ICM 2026 Invited Section Lecture - Sam Raskin

Value of the Information & Strength of the Argument

The talk offers substantial value by providing a coherent narrative connecting classical number theory to cutting-edge work in geometric Langlands. The argumentation is rigorous, with clear logical progression from historical conjectures to modern theorems. Raskin carefully distinguishes between established results and conjectural ideas, and he explicitly states the dependencies on previous work, such as Lafforgue’s and Deligne’s. He also highlights the subtlety of the naive Ramanujan conjecture and the need for Arthur’s refinement. The presentation is well-structured, and the speaker effectively conveys the depth of the subject while maintaining mathematical precision.

Scientific Rigor, Source Quality, Title Accuracy

The scientific rigor is high; the lecture is based on peer-reviewed results and formal collaborations. Raskin cites specific theorems and names the key contributors, including Deligne, Mordell, Laurent Lafforgue, and Vincent Lafforgue. The sources are appropriate and intersect with the established literature. The title accurately reflects the content: it is an invited section lecture at ICM 2026 delivered by Sam Raskin. However, the title is non-descriptive of the subject, being the speaker’s name and session type; the actual topic is only revealed through the video description. This could be considered a minor mismatch, but it is inherent to the formatting of ICM lectures.

The title accurately reflects the content as an invited lecture at ICM 2026, though it is generic and could apply to many lectures. The talk delivers on the announced topic.

9/10

The talk is a rigorous presentation of recent research results by a leading expert, based on established work in the Langlands program. The content is technically sound and well-referenced.

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

• 00:00Introduction to the talk and the Ramanujan conjecture.
• 03:00Discussion of the classical conjectures and the Langlands group.
• 07:00Setup for function fields: curves, reductive groups, and G-bundles.
• 12:00Hecke operators and cuspidal spaces.
• 18:00The Weil group, purity, and Laurent Lafforgue's theorem.
• 24:00Main theorem: Arthur parameters for unramified automorphic forms.
• 30:00Proof overview and role of geometric Langlands.

• Ramanujan–Petersson conjecture — Historical background and Deligne's proof mentioned in the talk.
• Laurent Lafforgue's proof of Langlands for GL_n — Referenced as the source of the purity theorem and the Langlands conjecture for GL_n over function fields.
• Geometric Langlands program — Mentioned as the underlying framework for the proof of the Arthur-Ramanujan conjecture.