What is Mathematics?

Lee Peng Yee21 November 2014

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

Calculus: Taylor series

An example:• + for all • +fornear 0

• If we can expand, we can approximate• To expand, a function must be infinitely

differentiable

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

Analysis: Fourier series

• Sawtooth function: for and defined periodically over • Fourier series approximates a given function

globally, whereas Taylor series locally

Functional analysis

• Convergence of Fourier series is a problem• It was solved by Hilbert space • Taylor series converges to the given function

uniformly inside the interval of convergence• Fourier series converges under the norm in

Mother functions

• Taylor series lives in an infinite dimensional space with coordinates

• Fourier series lives in a space with coordinates

• x and are called mother functions

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

Haar wavelet

• Mother function for , –1 for . 0 otherwise• Coordinates for all t

Properties of Haar wavelet

• Haar wavelet lives in • The Haar coordinate system is orthogonal• Haar wavelet converges to a given function

locally and globally

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

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

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)

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

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

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

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

END

pengyee.lee@nie.edu.sg