Sam Raskin - 6/6 Some Aspects of the Geometric Langlands Program

Sam Raskin - 6/6 Some Aspects of the Geometric Langlands Program

by Sam Raskin

Formal & Physical Sciences Mathematics PBMGeometryPBMWAlgebraic geometry
🎙 Sam Raskin 👥 79K 📅 March 19, 2026 ⏱ 131 min 👁 1K 📄 lecture course 🧭 2026-08-02
Available in: English (current) Français

Keywords

geometric Langlandsautomorphic formsspectral sidenilpotent singular supportexcursion algebra

Summary

In this sixth and final lecture of the series, Sam Raskin continues his discussion of the geometric Langlands program, focusing on the spectral side and recent joint work with Dennis Gaitsgory and others. He begins by recalling the main theorem from the previous lecture: for a smooth projective curve over a field of characteristic p, there is an equivalence between the category of sheaves with nilpotent singular support on the moduli stack of G-bundles and the category of ind-coherent sheaves on the stack of restricted local systems. He then addresses a question from a previous session about the trace of Frobenius map, explaining that the isomorphism implies the space of compactly supported automorphic forms is isomorphic to global sections of the dualizing sheaf on the stack of arithmetic local systems. He states three key results: the right-hand side is concentrated in cohomological degree zero and is reduced; the natural map from the structure sheaf to the dualizing sheaf is injective; and the unit element corresponds to the Poincaré function. He introduces a filtration on the category of ind-coherent sheaves indexed by nilpotent orbits, which induces a filtration on the space of automorphic forms. He discusses examples, including the case of SL2, and hints at connections to Arthur’s multiplicity conjectures and Eisenstein series. The lecture is highly technical and assumes familiarity with advanced topics in algebraic geometry and representation theory.

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Critical Evaluation

This lecture is a masterclass in advanced mathematics, delivered by a leading expert in the field. Sam Raskin’s presentation is rigorous and detailed, providing deep insights into the geometric Langlands program. The content is highly technical, assuming a strong background in algebraic geometry, representation theory, and category theory. The speaker carefully navigates complex concepts, such as ind-coherent sheaves, singular support, and trace of Frobenius, and connects them to concrete questions about automorphic forms. The lecture is part of a series, so it builds on previous material, but it also stands alone in its focus on the spectral side. The argumentation is solid, with clear logical progression and attention to subtle points, such as the necessity of right adjoints in the trace formalism. The sources are not explicitly cited in the lecture, but the work is presented as joint with Dennis Gaitsgory and others, indicating a strong research foundation. The title accurately reflects the content, and the lecture fulfills its promise to discuss recent work and partial results. The main limitation is the lack of explicit references, which is common in lecture settings. Overall, this is an excellent resource for researchers and graduate students specializing in the Langlands program, offering a unique perspective on current developments. The audience interaction adds value, as it clarifies potential misunderstandings. The lecture does not include any promotional content, and the focus remains purely scientific. The technical depth is exceptionally high, making it unsuitable for a general audience, but for the intended audience, it is a valuable contribution.

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Title / Content Match

The title accurately reflects the content: the sixth lecture in a series on aspects of the geometric Langlands program, delivered by Sam Raskin.

Quality & Reliability

8/10

Lecture by a leading expert (Sam Raskin, Yale) at IHES, part of a series on the geometric Langlands program. The content is highly technical and rigorous, with detailed mathematical arguments. The speaker is a recognized researcher in the field, and the presentation is part of an institutional seminar. However, the video is a lecture, not peer-reviewed, and the transcript contains informal remarks and audience interactions.

Key Moments

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

Cited Sources

  • Carmin.tv — Video platform for mathematics, mentioned in the description as hosting the lecture.

Concurring Sources

  • Carmin.tv — Platform hosting the lecture, consistent with the content.

Contribution & Novelties

This lecture presents recent research results in the geometric Langlands program, specifically focusing on the spectral side. The speaker introduces a filtration on the category of ind-coherent sheaves indexed by nilpotent orbits, which provides a new tool for understanding the structure of automorphic forms. This approach offers a novel perspective on the relationship between the geometric and arithmetic sides of the Langlands correspondence. The lecture also addresses open questions and conjectures, such as Arthur’s multiplicity conjectures, and suggests potential avenues for future research.

Pour aller plus loin :

  • Geometric Langlands correspondence — Overview of the geometric Langlands program.
  • Ind-coherent sheaves — Definition and properties of ind-coherent sheaves.
  • Nilpotent cone — The nilpotent cone in Lie algebra theory.
  • Arthur’s multiplicity formula — Arthur’s conjectures on multiplicities of automorphic forms.

128 words

Radar Profile

The radar profile shows very high scores in quantity and quality of information, reflecting the depth and rigor of the lecture. The technical level is maximal, indicating a highly specialized audience. The reliability score is slightly lower due to the informal nature of a lecture and lack of explicit citations, but still high given the speaker's expertise.

Reliability 8/10

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