It is proposed to study several problems in Probability. One class of problems concerns central limit theorems for complex objects like nearest neighbor statistics and linear statistics of eigenvalues of large dimensional random matrices. The PI has invented a new method that works by exploiting a hitherto unknown connection between normal approximation and concentration of measure, two different branches of probability theory. A second class of problems involves fair allocations of Lebesgue measure to discrete point processes in Euclidean spaces and manifolds. Finally, a third line of investigation pursues a method of connecting the analysis of interacting Brownian particles with the geometry of convex polytopes.
The key focus of the project is on Central Limit Theorems. CLT's, as they are popularly known, are one of the founding pillars of Probability Theory and arguably its most widely used tool in the applied sciences, finding everyday applications in fields ranging from Statistics to Bio-informatics, Computer Science to Economics. Although much is known, there are still many unsolved questions. In fact, the PI's investigation into the theory of Central Limit Theorems was initiated by an open question raised by Peter Bickel, an eminent Berkeley scientist working in the area of high dimensional data analysis. The PI now has a new technique for proving CLT's that has not only solved the open question, but has yielded and promises to yield much more.
