What have mathematicians done for us? Chris Budd

A common scenario You meet someone at a party and (foolishly) Tell them you are a mathematician Their immediate reaction is to … Panic Leave quickly Tell you that .. Mathematics is completely useless Mathematicians are souless geeks All mathematicians are mad!

In response you can … Deny that you are a mathematician Not go to any more parties Agree that mathematicians are evil or …. Say that The modern world would not exist without maths With maths you can tell the future and save lives Maths lies at the heart of art and music Mathematicians organise great parties

Some ways that maths has changed the modern world Maxwell: Electromagnetism … radio, TV, radar, mobile phones Linear algebra, graph theory, SVD ... Google Error correcting codes

Where it all started …. What mathematicians did to tell the future?

Conic section Appollonius Kepler

1600 Galileo watched a pendulum swing and realised that it was governed by predictable laws Swing time of the pendulum was constant Regardless of how it was pushed Or where it was Or when Nature had order and pattern and followed rules

Newton

1686 In the Principia Newton showed that this order and pattern could be expressed by using mathematics Pendulum equation

Key idea …. An astonishing discovery!!!!!!! Write down the equations describing a physical system Solve the equations Predict the future An astonishing discovery!!!!!!!

Newton’s law of gravitation Neptune: discovered by maths

Navier-Stokes equations Weather forecasting

Prediction of the tides (and Easter) Apparent motions of the sun and moon are not closely related … tides only repeat every 19 years Bombay tides 1872 h(t) t

Fourier: decompose h(t) into periodic functions Calculate coefficients and add up using an analogue computer Kelvin

Kelvin Tidal predictor US Tidal predictor

Laplace’s prediction … If we were to know the exact position of every particle in the universe then we could predict the future with certainty

The Double Pendulum Motion can be Periodic in phase : predictable Periodic out of phase : predictable Chaotic : unpredictable

Newton’s laws apply to the double pendulum! Angle of top part Angle of bottom part

Chaos Simple rules can lead to complex and unpredictable behaviour Poincare: discoverer of chaos

Motion of the asteroids: will the human race survive?

What mathematicians have done to save lives? Florence Nightingale

Mathematics can look inside you Modern CAT scanner CAT scanners work by casting many shadows with X-rays and using maths to assemble these into a picture

Measure attenuation of X-Ray R(ρ, θ) X-ray Source X-Ray Detector Object Density f(x,y) ρ : Distance from the object centre θ : Angle of the X-Ray Measure attenuation of X-Ray R(ρ, θ)

Also used to X-ray mummies Detect land mines If we can measure R(ρ, θ) accurately we can calculate The density f(x,y) of the object at any point Radon 1917 Also used to X-ray mummies Detect land mines

What have mathematicians done for music? Was Bach a mathematician?

Music is based on the idea of the scale … Some musical notes sound better when played together than others For example the notes C and G (a perfect 5th) And the notes C and E (a perfect 3rd)

Reason was discovered by Pythagaros Length of strings giving C and G, and C and E, were in simple fractional proportions C:G … 3/2 C:E … 4/3

Led to the Just Scale .. Sequence of notes with frequencies in simple rational relationships Problem: Keyboard instruments could only be tuned for one key Solution: well tempered scale arranges notes so that adjacent semi-tones have frequencies in the same proportion 12 semi-tones in the scale of one octave Frequency doubles in one octave

If is the frequency ratio then Good news: All scales equally good Bach .. Well tempered clavier Bad news: Is irrational, no scale is in perfect proportion .. But we can live with this!

What have mathematicians done for parties? Maths can help to cook food

A C E But who should you invite? You have five friends, Annabel, Brian, Colin, Daphne, Edward Want to invite three to a party Annabel hates Brian and Daphne Brian hates Colin and Edward Daphne hates Edward A C E Who do you invite?

Method Now have 200 friends and want 100 to come to a party Have a book saying who hates who Who do you invite? Choose a random party See if there are any clashes If so repeat steps 1 and 2 till you stop Method How long does this take? Parties to check Takes a high speed computer Years

Challenging problem in combinatorial optimisation Lots of other related problems of great commercial importance eg. Scheduling, SAT NAV Great modern advance combining maths, physics and biology ….. Can ‘solve’ using stochastic methods such as simulated annealing, genetic algorithms Constantly improve the party to reduce the number of clashes, but occasionally accept a party with more clashes Works for a party and many other problems

Conclusion …. your Life Soul Party Future Are safe in the hands of a mathematician