ICM 2026 Invited Section Lecture - Jean Marc Schlenker

Value of the Information & Strength of the Argument

The value of the information is high, as the lecture synthesizes a wealth of classical and recent results in hyperbolic geometry, offering a coherent perspective on the interplay between boundary data, volumes, and several deep conjectures. The argumentation is solid: the speaker systematically introduces definitions, states theorems with attribution, and explains the logical chain from polyhedral cases to manifolds with topology, always emphasizing the dualities and conjugate variables. The presentation is grounded in well-established references (Ahlfors, Bers, Bonahon, Thurston) and clearly delineates proven results from open questions, strengthening its credibility.

Scientific Rigor, Source Quality, Title Accuracy

The scientific rigor is commendable: the lecture is precise, with careful wording, and the speaker frequently attributes results to their authors. The sources, though not explicitly listed with URLs, are the classic works of the field, and the description mentions the AdS/CFT correspondence and Loewner energy as recent connections. The title exactly matches the content—it is an invited lecture by the named speaker at ICM 2026. The transcript, while containing some hesitations and informal phrasing, does not compromise the mathematical accuracy. Overall, the rigor is high, and the title is fully adequate.

The title accurately describes the content: a invited lecture at ICM 2026 by Jean Marc Schlenker on hyperbolic manifolds.

9/10

Presentation by an expert at a major international congress; content is rigorous and up-to-date, but some spoken statements are not fully formalized in the transcript.

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

• 00:00Introduction and overview of the talk structure.
• 05:00Classical convex bodies in H^3: Alexandrov's theorem and dual metric.
• 12:00Schläfli formula and the definition of dual volume.
• 20:00Convex domains in hyperbolic manifolds; definitions and quasi-Fuchsian example.
• 28:00Boundary data parameterizations: Ahlfors–Bers and Thurston's conjectures.
• 35:00Renormalized volume, connections to AdS/CFT and Loewner energy.

• Alexandrov's theorem on convex polyhedra in hyperbolic space — Mentioned as a classical result for realizing spherical metrics with cone singularities.
• Schläfli formula — Discussed as the first variation formula for the volume of a polyhedron.
• Ahlfors–Bers theorem on quasifuchsian manifolds — Used to parameterize convex co-compact metrics by conformal structures at infinity.
• Bonahon's formula for the variation of volume of the convex core — Generalization of Schläfli's formula to pleated surfaces and bending laminations.

• Thurston's Three-Dimensional Geometry and Topology — Provides foundational ideas for convex cores and pleated surfaces.