Mathematics Abstracts of Presentations

Some recent works on Baire class one functions and problem posing strategies involved

Zhao Dongsheng

Good problems play the central important role in both mathematical research and learning. Being able to find and pose new problems is one of the key skills for researchers in the discipline of mathematics. In this talk, by reviewing recent work on the definition of Baire class one functions, I will try to illustrate some strategies of finding and posing research problems in mathematics.

Mathematical research - Problem posing, problem solving and making connections

Nai Yuan Ting

Throughout my five years of study as a part-time PhD research student, I have learnt that there are three areas that are very important in mathematical research: problem posing, problem solving and making connections.

When posing problems, it is important that the statement of the problem must be phrased in such a way that it does not restrict the scope of the problem, but yet precise enough so as to avoid ambiguity. Through reading journal articles and attending oversea conferences, I have learnt from my supervisor and other mathematicians as to how problems can be posed and how problems are being posed. At conferences, mathematicians are interested in the “big ideas” and “intent” of the paper presentation so that they can transfer such “big ideas” and “intent” to other areas of research as well. Some examples will be given to illustrate the above.

As a teacher currently teaching Mathematics at a junior college, I have always taught my students Polya’s problem solving framework as well as the heuristics used. As a research student trying to come up with conjectures for the problems posed, I find myself sometimes guilty of not using some of the heuristics such as observing patterns and working with simpler cases. Besides the use of heuristics in problem solving to come up with the conjectures, I also realise the importance of using certain tools and techniques when trying to prove certain results. I will give examples of some of these tools and techniques.

Most importantly, I have learnt that there are inter-relation between the various areas of Mathematics; this is especially so for the areas of topology, algebra and order theory. It is necessary to make connections between the area of my research with other areas of Mathematics so as to make the results more meaningful. I will share examples of such mathematical results, and will conclude the presentation with some discussion on “point-free” Mathematics.

An Identity via Arbitrary Polynomials

F.M. Dong, W.K. Ho, T.Y. Lee

In this talk, we present a family of combinatorial identities. Applying these identities, we proved two conjectures posed in [Thomas P. Dence, Some Half-Row Sums from Pascal's Triangle via Laplace Transforms, The College Mathematics Journal 38(2007), 205-209].

On a new convergence class in k-bounded sober spaces

Hadrian Andradi

Abstract. Recently, J. D. Lawson encouraged the domain theory community to consider the scientific program of developing domain theory in the wider context of T_0 spaces instead of restricting to posets. We respond to this calling by proving a topological parallel of a 2005 result due to B. Zhao and D. Zhao, i.e., an order-theoretic characterisation of those posets for which the lim-inf convergence is topological. We do this by adopting a recent approach due to D. Zhao and W. K. Ho by replacing directed subsets with irreducible sets. As a result, we formulate a new convergence class on T_0 spaces called Irr-convergence and established that this convergence class I on a k-bounded sober space X is topological if and only if X is Irr-continuous.