The Cahn-Hilliard equation, originally developed in the field of materials science, finds nowadays applications in both physics and biology. In physics, this PDE is used to model the phase separation phenomena, such as the formation of patterns in binary mixtures. In biology, the Cahn-Hilliard equation has been applied to study various phenomena, including cell-cell adhesion, tumour growth, and pattern formation in biological tissues. The Cahn-Hilliard equation is also a source of interesting problems for mathematicians. Its analysis presents several intricate mathematical challenges that intrigue researchers across mathematical and numerical analysis. One major issue lies in establishing the well-posedness of solutions, as the equation's degeneracy and fourth-order nature imply a lack of maximum principle. The degeneracy in the fourth-order term is also a source of difficulties for numerical simulations. In recent years, these challenges have spurred the exploration of advanced mathematical tools, such as the application of de Giorgi's method to prove so-called separation property, analysis of the nonlocal approximations to demonstrate the validity of singular limits (for instance, the high-friction limit) or the concept of varifold solutions to study sharp-interface limit and establish connection with the Hele-Shaw flow.

The week-long conference on “The Cahn-Hilliard equation - recent advances and new challenges” is planned to take place from the 21st to 26th April 2024 at the European Centre for Geological Education in Chęciny, Poland. The Centre is beautifully located in the Holy Cross Mountains (Góry Świętokrzyskie) in southern Poland. More information on https://crossing.icm.edu.pl/conference/