Howard Blair
(Syracuse University)

"Differential Calculus on Convergence Spaces and Models of Hybrid Computation"

A system of ordinary differential equations can be taken as a system of axioms to be interpreted on a huge collection of spaces called Convergence Spaces. A problem at the center of this approach is to have a notion of what a differential is that is in agreement with the notion in already understood contexts, but is native to the space(s) in question. This problem is solved with the differential calculi that exist throughout the entire Cartesian closed category CONV of convergence spaces and beyond - such spaces include all topological spaces, reflexive digraphs and ramified hybridizations of these spaces. The calculi include that of classical analysis on real and complex vector spaces and function spaces. Further, under two additional constraints on differentials involving group actions on the spaces in CONV, a maximum differential calculus throughout CONV is uniquely determined. We use the convergence space differential calculi to set up a scheme for models of computation. Instances of the scheme involve arbitrary convergence spaces. We discuss the matter of computability from a point of view that is native to the spaces involved in each instance of the scheme.

The Colloquium is supported by generous contributions from the Bloomberg, Information Builders, Inc. and Netlogic, Inc.