More than just width, breadth and height When we think of geometry, most of us consider the three-dimensional (3D) space defined by the x, y and z axes that enables us to define everyday structures. After all, when measuring a new desk, we look at its width, breadth and height to determine if its volume fits where we want it to. Industrial Technologies © Shutterstock Scientists in areas from probability theory to quantum physics and statistical mechanics investigate mathematical problems involving geometric spaces of dimension much higher than three, theoretically approaching infinity. This calls for complex numerical methods to simplify understanding of their solutions. The ‘Geometric phenomena in high-dimensional probability distributions’ (GPHDPD) project is based on the idea that, contrary to popular opinion, when viewed correctly high dimensionality may yield simplicity and order rather than complications. During the first reporting period of the GPHDPD project, the researchers connected two seemingly disconnected theories related to high-dimensional convex geometrical spaces. The first is related to the slicing problem, essentially defining mathematically a plane-like section (hyperplane) that cuts an n-dimensional convex body with specific mathematical criteria. The second has to do with a fundamental theorem of mathematics, the Central Limit Theorem, which states that as the number of samples from any population increases, the probability distribution of the means will approach a normal distribution (the so-called bell-shaped curve). The researchers have managed to prove that the second implies the first, an important and quite unexpected result. In addition, the researchers have to date provided proof of an important mathematical theorem regarding simplification of high-dimensional probability measures as related to approximate symmetries and so-called nearly radial margins. Thus, the GPHDPD project has so far contributed two new mathematical proofs related to geometric phenomena in high-dimensional probability distributions, with widespread application to probability and statistics in numerous (infinite?) fields.
