J. D. Hamkins, * Indestructibility Phenomena of Large Cardinals,* National Science Foundation, program grant, NSF DMS 9970993, July 15, 1999 to June 30, 2003.

**Summary Abstract: **Professor Hamkins will continue his research on the indestructibility phenomena of large cardinals. This work, lying at the common focus of two broad set-theoretic research efforts, namely, forcing and large cardinals, seems particularly promising in light of the recent advances that have emerged from his earlier work with gap forcing and the lottery preparation. Specifically, Professor Hamkins will investigate a series of open questions concerning the extent to which indestructibility is possible with various kinds of large cardinals, with a particular focus on the strongly compact cardinals. In addition to this work, he will continue his work on two other projects, the automorphism tower problem and the new theory of infinite time Turing machines, a model of infinitary computation. Professor Hamkins’ research involves the study of the sublime, inaccessible notions of mathematical infinity, which have fascinated mathematicians for centuries. In the twentieth century, particularly in the past thirty years, set theorists have gained a profound understanding of the largest of these transfinite numbers, large cardinals. Professor Hamkins has been particularly interested in how large cardinals are affected by forcing, the technique invented by Paul Cohen by which set theorists have had glimpses into alternative mathematical universes and realized the rich diversity of mathematical possibility. Much of his work therefore lies in the common focus of two major set-theoretic research efforts, namely, forcing and large cardinals. He will endeavor to investigate the curious indestructibility phenomena of these cardinals, by which their largeness survives in a wide variety of mathematical universes.