Real-valued functions have been extensively studied and play an immense role in many branches of science. A typical example is the temperature considered as a function of a point varying on the surface of the earth. By contrast, the theory of maps with values into spheres has not yet been sufficiently developed. A typical example of such a map is the orientation of a compass needle as a function of a point varying on the surface of the earth. The orientation varies regularly except at the North Pole and the South Pole. Such points are called the singularities of the map. The reason why mathematicians work with Sobolev maps (rather than smooth maps) is precisely to allow maps with singularities. The simplest example consists of maps with values into the unit circle (a.k.a. unimodular maps). They occur naturally in many physical phenomena (e.g., liquid crystals, superconductors). It turns out these classes of maps have an amazingly rich structure from the point of view of analysis, geometry, and topology. This is one topic to be investigated in this project. The position of a point on the unit circle is determined by an angle (called the lifting of the original map). Any unimodular map admits plenty of liftings (since the angle is measured modulo two-pi). The project will study "optimal" liftings in the sense that they have least total variation. A lifting is usually discontinuous (even if the original map is smooth): every traveler crossing the International Date Line is aware of this discontinuity! The principal investigator proposes to classify all optimal liftings of a given unimodular map using only the geometry of its singular set. More precisely, he and his collaborators conjecture that there is a one-to-one correspondence between optimal liftings and minimal surfaces spanned by the singularities. They have been able to establish the conjecture in many two-dimensional cases. For example, if the map has precisely two singularities located at the North and South Poles, optimal liftings are classified by the meridians. Another important topic is the uniqueness of liftings. The problem reduces to the following question: Given a function taking only the values 0 and 1, what additional assumptions imply that the function is constant? The standard condition is that the function be continuous (but this excludes many important physical problems). The principal investigator has been able to derive the same conclusion for a much wider class of functions, but a general condition is still missing. A key ingredient is a new formula that provides a way of approximating total variation by nonlocal functionals involving no derivatives.
The principal investigator has recently learned that some of the tools that will be either improved or developed from scratch in the process of carrying out this project could be useful in image processing. The need for efficient image restoration methods has grown with the massive production of digital images often taken, or transmitted, in poor conditions (e.g., by UAVs). Likewise, to achieve the best possible diagnosis it is important that medical images be sharp, clear, and free of noise. The analysis of fine structures (e.g., micro-calcifications detected in mammograms) is one of the major challenges faced in medical image processing. Blurred and distorted images need to be restored and enhanced before one can extract reliable information. Over the past twenty years sophisticated mathematical techniques have been used in this field. The principal investigator has established contacts with leading experts who surmise that his and his collaborators' discoveries may lead to more efficient algorithms used effectively in concrete situations.
