Speaker: Prof. Ho Weng Kin
Title: Non-Hausdorff Topology
For many people working with topology, especially those in the fields of real analysis and functional analysis, a separation weaker than T2 would commonly be accepted as exoteric, if not, function no more than a passing counterexample. However, recent years saw the development of domain theory which has key applications in computer science, and this important development opened up the field of non-Hausdorff topology. Indeed I subscribe to the Jean Goubault-Larrecq’s slogan which he puts in his book (Non-Hausdorff Topology and Domain Theory): “Domain Theory is Topology done right”.
In a series of four lectures, I would attempt to touch on domain theory and non-Hausdorff topology, making connections between these. Because of the breadth and depth of this field, what I do has to be very selective. The journey begins by looking retrospectively at D. S. Scott’s motivation of domains as an approximation structure for the denotational semantics of untyped lambda calculus. Key features here include the way-below relation and hence the notion of a continuous dcpo, i.e., a domain, and of course the Scott topology. There are always two-sides of the domain-theoretic coin: the order-theoretic side, and the topological side. Two main results will be mentioned to shed light on the links between these two sides, i.e., one that characterizes the continuous lattices in their Scott topology as the injective T0 spaces, and the other one the Scott Convergence Theorem. Other categorical aspects on the order-theoretic side will be mentioned, particularly, some notable dualities and equivalences. In the passing, we shall scrape the surface of recursive domain equations, allowing us to come back to our initial semantical motivation. Interesting properties of the Scott topology will be touched upon in this course in relation to continuous dcpo’s, e.g., sobriety of the Scott topology. Sober spaces will be a very important point of our four-day discussion.
The special feature in this lecture series is my attempt to raise to the audience’s attention some research areas in domain theory that can be followed up, especially amongst the younger generation of domain-theorists. Achim Jung once remarked that in domain theory the easy problems have all been solved, and the ones of medium difficulty were also more or less settled; what remain now are the hard ones – and they are really hard. Open questions, together with recent development, will be ‘sprinkled’ along the way to entice young minds to have a go at them! Alongside with the development of the lecture will be frequent mention of my own view of research methodology that, I hope, is useful to the audience.
Short biography of Professor Ho Weng Kin
HO Weng Kin received his Ph.D. in Computer Science from The University of Birmingham (UK) in 2006. His doctoral thesis proposed an operational domain theory for sequential functional programming languages. He specializes in programming language semantics and is dedicated to the study of hybrid semantics and their applications in computing. Notably, he solved the open problem that questions the existence of a purely operationally-based proof for the well-known minimal invariance theorem of (nested) recursive types in Fixed Point Calculus. His research interests also include domain theory, exact real arithmetic, category theory, algebra, real analysis and applications of topology in computation theory.
His research interests in mathematics and computer science include domain theory, exact real arithmetic, category theory, algebra, real analysis, applications of topology in computation theory and computer science education.
His research interests in mathematics education include the use of computer and video technology in the teaching and learning of mathematics, flipped classroom pedagogy, mathematics problem solving, computational thinking.