A central theme of this project is the study of geometric properties of convex bodies based on information about their sections and projections. This branch of convex geometry is called geometric tomography. An important instance of this theory is x-ray tomography, which has numerous applications in science, medicine and engineering. The PI has developed a new approach to geometric tomography in which the geometric properties of convex bodies are expressed in terms of integral transforms, enabling the use of certain analytical methods to solve geometric problems. In this project, the PI plans to further develop these techniques and apply them to a range of problems at the interface between convex geometry, functional analysis, harmonic analysis and probability. For example, can one find an algebraic equation whose solutions are sections of a given solid? Can one estimate the volume of a solid from data involving areas of certain sets of sections or projections of this solid? Which random variables are stable, i.e. have the property that the sums of several copies of these variables always reproduce the same variable up to a constant? The PI will continue to work with students and early career stage mathematicians, to introduce them to this evolving area of research. The problems considered in this proposal connect several areas of mathematics - convex geometry, functional analysis and probability. However, the strategy of solution is common for most of the results - the q