1. What is Mathematics? Lee Peng Yee 21 November 2014

2. What is mathematics? A book by Courant and Robbins (1941) Last two chapters: calculus Calculus was the palace of modern math Now calculus is the gate to the palace

3. Calculus: Taylor series

4. Power series Taylor series is one special case We use it to solve differential equations We used to express solutions in closed form Also by special functions and huge volumes of tables

5. Analysis: Fourier series

6. Functional analysis

7. Mother functions

8. Further comparison Taylor series / Fourier series Coefficients: by differentiation / by integration Coordinates: non-orthogonal / orthogonal Better approximation: locally / globally Question: Find a series … … both locally and globally

9. Haar wavelet

10. Properties of Haar wavelet

11. Wavelets Haar wavelet is one special case “Whatever your curves, we’ve you wrapped” Wavelets help send images fast and far away among other applications

12. Stochastic equations Nobel prize was awarded for the works on the Black-Scholes equation Stochastic equations contain nondeterministic terms called noises Hence possible to model events in the financial world

13. Wiener and Itô Norbert Wiener (1894 – 1964) provided the concept known as Wiener process Kiyoshi Itô (1915 – 2008) developed the operational tools French school led by Paul-André Meyer (1934 – 2003)

14. Big data analytics Latest challenges due to the size and the speed of trade data Big data technology is to collect, analyse and make use of the data Big data is not well-defined

15. Big data Singapore Two examples: A bank with 25 million transactions a month uses the technology to plan for ATM cash reloading Singapore Civil Defence Force is looking into using the technology to slash response times for ambulance and fire engines

16. Modern history of mathematics Calculus helped make better locomotives Analysis was a product of the nineteenth century Mathematics is being used in the non- traditional areas The latest: big data

17. What we miss Statistics, computational mathematics. discrete mathematics … … Discrete mathematics came in only after World War II Now we have moved from the world of telephones to that of internet Mathematics outside is not the same as inside the classroom

18. END pengyee.lee@nie.edu.sg