Cardinal Characteristics and Related Topics

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ABSTRACTRecent work in the theory of cardinal characteristics of the continuum has indicated that one can get more detailed information by working, not with the cardinal characteristics themselves, but with certain relations associated with them. This applies especially in cases where these relations are Borel sets. Part of the planned research concerns the question "Which cardinal characteristics are associated with Borel relations?" For many characteristics the answer is known to be positive; for others it appears to be negative, but it isn't yet known to be negative for any particular characteristic. The investigator hopes to close this gap by proving that certain specific characteristics are not associated with Borel relations. A second aspect of the planned research concerns the sequential composition of relations, which occurs in many theorems and proofs about cardinal characteristics. A priori, sequential compositions cannot be Borel relations, but the investigator intends to extend the theory of Borel relations, using tools from topos theory, so as to cover sequential composition. The research also includes questions relating computability theory to cardinal characteristics. Finally, the investigator also plans to study the use of the groupwise density number (a characteristic introduced some years ago by Laflamme and the investigator) in partition theorems.Cardinal characteristics of the continuum constitute a significant area of contemporary research in set theory. Not only are they of interest for their own sake, but for the last two decades they have played a role in applications of set-theoretic methods to general topology. More recently, there have been applications (including some due to the investigator) to other parts of mathematics, particularly algebra. The investigator and graduate students will extend this theory in several directions. One direction concerns connections with classical descriptive set theory, dealing with relatively easily definable relations involving real numbers. A second direction connects this theory with recursion theory, the study of what is (and what is not) computable in principle. A third direction establishes connections with combinatorial information about infinite sets and structures.