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Most of my research has been collaborative research on a few topics lying at the intersection of general topology, category theory, logic, and to some extent formal mathematics tangential to computer science.
I have a special interest in the category TopSys, whose objects are topological systems, a common generalization of topological spaces and locales, and whose morphisms are a suitably generalized version of continuous functions. For my dissertation, I analyzed concepts like “sub-system of a topological system.” There is much to be done in the area of topological systems, both in generalizing results from the theory of topological spaces, and also in developing a the theory of topological systems in the case where it differs.
I am particularly interested in generalizing the concept of discontinuity to the setting of topological systems.
Other recent research has focused on lattice-valued topology, its relationship to topological systems theory, and forming a link between the concept of topological system and that of institution in computer science.
Crazy Ideas
I occasionally come up with crazy research ideas that lie outside my training and expertise. I am currently looking for collaborators for two crazy ideas.
The first of these I call “Fibered Quantum Gravity.” Dov Gabbay developed a broad method of combining logical systems. I would like to propose formalizing the causal systems describing general relativity and quantum mechanics, then fibring these causal systems together, and seeing what happens.
The second idea is to prove a meta-meta-theorem in mathematical logic along the lines of “Every consistency proof [of statements in some specified deductive system] is essentially semantic [i.e., at some point in the proof, maybe in a hidden way, we talk about models].” It is extremely complicated to even figure out how to write down a conjecture formally, let alone prove any results.
Mathematical Linguistics
Finally, I have a long-standing interest in mathematical linguistics, and welcome input along the lines of “The Ten Mathematics Problems Linguists Most Want Solved,” if anyone actually has such problems they want solved.
Contact Jeff at jdennist@kent.edu
View a list of publications here: Denniston Publications
