ICM 2026 Invited Section Lecture - Hong Wang

Value of the Information & Strength of the Argument

The value of the lecture lies in its synthesis of two active research areas—restriction theory and geometric measure theory—via a common combinatorial core: the incidence problem for tubes. The argumentation is logical and coherent, progressively building from Fourier analysis and convex sets to wave packet decomposition and the high-low method. The speaker clearly explains why incidence estimates imply bounds on Hausdorff dimension of projected sets, and how the high-low method provides such estimates in two dimensions. While the lecture is primarily a survey, the presentation of the high-low method as a unifying technique offers a fresh perspective that is pedagogically effective for an expert audience.

Scientific Rigor, Source Quality, Title Accuracy

The lecture demonstrates high scientific rigor, with precise mathematical definitions and references to known results (e.g., the Bochner-Riesz conjecture, the Stein-Tomas argument, and works by Bourgain, Guth, and Solomon). No external sources are explicitly cited in the transcript, but the content aligns with established literature in harmonic analysis. The title matches the content exactly: the talk centers on orthogonal projections to restricted families and their link to incidence and restriction theory. There are no comments provided for analysis, so public reception cannot be assessed.

The title accurately reflects the content, which focuses on restricted orthogonal projections and their connection to restriction theory and incidence geometry.

9/10

The lecture is delivered by a leading researcher at an ICM invited section, presenting a rigorous survey of recent results in harmonic analysis and fractal geometry. The technical arguments are standard and appear consistent with established literature, though not verified in detail.

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

• 00:00Introduction: overview of restriction theory and projection estimates, and their common connection to incidence problems for tubes.
• 05:00Fourier transform of convex sets and the uncertainty principle; key example of rectangular boxes and dual convex sets.
• 12:00Restriction problem on the sphere and the role of wave packet decomposition; representing functions with frequency support on curved sets.
• 17:00Definition of the incidence problem for tubes; comparison with point-line incidences and the challenge posed by intersecting tubes.
• 25:00Orthogonal projections and reduction of projection estimates to incidence bounds; double-counting argument to obtain dimension bounds.
• 33:00High-low method in two dimensions: splitting frequency support to get incidence estimates via Cauchy-Schwarz and geometric observations.