17 From: cwood@nsf.gov (Carol Wood) at NOTE 5/24/96 8:17AM (2834 bytes: 52 ln) To: jwhitehu at nsf11, ahorton at nsf11 cc: cwood@nsf.gov at NOTE Subject: DuBose/Burke abstract (UNLV, 9626212) Message Contents Text item 1: Text Item Date: Wed, 22 May 1996 14:58:28 -0700 (PDT) From: DERRICK DUBOSE To: Carol Wood Subject: abstracts Mime-Version: 1.0 Dear Carol, below are the two abstracts (finally). I shall be waiting for any comments, suggestions, etc. On my portion, I was somewhat general, instead of specifically indicating that in particular, I am interested in the determinacy strength of mice with so many measurables above some fixed number of Woodin cardinals. I guess this is fine, being general, and better satisfies the purpose of the abstract?? Thanks in advance for any comments, Derrick DMS-9626212 Derrick DuBose/Douglas Burke Douglas Burke and Derrick DuBose are involved in this project. The research of Douglas Burke is primarily concerned with generic embeddings in set theory. These embeddings have been instrumental in using new axioms (large cardinals) to prove statements that are undecidable in the usual, basic axioms of set theory. In particular, they have many applications in descriptive set theory and combinatorial set theory. Burke has two long term goals in his research program: (i) to apply large cardinals and generic embeddings to combinatorial questions (and apply combinatorial results to questions about generic embeddings), and (ii) to extend the connection between large cardinals and definable well-orderings of the real numbers to larger classes of definable sets. Derrick DuBose has been engaged in establishing correspondences between inner models closed under sharp functions and the determinacy of classe s near the bottom of the analytic hierarchy. He intends to establish similar correspondences involving classes higher up in the analytic hierarchy. He will also continue to investigate moderate determinacy assumptions and the determinacy strength of small "mice" and of mild large cardinal properties. During the last fifty years, many natural mathematical questions have been shown to be independent of the usual axioms of set theory. In order to decide such questions, new axioms have been introduced. Of particular importance are Large Cardinal Axioms and Determinacy Hypotheses. The Large Cardinal Axioms are axioms of infinity, whereas determinacy hypotheses state that certain definable infinite games have winning strategies. Surprisingly, a strong connection exists between these two groups of axioms. Also both are important in that they decide many properties about sets of real numbers and other ``small'' sets.
