Thursday 23 November 2023, 15:00-16:00
LG10 Old Gym
Compact Hausdorff spaces are the topological abstraction of the unit interval [0,1] (in a sense that can be made precise). Let us now equip the unit interval with the "denominator map" den: [0,1] -> N that maps a rational number to its denominator and an irrational number to 0. We characterize the abstraction of [0,1] that takes into account both the topology and the denominator map. The reason why we were interested in this problem is that one can show that the resulting structures form a category that is categorically dual to the category of Archimedean metrically complete Abelian lattice-ordered groups. This is a joint work with V. Marra and L. Spada: https://arxiv.org/pdf/2210.15341.pdf.
Thursday 9 November 2023, 15:00-16:00
LG10 Old Gym
The probabilistic powerdomain was introduced by Jones and Plotkin in 1989 and some fundamental properties were proved in the PhD thesis of Jones. Several authors were able to establish a close link between this construction and Borel measures on fairly general topological spaces. From the point of view of semantics of programming languages, however, one would like to know whether the construction can be restricted to one of the Cartesian closed categories of continuous domains. This remains an open problem. In this talk I will present some of the background and some of the recent progress towards a positive solution.
Thursday 26 October 2023, 15:00-16:00
LG10 Old Gym
In classical mathematical analysis, the topologies that are commonly used are Hausdorff. In contrast, over partial orders, the topologies that capture the concept of approximation are typically non-Hausdorff. Continuous domains are a special class of partial orders that were introduced by Dana Scott (in the late sixties) as a mathematical model of computation. Domain theory has enriched computer science with powerful methods from order theory, topology, and category theory, and domains are ideal for analysis of robustness, soundness, completeness, and computability. Many concepts of mathematical analysis have been recast in domain theory, e.g., dynamical systems, iterated function systems, measure theory, non-smooth analysis, stochastic processes, and solution of ODEs. The aim of the talk is to present a brief introduction to domain-theoretic mathematical analysis together with some key results. This will be followed by a discussion of some open problems, e.g., solution of PDEs. A common challenge in addressing these problems is handling topological spaces which do not have favourable properties (e.g., local compactness, core-compactness, etc.) that are necessary for domain constructions. An overview of recent results in this area will be presented as well.