Professor Loeb's project involves the use of nonstandard models in real analysis and related areas. Fundamental in his approach is the use of standard measure spaces defined on nonstandard point sets. These spaces make finite combinatorial methods available in research on infinite probability spaces. Another facet of his work deals with nonstandard hulls (standard, infinite-dimensional vector spaces formed from nonstandard ones) of vector lattices and their role in measure theory and integration. He will also continue research on an ideal boundary, analogous to the Martin boundary, that he has developed for general potential theories. All of mathematical analysis is built upon the structure of the system of real numbers, which are identified as points on a two-way infinite line. It is commonsensical and fundamental in this system that every non-zero real number lies at a finite positive distance away from zero, but sometimes there is a powerful intellectual temptation to consider numbers infinites- imally close to zero or infinitely far away. Yielding to this temptation, one can construct a coherent system, called the nonstandard reals, that includes infinitesimals and infinities, and thence proceed to do nonstandard analysis. Professor Loeb's research involves starting with problems in standard analysis, such as appear for instance in probability theory and mathematical physics, using nonstandard methods to gain some elbow room, and then getting back to a solution of the original problem in standard terms.
