Ana Caraiani - Igusa Stacks IV

This lecture is a masterclass in arithmetic geometry, delivered by a leading expert. The content is highly technical and assumes a deep background in Shimura varieties, p-adic geometry, and the Langlands program. The speaker does an excellent job of motivating the study of intersection cohomology, connecting it to classical conjectures and recent developments. The presentation is logically structured, moving from definitions to motivation to the main theorem (the Igusa stack diagram) and its applications. The speaker is careful to attribute results to the appropriate researchers, demonstrating scholarly rigor. The use of the Igusa stack diagram to decompose the cohomology of Shimura varieties is a powerful new tool, and the lecture provides a clear overview of its potential. The discussion of Arthur’s conjectures and the recent work of Koshikawa and Shin situates the research within a broader framework. The lecture is not self-contained, but that is expected for a mini-course. The technical level is extremely high, but the speaker’s clarity and organization make it accessible to experts. The title accurately reflects the content, and the lecture delivers on its promise to discuss applications to intersection cohomology. Overall, this is an outstanding lecture that will be of great value to researchers in the field.

The title accurately reflects the content: the fourth lecture in a series on Igusa stacks, focusing on applications to intersection cohomology of Shimura varieties.

9/10

Talk by a leading expert in arithmetic geometry, presenting recent research in collaboration with Linus Hamann and Mingjia Chen. The content is highly technical and rigorous, with references to published works and ongoing research. The presentation is clear and well-structured, though it assumes a high level of expertise.

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

• 00:00Introduction and setup: Shimura datum, coefficient ring, and goal to study non-compact Shimura varieties.
• 02:00Definition of intersection cohomology via middle extension on the minimal compactification.
• 05:00Discussion of structures on intersection cohomology: Hecke action, Galois action, Hodge and Lefschetz structures.
• 10:00Motivation from Arthur's conjectures: systems of Hecke eigenvalues and their range of degrees.
• 15:00Connection to L2 cohomology via Zucker's conjecture and Borel-Casselman computation.
• 20:00Recent work of Koshikawa and Shin on conjectures for the range of degrees in terms of representation-theoretic conditions at p.
• 25:00Introduction of the Igusa stack diagram and its role in studying cohomology.
• 30:00Applications to torsion-vanishing, Eichler-Shimura relations, and Ihara's lemma (preview).
• 35:00Discussion of relative intersection cohomology of the Igusa stack (joint work with Hamann and Chen).

• Carmin.tv — Video platform for mathematics, hosting this lecture.

• Koshikawa and Shin, 'On the cohomology of Shimura varieties' — Recent paper formulating conjectures on the range of degrees for cohomology of Shimura varieties.