Quasicrystals are quasiperiodic crystals that are strictly aperiodic. Their discovery put an end to the age-old paradigm that long-range order—in the positions of atoms in a material—necessarily implies periodicity. In the years that have passed, we have seen substantial development of experimental tools, mathematical and theoretical ideas, as well as numerical methods, for treating these complex structures and addressing their challenging scientific questions. Yet, quite a few fundamental questions remain open [1]. The community has not met in a long time and is eager to present new findings and discuss some of the outstanding open questions. Some of these are:

1. The basic crystallographic problem – ״where are the atoms?״ – has seen many advances, with the use of high-dimensional crystallography and the advent of so-called charge-flipping algorithms, as well as some ad-hoc clever ideas [2]. But novel approaches are necessary to tackle the general phase problem for aperiodic crystals. New experimental tools, such as X-ray holography, and novel surface probes, will be discussed, as well as ideas for potentially new theoretical and algorithmic approaches for structure determination.
2. Quasicrystals are observed today (almost) everywhere, from the atomic scale as in metallic alloys to the mesoscopic scale in soft matter, and even at larger scales in metamaterials and fluid dynamics experiments. Some of the most recent additions are 2-dimensional dodecagonal oxide layers [3] and hexagonal and dodecagonal quasicrystals that form in bilayer (and trilayer) graphene [4]. These and other novel systems will be presented and discussed, as they offer new perspectives for studying the basic open questions of thermodynamic stability, formation and growth, and the dynamics of excitations like phonons and phasons, and of structural defects like dislocations.
3. The study of electronic structure in quasicrystals is regaining attention with the discovery of superconductivity [5] and ferromagnetism [6] in quasicrystals, and the observation of Dirac electrons in dodecagonal bilayer graphene [7], as well as renewed theoretical interest in correlated electrons, quantum criticality, many-body localization, and the topological classification of electronic bands in quasicrystals. All of these pose new puzzles that will be addressed in the workshop.
4. Research in the mathematics of long-range aperiodic order has made rapid progress in recent years, in aspects such as tiling theory, diffraction theory, spectral theory of quasiperiodic operators, and their topological nature. These will be presented in a manner that should be accessible to the broad community, which often finds the mathematical literature inaccessible.

It goes without saying that numerical methods are essential for the study of quasicrystals. These include ab initio total energy calculations (e.g., for determining the thermodynamic stability of candidate phases), band-structure calculations on larger and larger periodic approximants, structure solution and refinement of aperiodic crystals (e.g., Jana2020), as well as classical density functional, molecular dynamics and Monte Carlo simulation. All of these will appear predominantly in the workshop, along with novel approaches like the recent use of machine learning for the discovery of new quasicrystals [8].