Logic, Sets, Categories, and Applications

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The proposer continues his research on applications of set-theoretic methods to abelian group theory, with particular emphasis on products of infinite cyclic groups and related groups. In addition to applying set-theoretic ideas to this branch of algebra, he is looking for new set-theoretic concepts and results motivated by algebraic problems. He is exploring the connection between universal algebra (in Lawvere's category-theoretic formulation), the theory of classifying topoi, and the theory of unification (which plays a prominent role in some areas of computer science). The connection is already understood in the case of absolutely free algebras, but further exploration is needed in the more general case of algebras subject to nontrivial identities. Unification in the presence of identities is in general a troublesome topic, and it is hoped that the connection with other, better understood areas can clarify it. The proposer is also studying a topos-theoretic approach to extending the notion of Borel functions to higher types. Such an extension is useful for the theory of cardinal characteristics of the continuum.Borel functions at the base type (real numbers) already play a major role in this theory, but so does an operation of sequential composition that necessarily leads to higher types. The proposer is also studying several questions at the border between constructive logic, category theory, and game semantics. This research includes an attempt to amplify the connection between games and free (or co-free) bicomplete categories, axiomatization of game-semantical validity, and a game-theoretic principle that may be fruitfully added to intuitionistic logic. Finally, the proposer is studying several questions in set theory, most of which are connected with ultrafilters and in particular with the equivalence relation of near coherence on ultrafilters. There are also some questions in infinite combinatorics growing out of considerations in abelian group theory. The project builds on the proposer's past experience in several areas of mathematics --- set theory, category theory (especially topos theory), abelian group theory, game semantics, constructive mathematics, and computer science --- in order to address problems bridging these areas. Almost all of the topics in the proposal connect two or more of these areas. In each case, it is reasonable to expect that ideas and styles of reasoning from one area can benefit other areas. Among the potential outcomes of this research are: (1) a better understanding of unification, which is an essential ingredient of logic programming, (2) a clearer view of constructive mathematical reasoning as based on strategies in certain sorts of debates, and (3) applications of a context that resembles traditional set theory in many ways yet avoids the phenomena sometimes regarded as pathological.