NSF Award
2448416
This project concerns the structure of two important and closely related mathematical objects: translational tilings and orthogonal systems. A translational tiling is a covering of a space using translated copies of certain building blocks, called the “tiles”, without overlaps. The focus of this project is on the major question in the area: which are the possible ways that a space can be tiled. The structure of tilings has been extensively studied over the years by many mathematicians; among the most famous and significant contributors are the mathematical physicist Roger Penrose and the amateur mathematicians Robert Ammann and Maurits Cornelis Escher. This mathematical study has found many important applications to other sciences. In particular, it led to one of the most significant breakthroughs in physical science: the discovery of natural quasicrystals, physical solids whose atoms’ arrangement is not periodic. Since aperiodic tilings (i.e., pavings of space which cannot be “determined” by how they look in an arbitrarily large but bounded part of the space) serve as the mathematical models of quasicrystals, the discovery of those aperiodic forms in nature by Dan Shechtman (for which he won the Nobel prize), was based on the mathematical discovery of aperiodic tilings. Orthogonal systems are of importance to several branches of mathematics, including number theory, algebraic geometry and analysis. One of their advantageous properties is that they can facilitate an amenable decomposition of a function (e.g., describing light or sound waves) into more manageable pieces with a high level of mutual independence. Studying these pieces separately can yield insight into the physical properties of a system such as dominant frequencies. The research carried out (in conjunction with collaborators) consists of revealing connections to a range of different areas of mathematics and building on past results in those areas, as well as developing novel methods and techniques.<br/><br/>The first part of this project will be devoted to the study of the structure of translational tilings. The main goal is solving the well-known periodic tiling conjecture, which asserts that any bounded measurable tile of the Euclidean space must admit at least one periodic tiling. This conjecture is known to hold for tilings of the real line, and there are some partial results towards it in higher dimensions. However, the periodic tiling conjecture has not been settled yet in dimensions two and higher. Over time it has become apparent that in many respects translational tiles “behave like” domains whose Hilbert space (i.e., the space of quadratically integrable functions that are supported on the domain) admits an orthogonal basis of exponential functions. Thus, a second area of focus is on the structure of frequency sets of orthogonal systems of exponentials in two different settings: in time-frequency spaces (Gabor bases) and in Hilbert spaces of certain domains. This part of the project aims to find new approaches to the latter mentioned studies of orthogonal systems, building new bridges between number theory, algebraic geometry and Fourier analysis.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
