The IRP on Modern Fourier Analysis moves forward with its main conference at CRM

For one intense week, the CRM auditorium became a meeting point for some of the most active researchers in modern Fourier analysis. From June 2 to June 6, 2025, the Conference on Modern Trends in Fourier Analysis brought together senior experts, young mathematicians, and PhD students from across the globe to explore the latest developments in the field.

The conference, part of a two-month Intensive Research Programme, gathered more than 100 participants from Europe, North America, and Asia. The scientific and organising committee was led by Dmitriy Bilyk (University of Minnesota), Emanuel Carneiro (ICTP), Diogo Oliveira e Silva (Instituto Superior Técnico), Betsy Stovall (University of Wisconsin–Madison) and Sergey Tikhonov (ICREA, CRM).

The invited talks spanned a wide spectrum of current directions. Hong Wang (Courant Institute) discussed restriction estimates using decoupling theorems and incidence geometry, offering new partial progress toward Stein’s conjecture via Kakeya-type estimates in dimension three. Nir Lev (Bar-Ilan University) explored Schauder frames of translates in , revisiting fundamental questions about bases generated by translations of a single function.

Jill Pipher (Brown University) presented a joint work on regularity problems for time-varying parabolic equations, addressing boundary behaviour on Lipschitz domains using advanced tools such as Hardy spaces for parabolic operators. In a related vein, Eugenia Malinnikova (ETH Zürich) connected uncertainty principles with Schrödinger evolutions, including recent results on spectral inequalities and Logvinenko–Sereda-type theorems.

Several talks touched on fractal geometry and geometric measure theory. Krystal Taylor (Ohio State University) surveyed progress on the Buffon circle problem, investigating projections of fractal sets and open questions around Favard length decay. Yumeng Ou (University of Pennsylvania) addressed pinned distance and tree sets, proposing refined conditions related to Falconer-type conjectures. Svitlana Mayboroda (ETH Zürich) discussed free boundary problems for partially reflected Brownian motion, presenting results that challenge prior expectations on the dimension of Robin harmonic measures.

Other invited speakers contributed advances on classical harmonic analysis problems. Alex Iosevich (University of Rochester) linked restriction phenomena with signal recovery in finite settings through Bourgain and Talagrand-type estimates. Aleksei Kulikov (University of Copenhagen) described the spectral behaviour of one-dimensional time-frequency localisation operators, revealing sharp eigenvalue distributions with phase transitions.

Mateus Costa de Sousa (BCAM) surveyed recent progress in Fourier uniqueness theory and phase retrieval problems. Giuseppe Negro (IST Lisboa) introduced a new family of Fourier restriction inequalities where constant functions extremize the problem, strengthening long-standing conjectures about optimal constants. Kristian Seip (NTNU) characterised extremal functions of minimal  norm among exponential type functions. Shahaf Nitzan (Georgia Tech) revisited Riesz basis bounds for exponential systems, improving estimates in classical theorems of Avdonin and Levin. Finally, Jose Ramon Madrid Padilla (Virginia Tech) discussed isoperimetric inequalities on the hypercube, highlighting recent interactions between analysis, combinatorics, and geometry.

Beyond the invited lectures, the afternoon sessions featured an extensive program of contributed talks, with presentations by more than 40 early-career researchers. The multiple informal discussions allowed participants to exchange ideas and build new collaborations in a vibrant research atmosphere.

Many participants will continue their discussions next week at the complementary conference in El Escorial (Madrid), keeping alive the momentum generated during the CRM gathering. The Intensive Research Programme will continue throughout June, with ongoing seminars and several visiting researchers staying at CRM to deepen collaborations and advance their projects.