Mathematics Abstracts of Presentations

Erdos-Ko-Rado Combinatorics

Dr. Ku Cheng Yeaw (SPMS - NTU, Singapore)

The Erdos-Ko-Rado theorem is a cornerstone result in extremal combinatorics that gives the size and characterization of the largest intersecting families of k-subsets of a finite set. The theorem has been extended to many other combinatorial and algebraic objects. In this talk, I will provide a glimpse into this fascinating area, and some problems I find interesting, particularly for permutations and perfect matchings.

Multivariate Statistical Methods for Discriminating Geographical Origins of Red Wines using Spectroscopic Data

B Viveka (National Institute of Education, Singapore)

This research delves into the utilization of multivariate statistical methods, specifically Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA) to distinguish the geographical origins of red wines using high-dimensional spectroscopic data. Focusing on Merlot, Cabernet Sauvignon, and Pinot Noir varietals sourced from Australia and France, this study addresses the critical need for accurate wine authentication and provenance determination in the global market. By harnessing the inherent chemical signatures captured through spectroscopic analysis, coupled with advanced statistical techniques, this research aims to develop interpretable classification models capable of effectively discriminating between wines of different origins. The integration of PCA and LDA, along with the rigorous model evaluation and selection afforded by cross-validation, facilitates the creation of reliable predictive models that enhance the authenticity and traceability of red wines. Through empirical validation and analysis, this study contributes to the advancement of methods for ensuring quality assurance within the wine industry.

Pointless Topology

Tan Zheng Han, Hans (National Institute of Education, Singapore)

The word for “topology” comprises two Greek root words “place” or “location” (τόπος) and “study” (λόγος). Classically, point-set topology is concerned with notions that articulate “getting near to a location/point” or “neighbourhood of a location/point” using the concept of open sets. More formally, a topological space X is a set endowed with a distinguished collection t of subsets of X – called open sets – that satisfies three basic axioms that are distilled from those arising from the usual Euclidean topology on the real line, i.e., closure under the formation of finite intersections and arbitrary unions. My talk shows how one can still talk about topological notions without referring to the elements or points of the space, hence the title “pointless topology”. Ironically, we do so by defining “generic points” of the lattice WX of opens of the underlying space X, which are precisely the completely prime filters of WX. This pointless approach is crucial if one pursues the line of constructive mathematics.

Topology and the Infinitude of the Primes

Leong Chong Ming (Nanyang Junior College, Singapore)

The infinitude of the primes is one of the most proved results in mathematics. It is interesting to note that there are several different proofs of this famous result based on entirely different domains of mathematics. Euclid probably gave the first proof by a clever construction which is still relished by mathematics enthusiasts today. One ingenious topological proof was given by Furstenberg in 1955 while he was still an undergraduate. What is surprising about this beautiful proof is the ‘unreasonable’ connection between topology and number theory. In this talk, I shall attempt to re-enact the topological proof supplied by Furstenberg which according to the famous Hungarian mathematician Paul Erdos, was “a proof from the book”.

The Scott Topology

Ho Weng Kin (National Institute of Education, Singapore)

Point-set topology is a mathematical theory that formalizes convergence and approximation by capturing what it takes for points to get near to a location – reminding us of its Greek etymology: “Topology” in Greek means the study (logoz) of location (tottoz). First courses in topology mostly assume the strongest degree of separation, focusing only on Hausdorff spaces, i.e., spaces in which every two distinct points can be told apart using disjoint open neighborhoods containing each. For a long time, non-Hausdorff topologies are misconstrued as unnatural and hence unimportant. In this talk, we introduce the Scott topology which stands as the most important non-Hausdorff topology, possessing many interesting and fundamental mathematical properties; thus debunking the myth that non-Hausdorff topologies are esoteric.