ICM 2026 Invited Section Lecture - Jessica Fintzen

ICM 2026 Invited Section Lecture - Jessica Fintzen

by Jessica Fintzen · View the channel on PSI

Formal & Physical Sciences Mathematics PBMathematicsPBGGroups and group theory
🎙 Jessica Fintzen 👥 61K 📅 September 28, 2026 ⏱ 41 min 👁 499 📄 expert opinion 🧭 2026-10-06
Available in: English (current) Français

Keywords

p-adicrepresentationssupercuspidaldepth-zeroreductive groups

Summary

The video is an invited ICM 2026 section lecture by Jessica Fintzen, presenting the theory of representations of p-adic groups and their connections to finite groups of Lie type. She begins by introducing p-adic numbers and their topology, then defines p-adic groups (e.g., GL_n(Q_p)) and smooth representations. Using an explicit example (SL_2(Q_p)), she illustrates how supercuspidal representations are constructed via compact induction from finite group representations. The talk then surveys recent progress on describing the entire category of smooth representations: under a mild tameness condition (p not dividing the order of the Weyl group), the category decomposes into a product of blocks, each equivalent to the depth-zero representations of a smaller p-adic group. This reduction to depth-zero simplifies many problems, as depth-zero representations are governed by explicit Hecke algebras and finite group representations. She also mentions joint work with Jean-François Dat extending these results to representations over general coefficient rings (e.g., algebraically closed fields of characteristic ℓ ≠ p, or integrally closed rings). The talk is accessible to a broad mathematical audience while conveying the current state of the art.

180 words

Critical Evaluation

Value of the Information & Strength of the Argument

The talk provides substantial value by clarifying a highly technical area through a concrete example and a clear conceptual framework. The argumentation is well-structured: starting from foundational definitions, building to the construction of a representative supercuspidal representation, and then presenting the major categorical decomposition theorem. The speaker effectively justifies the importance of the results via applications to the Langlands program and automorphic forms. The logical flow is strong, with transparent motivations at each step.

Scientific Rigor, Source Quality, Title Accuracy

The scientific rigor is high: the speaker is an established researcher, and the results presented are based on preprints and collaborations that are explicitly named. The description lists the relevant joint preprints (with Alder, Mishra, Ohara, and with Dat). The title accurately corresponds to the content, as the lecture indeed covers representations of p-adic groups and their reduction to finite group representations. The presentation is careful with technical hypotheses (e.g., tameness conditions) and distinguishes between complex and general coefficient cases.

169 words

Title / Content Match

The title accurately reflects the content: an invited lecture on representations of p-adic groups and their connection to finite groups.

Quality & Reliability

9/10

Lecture by a leading expert in the field, clear exposition of advanced material, based on recent original research and preprints.

Key Moments

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

Cited Sources

  • Preprint on reduction to depth-zero (with J. Alder, M. Mishra, K. Ohara, August 2024) — Mentioned in the talk as joint work providing the complex coefficient case.
  • Preprint on general coefficient rings (with Jean-François Dat) — Mentioned as ongoing work extending the results to more general coefficient rings.

Contribution & Novelties

The talk presents a significant recent result: a decomposition of the category of smooth representations of p-adic groups into blocks equivalent to depth-zero representations of smaller groups. This yields a powerful reduction-of-problems strategy. The presentation also highlights new work with Dat on general coefficients, which interpolates different mod-ℓ settings.

Pour aller plus loin :

  • P-adic number — Foundational concept for the talk.
  • Langlands program — Major motivation and application.
  • Supercuspidal representation — Building blocks discussed in the lecture.
  • Hecke algebra — Key tool for depth-zero blocks.

86 words

Radar Profile

The radar profile shows very high scores in quality, technical level, and reliability, with a slightly lower but still solid quantity of information. This indicates a dense, expert-level presentation that is both reliable and technically rich.

Reliability 9/10