ICM 2026 Invited Section Lecture - Joaquim Ortega Cerdà

Value of the Information & Strength of the Argument

The lecture provides significant value by offering a complete solution to a longstanding extremal problem, including new characterizations and numerical evidence. The argumentation is rigorous, with clear statements of theorems and proofs. The speaker effectively connects the problem to diverse applications, demonstrating its central importance in harmonic analysis and related fields. The presentation builds on the history of the problem and clearly delineates new contributions from prior work.

Scientific Rigor, Source Quality, Title Accuracy

The scientific rigor is high; the speaker is an established expert and the results are presented with precise statements and references to prior work (Hörmander, Bernhardsson, Carneiro, etc.). The sources used are appropriately cited within the lecture. The title is somewhat generic but accurately reflects the context of an invited lecture at ICM 2026; it does not claim more than the content delivers.

The title indicates the context (Invited Section Lecture at ICM) and the speaker, while the content is a focused mathematical presentation on Fourier analytic extremal problems; the title is appropriate though generic.

9/10

High-quality mathematical lecture delivered by a leading researcher, presenting original results with rigorous proofs and connections to prior literature.

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

• 00:30Introduction to the Paley-Wiener spaces and the extremal problem.
• 03:00Case p=2: reproducing kernel and sinc function as solution.
• 07:00Motivations from number theory, operator theory, and combinatorial problems.
• 11:00General properties of extremal functions: symmetry, zeros, uniqueness for p>1.
• 15:00Focus on p=1: connection to Hörmander-Bernhardsson function and oscillator representation.
• 20:00Main theorem: dual characterization via differential and functional equations.
• 25:00Asymptotic expansion of zeros and coefficients; numerical methods.
• 30:00Discussion of the Taylor expansion of the extremal function and applications.

• Hörmander & Bernhardsson (1993) on extremal problems — Cited as providing precise bounds for C1 and conjecturing the existence of the extremal function.
• Collaborations with O. Brevig, A. Chirre, K. Seip — Joint work on the extremal problem, including the main results presented.
• Collaboration with A. Bondarenko, D. Radchenko, K. Seip — Additional joint work on the same problem.
• Carneiro & Milinovich on gaps between primes — Application to number theory, relevant to the constant C1.
• Ben Green's problem in additive combinatorics — Connection to the p=4 case and minimization of self-correlation.