9503375 T.P.Hill Abstract This research project involves several topics related to the mathematical theory of probability including: optimal stopping theory -- the study of stopping a stochastic process to achieve some goal such as maximizing the expected value of the process at the time of stopping, with emphasis on finding good stop rules based on incomplete information; partitioning problems -- the study of bisection (e.g., ham-sandwich theorems), fair-division (cake-cutting inequalities) or general partitioning (Lyapounov convexity) of an object as well as applications to hypothesis testing and game theory; empirical distribution theory -- the study of estimating an unknown distribution based on random samples from that distribution, with emphasis on new alternative weight functions; convergence and invariance principles -- the study of representations of measures and their relationships to limit laws; and the significant-digit problem -- the study of non-uniformity in significant digits of empirical data, and its applications to mathematical modelling and computer design. This project is mostly theoretical, but has potential applications in statistics, economics and computer science, as well as in other areas of pure mathematics such as measure, integration and function theory. This research project involves several topics in mathematics related to the theory of probability. These topics include: the theoretical aspects of controlling or stopping a random process such as pollution-level fluctuations or inflation rates; the theory and application of fair-division problems in which an object such as a parcel of land or other assets must be divided and distributed among several parties; the study of estimating the distribution of an unknown population based on random samples and observations; and general theories about long-range behavior of random processes such as population growth or global temperature fluctuations. This pro ject is primarily theoretical, with direct applications to statistics and other areas of mathematics, and potential applications to other areas of science such as economics, operations research, and computer technology.
