This mathematics research project by Alessio Figalli is focused on several problems in the calculus of variations and partial differential equations. These include the optimal transport problem, the issue of stability in functional inequalities, and the Mumford-Shah functional. The optimal transport problem consists of finding the least expensive way to transport a distribution of mass from one place to another. In addition to being a natural problem in the calculus of variations, it is also related to partial differential equations, Riemannian geometry, and probability. The issue of stability in functional inequalities consists of understanding whether a minimizer of some inequality is stable in some suitable sense. This is an important issue in order to understand and/or predict the evolution in time of a physical phenomenon. For instance, quantitative stability results are used to quantify the rate of convergence of a given physical system to its steady state, and they can also be used to understand the extent to which the system changes under the influence of external factors (for instance, external forces). The Mumford-Shah functional is a classical model in image segmentation which is used to extract from a blurry image the meaningful discontinuities (which correspond to edges of objects, shadows, and overlapping objects). The regularity properties of minimizers of the Mumford-Shah functional are still far from being understood, and understanding the smoothness of the interfaces and their topological properties is an important and challenging problem.
All the problems investigated in this mathematics research project by Alessio Figalli have important applications in other areas of sciences. For instance, the optimal transport problem is a fundamental problem in economics, with further applications to meteorology, biology, and population dynamics; the Mumford-Shah functional, which is studied in this project, has applications to image processing (it allows to extract good images out of blurry ones). Some of the problems in this project will be used in the training of undergraduate students, graduate students and postdoctoral fellows. Several of Figalli's PhD students and postdoctoral fellows will engage in research in these areas, and the results obtained will be widely disseminated via the publication of research papers and lecture notes, as well as through the development of courses and seminars.
