1. What have mathematicians done
for us?
Chris Budd

2. 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!

3. 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

4. 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

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

6. Conic section
Appollonius
Kepler

7. 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

8. Newton

9. 1686
In the Principia Newton showed that this order and pattern could be expressed by using mathematics
Pendulum equation

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

11. Newton’s law of gravitation
Neptune: discovered by maths

12. Navier-Stokes equations
Weather forecasting

13. 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

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

15. Kelvin Tidal predictor
US Tidal predictor

16. 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

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

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

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

20. Motion of the asteroids: will the human race survive?

21. What mathematicians have done to save lives?
Florence Nightingale

22. 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

23. 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)

24. 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

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

26. 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)

27. 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

28. 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

29. 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!

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

31. 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?

32. 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

33. 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

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