Enrichment Lectures

This is the list of enrichment lectures proposed by Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University. The aim is to promote mathematics by showing its beauty and relevance to JC/poly students.

For teachers: you can invite us over to your school to conduct one of these lectures and tell your students about NTU.

easy level - suitable for strong JC students

medium level - suitable for strong science JC students

advanced level - suitable for strong JC students with a particular interest to maths (requires understanding of things like proofs by contradiction)

Contact: Dr Tang Wee Kee (Outreach Chair, Division of Mathematical Sciences), weekeetang@ntu.edu.sg

The Hairy Ball Theorem

The hairy ball theorem states that when you try to comb the hair on a hairy ball there will always be at least one place where the hair sticks up -- it is impossible to comb it flat everywhere. Mathematically, this is due to a topological property of "vector fields" on a sphere (with the hair on the ball being an example of such a vector field). It has interesting real world consequences: For example, in a simplified model of the Earth's atmosphere it implies that there must always be at least one cyclone somewhere in the world at any time. There is also an application to a practical problem encountered in computer graphics, namely to construct an algorithm that produces a vector perpendicular to any given vector in 3-dimensional space -- the theorem says that it is impossible to do this in a continuous way.

This talk gives an introduction to the concept of vector fields on the sphere (and other surfaces such as the torus) and discusses their topological properties and the real world consequences of these properties mentioned above.

70 min, whiteboard lecture, advanced level

Counting Rationals

We know that there are infinitely many rational numbers. Is it possible to list all of them down in a sequence? If so, what are the different ways to list them down? In this talk, we shall demonstrate a few different ways of listing them, one of which involves the Stern sequence (also known as the Calkin-Wilf sequence). Only basic knowledge from high school mathematics is required for this talk.

45 min, lecture, advanced level

Mathematics of Transportation

The efficient allocation of resources is essential in the sustainability of a global metropolis such as Singapore. At the core of this lies the theory of transportation - the study of optimal transportation and allocation of resources such as material, goods and energy. The mathematical study of transportation theory began as early as the 1920’s, with major advances made during the Second World War by several renowned mathematicians including George B.

Dantzig (1914--2005) and Leonid V. Kantorovich (1912-1986). In this talk, we study the mathematics behind a problem in transportation theory: the maximum flow problem. The maximum flow problem is a problem of optimal resource movement, seeking to maximize the total flow across a network of links with capacities.

50 min, lecture, medium level

Stable Matching: How to Match Men and Women Fairly

Given a set of men and women where each individual has a preference over the opposite sex, how can one match men and women in a certain fair way such that everyone can have only one partner? We will introduce the notion of stable matching to solve the problem. Stable matching has a variety of applications in practice, e.g., school admission, class registration, and social dating.

30 min, lecture, medium level

Braids in Maths and Sciences

We'll explain how braids that you see in hair styling occur in mathematics, astronomy, physics, engineering.

30 min, lecture, easy level

Applications of complex numbers

1) Historical review of how mathematics contests in Renaissance Italy led to the discovery of general solutions of cubic and quartic equations and eventually complex numbers as a necessary tool. 2) Applications of complex numbers to aircraft design. Pre-requisite: basic knowledge of complex numbers.

35 min, lecture, medium level

Exacting Princess

We'll talk about Martin Gardner's recreational problem that gave rise to a new mathematical theory called optimal stopping of stochastic processes.

Here is the problem: Once upon a time in the land of Fantasia a princess decided to get married. 100 princes came to seek for her hand; and she intends to choose the best of them. She can compare princes - if she talked to any two of them, she would know which one is better. However, the 100 princes meet her one by one; once she's spoken to a candidate, she has either to accept or to reject him. The princess will only be satisfied with the best of the 100. If she chooses a fiance who is not the best of all, she backs out and goes into a convent. What is her chance to find the best prince and what is her optimal strategy?

50 min, lecture, advanced level

A Few Elementary Ways to Calculate Pi

In this talk we'll deduce a few beautiful formulae to calculate Pi as an infinite sum or an infinite product. We'll do it through a series of exercises that do not require any very advanced mathematics; simple trigonometry is enough.

This is, in fact a tutorial. It requires a small class and an active participation of the audience.

60-90 min, tutorial, advanced level

Introduction to Ideas in Discrete Geometry

We introduce some basic ideas in discrete geometry, relating the areas of polygons to their "discrete area " counterparts, which are combinatorial integer point enumeration algorithms for polygons in the plane. We also introduce some ideas which help us understand how the ancient fact that " the angles of a triangle sum up to 180 degrees" extends to higher dimensions. This extension is called the "Gram relations" for solid angles, which are the higher dimensional extension of 2-dimensional angles.

60 min, lecture, advanced level

A Game with Arithmetic Progressions

We play a two-person game in which each player picks an arithmetic progression among the integers so that there are no overlaps among any of the progressions. In the process, we learn about prime numbers, greatest common divisors, and some non-trivial facts about these fascinating number-theoretic objects. The players are encouraged to pair up and play the game in pairs, learning about these functions as they play the game.

60 min, tutorial, easy level

Formalizing the Liar’s paradox

In mathematics we are often asked to prove or disprove a certain statement. When mathematicians work they often set out to prove or disprove a certain conjecture. However, can this always be done? Given a mathematical statement, can we always find a proof or a refutation of the statement?

Kurt Godel proved the surprising “Incompleteness Theorems” in the early 20th century, which assert that given any reasonable system of axioms, we can always find some mathematical statement that can neither be proven nor refuted from the system of axioms; These are statements whose verification is impossible, or “independent” of the system. A strange, but interesting consequence is that if mathematics is non-contradictory, then it is impossible to prove so.

In this talk we will sketch the brief history of mathematical logic and talk about the famous Incompleteness Theorems. Only basic knowledge from high school mathematics is required for this talk.

60 min, lecture, advanced level

Introduction to Mathematical Finance

This talk is composed of an historical sketch of the development of mathematical finance over the past decades, followed by an informal introduction to stochastic processes and their role in the modelling of risky assets. In the last part of the talk we will review the fundamental concepts of arbitrage and hedging in a simplified way, and solve a hedging problem using systems of equations with two unknowns.

50 min, lecture, medium level

**Fibonacci Numbers in Nature **

The numbers in the famous Fibonacci number sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,... turn up
regularly in Nature, for example as the number of spiral arms in the seed patterns of sunflower
plants or in the cone patterns of pine cones. This talk will discuss the special properties of the
Fibonacci numbers and how their appearance in Nature can be understood as a consequence of
plants choosing the optimal strategy for maximizing their growth or seed production.

45 min, lecture, easy level

**Paper Folding and Geometrical Proofs**

How can you trisect an angle? It can be shown it's impossible to do this with ruler and compass alone, (using Galois theory) - so don't try it!!! But you may be able to find some good approximations using iterations. However, using paper folding, you can get accurate trisection of an acute angle.

In this workshop, we will learn how to use paper folding to make certain constructions which are not possible using the traditional ruler and compass constructions.

60 min, workshop, easy level