What have mathematicians done

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 .

• Write down the equations describing a physical
  system
• Solve the equations
• Predict the future

An astonishing discovery!!!!!!!
11
Newtons 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
US Tidal predictor
Kelvin Tidal predictor
16
Laplaces 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
Newtons 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
X-ray Source
Detector
X-Ray
Object Density f(x,y)
? Distance from the object centre ? Angle
of the X-Ray
Measure attenuation of X-Ray R(?, ?)
24
If we can measure R(?, ?) 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
CG 3/2 CE 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
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
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

• Future
• Life
• Soul
• Party

Are safe in the hands of a mathematician