannakian categories as a natural abstraction of representation categories of algebraic groups. Snowden discusses the historical roots in Deligne’s work and the dichotomy between moderate and super‑exponential growth. He introduces the concept of oligomorphic groups—permutation groups with finitely many orbits on all finite Cartesian powers—and explains how they can be used to construct new tensor categories via a measure that resolves the infinite sums in matrix multiplication. The construction, joint with Nate Harmon, yields tensor categories that are rigid, self‑dual, but generally not abelian, thus lying outside the classical pre‑Tannakian framework. The talk covers several examples, including the interpolation of symmetric groups and classical groups, and contrasts the approach with ultraproduct constructions. The main contributions are the introduction of measures on oligomorphic groups and the explicit construction of tensor categories from them, opening a new direction for classifying super‑exponential growth categories. The talk is aimed at specialists, requiring familiarity with category theory and representation theory.

182 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture offers substantial mathematical value by providing a systematic construction of pre‑Tannakian categories with super‑exponential growth, a relatively unexplored area. The argumentation is clear and rigorous: Snowden builds the theory step by step, starting from the motivation, defining the necessary concepts (pre‑Tannakian categories, oligomorphic groups, measures), and then presenting the construction. The reliance on known theorems (e.g., Deligne’s reconstruction) and the careful handling of technical issues (infinite sums, coherence) demonstrates a high level of precision. The talk also highlights open problems, such as classification and explaining examples, which underscores the exploratory nature of the field. The reasoning is logical and well‐supported, though it presupposes a strong background in algebra, which limits accessibility.

Scientific Rigor, Source Quality, Title Accuracy

The scientific rigor is high: the talk is based on original joint work with Nate Harmon, and it references standard results from Deligne and others. However, no external sources or links are provided in the description, and the talk relies on the audience’s familiarity with the literature. The title is perfectly adequate—it exactly describes the two central themes. The presentation is well‐organized, with clear definitions and examples, and appropriately situates the work within the broader research context. No comments were provided, so any public reception cannot be assessed.

216 words

Title / Content Match

The title precisely reflects the content, covering the intersection of oligomorphic groups and tensor categories.

Quality & Reliability

8/10

The lecture presents rigorous mathematical content, relying on well‑known theorems and original research, but lacks external sources or verification in the video itself.

Key Moments

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

Cited Sources

  • Joint paper with Nate Harmon — Core ideas of the talk are joint work with Nate Harmon, published about four years ago.
  • Cameron, P. – Oligomorphic Permutation Groups (book) — Cameron coined the term 'oligomorphic' and wrote an introductory book which Snowden recommends.
  • Deligne – Construction of Rep(S_t) — First example of a pre‑Tannakian category with super‑exponential growth, introduced by Deligne.

Contribution & Novelties

The lecture presents an original construction of tensor categories from oligomorphic groups, a novel approach that generalizes the interpolation method and provides a new source of examples with super‑exponential growth. The key new notions are measures on oligomorphic groups, which allow the resolution of infinite sums in the composition of transformation matrices, and the resulting category of ‘virtual permutation modules.’ This approach systematically produces rigid, self‑dual tensor categories that are typically not abelian, and it opens a pathway toward classifying pre‑Tannakian categories of super‑exponential growth.

Pour aller plus loin :

  • Tensor category — Provides the basic framework for the categories discussed, including symmetric monoidal structure.
  • Oligomorphic group — Defines the permutation groups with finitely many orbits on all finite products.
  • Fraïssé limit — A central tool for constructing oligomorphic groups, mentioned in the talk as a way to build many examples.

141 words

Radar Profile

The profile is heavily weighted toward technical level (9) while reliability and quantity are moderate (8). This indicates a high‑specialisation talk that goes deep into mathematical details but may not cover an extremely broad range of information. Quality information is also high, consistent with an expert lecture.

Reliability 8/10