Mathematics Abstracts of Presentations

What is the crank of a partition?

Toh Pee Choon

The partition function p(n) counts the number of ways an integer n can be written as a non-increasing sum of positive integers. The Indian mathematician S. Ramanujan discovered many fascinating properties of p(n), chief among which are the three congruences which are now known as Ramanujan's congruences. About 25 years later, the famous physicist Freeman Dyson, then an undergraduate at Cambridge, discovered a remarkable property concerning partitions which he called the rank of a partition. The ranks of partitions provided a combinatorial explanation of two of Ramanujan's congruences but not the third (and most difficult to prove) congruence. Dyson then conjectured the existence of what he called the "crank" of a partition which would explain the third congruence. This was one of the rare instances in the history of mathematics where a mathematical object was named before anyone had an example of what the object looked like. It took another 40 years before Andrews and Garvan discovered what is now known as the crank of a partition. In this talk, I will explain, with the help of many pictures, what are ranks and cranks of partitions.

Quasi-metric spaces and their corresponding poset of formal balls

K.M. Ng, W.K. Ho

First introduced by Weihrauch and Schreiber in 1981, the poset of formal balls can be used to represent metric spaces as a computational model. Many connections and characterizations of the metric spaces using the order structure of the formal balls were discovered over the years. For instance, in a classical result due to Edalat and Heckmann (1998), a metric space is complete if and only if its corresponding poset of formal balls is directed-complete. Analogous to the metric spaces, there are also interesting links between a quasi-metric space and its poset of formal balls. In this talk, we share some of these links and how we attempt to anchor on them and existing results in general order structures to revisit some of the traditional concepts such as the various completions on quasi-metric spaces.

A new topology on the Denjoy space

Dewi Kartika Sari, Peng-Yee Lee, Dongsheng Zhao

We define a new topology on the Denjoy space in terms of a neighbourhood system. The neighbourhood system is constructed from the complete metrizable space C[0,1] with additional properties. During the presentation we will illustrate the space and discuss it.

Chromatic equivalence classes of complete tripartite graphs

Ng Boon Leong

The chromatic equivalence class of a graph G is the set of graphs that have the same chromatic polynomial as G. We review previous research of chromatic equivalence classes of some families of graphs, in particular, complete tripartite graphs. We also find the chromatic equivalence class of the complete tripartite graphs K(1,n,n+2) for all n≥2, a result obtained by A/P Dong and me in 2015.

Rough set theoretic approach to domain theory

Ho Weng Kin

Rough set theory was introduced by Z. Pawlak in 1991 to formalize the notion of approximation in the context of data and information. The upper and lower approximations in rough set theory were recently modified to characterize the notion of continuity in posets by Li, Zhou & Zhou 2015. In this talk, we look at this new approach and investigate how domain theory can be studied through the lens of rough set theory.