ICM 2026 Fields Medal Award Lecture - Hong Wang

Value of the Information & Strength of the Argument

The lecture provides a high-value overview of recent breakthroughs in fractal geometry, specifically projection theorems and the Kakeya conjecture. Wang presents original results, including her proof with Kevin Ren of the projection theorem for sets of directions with positive dimension, and the recent proof of the Kakeya conjecture in R3 with Joshua Zahl. The argumentation is rigorous and well-structured, building from classical results to new theorems. She clearly explains the key ideas and the obstacles, such as the real-complex distinction, and how they are overcome using tools like the sum-product theorem and the analysis of branching functions. The presentation is dense but logically coherent, making it valuable for researchers in the field.

Scientific Rigor, Source Quality, Title Accuracy

The talk is scientifically rigorous, as expected from a Fields Medal lecture. Wang references several key theorems and results, including Marstrand’s projection theorem, the sum-product theorem, and the work of Bourgain, Katz, and Tao. She also mentions her own papers and collaborations. The sources are not explicitly cited with URLs, but the mathematical content is verifiable and consistent with known literature. The title accurately reflects the content, as the lecture is indeed a Fields Medal award lecture by Hong Wang on projection theorems for fractal sets. No comments were provided for analysis.

The title accurately reflects the content: a Fields Medal award lecture by Hong Wang on projection theorems for fractal sets.

9/10

Lecture by a Fields Medalist presenting original research, with rigorous mathematical content and references to established theorems. The talk is technical and assumes advanced knowledge, but the presentation is clear and well-structured.

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

• 00:00Introduction and welcome by the session chair.
• 01:00Hong Wang begins her lecture, introducing orthogonal projections and the main question.
• 03:00Introduction to Hausdorff dimension and the middle-third Cantor set.
• 05:00Statement of Marstrand's projection theorem.
• 07:00Presentation of the theorem on projections for sets of directions with positive dimension, and the sharpness example.
• 10:00Discussion of the failure of the theorem in complex spaces.
• 12:00Introduction to Kakeya sets and the Kakeya conjecture.
• 15:00Connection between Kakeya sets and orthogonal projections via slicing.
• 20:00Introduction of branching functions and uniform sets.
• 30:00Definition of sticky sets and Ahlfors-regular sets.
• 40:00Statement of the main theorem for Ahlfors-regular sets and proof strategy.
• 50:00Discussion of the sum-product theorem and the real-complex obstruction.

• Marstrand's projection theorem — Classical result cited as the starting point for projection theorems.
• Bourgain's sum-product theorem — Used to distinguish real and complex numbers in the proof strategy.
• Kakeya set conjecture — Mentioned as a related conjecture with connections to projection theorems.

• Marstrand's projection theorem — Classical result that the talk builds upon.
• Bourgain's sum-product theorem — Used to distinguish real and complex numbers in the proof strategy.