Where is mathematics
taking me?
An exciting ride into the future.
Bill Barton
Department of Mathematics, The University of Auckland
President, ICMI
(International Commission on Mathematical Instruction)
Hyderabad, India, August, 2010
Is there a future? (#1)
Apophis
Near Earth Objects




Apophis will not hit, but will pass by in 2013 close
enough for us to put a tracker on it.
In April 13, 2029, another asteroid will pass at the
height of satellites (30 000km).
In 2036 another will come past with a 1:250 000
chance of hitting the earth.
The mathematics involves generating a million
imaginary asteroids, tracking each one (a lot of
integration), and seeing how many will hit the earth.
Is there a future? (#2)
Ice Shelf Break-Up




In Antartica there are huge ice
sheets (as big as Andhra Pradesh).
They act like mirrors to reflect the
sun’s heat—and this affects the
climate all over the world.
These sheets break up under the
action of the waves.
We need to be able to predict how
quickly this happens.
Ice Shelf Break-Up




Mathematically, the problem involves writing down
the equations for wave motion in water and ice and
the way they interact.
These equations include derivatives.
Solving these equations simultaneously is very
complex.
Part of the mathematical task is to transform the
equations so they exhibit symmetry—making them
easier to solve.
Is there a future (#3): Our Hearts
Modelling Calcium in the Heart
Background:
spiral wave of calcium in
a real beating heart
Foreground:
Simulations of a calcium
wave model —
demonstrating the
development of spiral
wave instability
Calcium transfer
This occurs in
your heart, throat,
gut, or wherever
muscles contract.
Not to mention in
neurons in your
brain!
Riding into the future (#1)

Today’s airlines need to solve enormous and complex
problems on a daily basis.
 One of these is the
scheduling problem.
 How to allocate
(and re-allocate) the
many aircrews to
the hundreds of
flights each day ?
Air Crew Scheduling
Let us say we have nine flights between four airports as follows:
Flight From: To:
Depart Arrive
DEL
AI 1
DEL
HYD
08:00
09:00
AI 2
DEL
HYD
09:00
10:00
AI 3
DEL
MAA
10:00
11:30
AI 4
HYD
BOM
13:00
13:40
AI 5
HYD
MAA
12:00
13:00
AI 6
BOM
MAA
17:00
18:00
AI 7
MAA
DEL
16:00
17:30
AI 8
MAA
DEL
19:00
20:30
AI 9
MAA
DEL
20:00
21:30
HYD
BOM
MAA
Air Crew Scheduling
Let us draw the diagram differently, with each flight being a different point:
Flight From: To:
Depart Arrive
AI 1
DEL
HYD
08:00
09:00
AI 2
DEL
HYD
09:00
10:00
AI 3
DEL
MAA
10:00
11:30
AI 4
HYD
BOM
13:00
13:40
AI 5
HYD
MAA
12:00
13:00
AI 6
BOM
MAA
17:00
18:00
AI 7
MAA
DEL
16:00
17:30
AI 8
MAA
DEL
19:00
20:30
AI 9
MAA
DEL
20:00
21:30
1
4
2
5
3
6
7
8
9
Air Crew Scheduling
Now we draw a connection between any two flights that can have the same crew:
Flight From: To:
Depart Arrive
AI 1
DEL
HYD
08:00
09:00
AI 2
DEL
HYD
09:00
10:00
AI 3
DEL
MAA
10:00
11:30
AI 4
HYD
BOM
13:00
13:40
AI 5
HYD
MAA
12:00
13:00
AI 6
BOM
MAA
17:00
18:00
AI 7
MAA
DEL
16:00
17:30
AI 8
MAA
DEL
19:00
20:30
AI 9
MAA
DEL
20:00
21:30
1
4
2
5
3
6
7
8
9
Air Crew Scheduling
Now we create the mathematical model, a MATRIX.
A MATRIX is a table of numbers, in this case only 0’s and 1’s.
Each row represents a FLIGHT.
PATHWAY
Each column represents a CREW
SCHEDULE.
4
1
AI 1 1 1 1 1 1 0 0 0 0 0 0 0 0
AI 2 0 0 0 0 0 1 1 1 1 1 0 0 0
AI 3 0 0 0 0 0 0 0 0 0 0 1 1 1
2
5
AI 4 1 1 0 0 0 1 1 0 0 0 0 0 0
AI 5 0 0 1 1 1 0 0 1 1 1 0 0 0
AI 6 1 1 0 0 0 1 1 0 0 0 0 0 0
AI 7 0 0 1 0 0 0 0 1 0 0 1 0 0
3
AI 8 1 0 0 1 0 1 0 0 1 0 0 1 0
AI 9 0 1 0 0 1 0 1 0 0 1 0 0 1
6
7
8
9
Air Crew Scheduling
1
0
0
1
0
1
0
1
0
1
0
0
1
0
1
0
0
1
1
0
0
0
1
0
1
0
0
1
0
0
0
1
0
0
1
0
1
0
0
0
1
0
0
0
1
0
1
0
1
0
1
0
1
0
0
1
0
1
0
1
0
0
1
0
1
0
0
1
0
1
0
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
0
1
0
0
1
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
1
0
0
0
0
0
1
Our problem of Air Crew
Scheduling now looks like this.
But we do not want all possible
Crew Assignments—we only
want to cover each flight once.
So we must choose some
columns so that there is only
one 1 in each row.
Let us choose columns 3, 7,
and 12.
Air Crew Scheduling
We chose columns 3, 7, and 12.
1
0
0
1
0
1
0
1
0
1
0
0
1
0
1
0
0
1
1
0
0
0
1
0
1
0
0
1
0
0
0
1
0
0
1
0
1
0
0
0
1
0
0
0
1
0
1
0
1
0
1
0
1
0
0
1
0
1
0
1
0
0
1
0
1
0
0
1
0
1
0
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
0
1
0
0
1
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
1
0
0
0
0
0
1
But we could have chosen
columns 2, 9, and 11 instead.
In fact there are many possible
combinations.
Each combination has a cost.
So the mathematical task is to
find all possible combinations
and then choose the one with
the least cost.
Air Crew Scheduling
Fortunately finding all possible combinations,
and then minimising the cost, is possible, even
though for real airlines there are millions of
possibilities.
Doing it for such large matrices does depend on
some mathematical theory in linear algebra, still
being developed, for which the original theorem
was proven only 30 years ago.
Air Crew Scheduling
This is actually only half the problem solved.
We have found possible “Tours of Duty” for a crew—but
who is on the crew? Each possible crew member has certain
constraints (leave, place of residence, training).
So the second problem is to assign crew members to each
crew in the most efficient manner.
And, in reality, both problems should be solved together.
Riding into the future (#2)
“The Love Life of Buses”
Why do buses pair up?
Mathematical models help
us explain both why it
occurs, and what we can
do about it.
Passenger behaviour can
prevent this happening.
The Future (#1): Fish & Robots


Biological
mathematics is a
fast-growing field.
For example, we
want to know how
animals move to
help robotics and
understand their
evolution.
Modelling Fish



Claire has studied the
Amazonian knife fish.
It lives in muddy water, and
cannot see. It senses its food
using magnetic strips on its side.
This means that it must move
backwards and sideways in a
very odd way.
The fish’s
movement
is, in fact,
the most
efficient
possible to
reach that
position.
The future (#2): Google
How does Google order all the pages it finds when
you ask it to search for
something?
If I search for “Hyderabad
Schools” Google finds
over 2 million pages.
How does it decide which webpages to show on the
first screen?
Why does “Oakridge” get shown on the first screen, and
the “Hyderabad Parents Association” not appear until
screen 41?
The Google Search Engine
Again, let us draw a diagram of a simple case.
Imagine that Google finds five pages: A, B, C, D, and E.
Google also knows how pages
are linked to each other.
In this diagram I have drawn
those links as arrows.
For example, B links to two
other pages, A and C.
Now we create a matrix so that each row and column
represents one of the pages.
A number links the column page to the row page.
Can you see why I have used these numbers?
It is possible to multiply
matrices. If you multiply
this matrix by itself, then
you get another matrix
that tells you whether you
can get from one page to
another in TWO steps.
What do you notice ?
0.50 0.17 0.17 0 0.61


0
0.66
0.44
1
0.11


P 2  0.50
0
0.28 0 0.17


0
0
0.11
0
0




0.17
0
0 0.11
 0
Did you notice that all the columns add up to 1?
A list of numbers that add up to 1 should make you
wonder whether the numbers are probabilities—WHY ?
In this case, the figures are the probabilities that, if you
start on one page, then you will end up on another after
A
B
C
D
E
two steps.
For example, if you
start on page C, then
the probability that
you will end up on
page B is 0.44 or 44%
0.50 0.17 0.17 0 0.61A


0.66 0.44 1 0.11B
 0
P 2  0.50
0
0.28 0 0.17C

D
0
0
0.11
0
0



E
0.17
0
0 0.11
 0
P3 gives you the same information for three steps.
What happens if I try lots and lots of steps?
Amazingly, after I get to P30, anything higher gives me the
same result.
And…. ALL THE COLUMNS ARE THE SAME !
This happens because of the structure of the matrix. We
call it a stochastic matrix.
What it means is that the probability of ending up on a
particular page after a lot of steps is the same whichever
page you start from.
So Google just ranks the pages in order
from the most likely to the least likely.
In this case it would rank them:
B, A, C, E, D.
What is even more wonderful, is that, starting from just
the matrix P, we can get directly to that final column.
This is called an EIGENVECTOR of the matrix—you
will learn about it in an undergraduate university course
in linear algebra.
Of course doing this for a matrix with 2 million rows and
columns is not a trivial task !! You study how computers
do these calculations with minimal round-off errors in the
study of numerical analysis.
e  0.293 0.390 0.220 0.024 0.073
You have seen how matrices are useful in a variety of
contexts. They also can be used to represent systems of
equations—and if we use them with Iterative Equations,
then we can model change.
Environmental scientists use these to investigate
populations of endangered species.
For example, when forests on
the Pacific Coast of North
America were being logged
people became concerned
about the Spotted Owl.
We can model the life-cycle of the Spotted Owl in three
stages:
ji — the number of juvenile owls in time interval i
si — the number of sub-adult owls in time interval i
ai — the number of adult owls in time interval i
Now we write Iterative
Equations to show how
they progressed, making
field observations for the
coefficients.
jn 1  0.33an
sn 1  0.18 j n
an 1  0.71sn  0.94an
We can model the life-cycle of the Spotted Owl in three
stages:
ji — the number of juvenile owls in time interval i
si — the number of sub-adult owls in time interval i
ai — the number of adult owls in time interval i
Let us write these
equations so that
each equation uses
each variable (with
zero coefficients).
jn 1  0 j n
sn 1  0.18 j n
an 1 
 0sn  0.33an
 0sn
 0an
0 jn  0.71sn  0.94an
We can model the life-cycle of the Spotted Owl in three
stages:
ji — the number of juvenile owls in time interval i
si — the number of sub-adult owls in time interval i
ai — the number of adult owls in time interval i
Now let us write
these equations
in a matrix form,
the elements of the
matrix are the
coefficients.
j n 1   0
0
0.33j n 
  
 
0
0 sn 
sn 1  0.18

0.71 0.94
an 1
 
 0

an 

We can use the properties of the matrix to determine what
will happen in the long term to the population of the
Spotted Owls, and also we can compare this with what
happens to other populations (for example, where there is
no logging).
j n 1   0
0
0.33j n 
  
 
0
0 sn 
sn 1  0.18

0.71 0.94
an 1
 
 0

an 

The future (#3): Knots
5-component link with Jones polynomial (√t + 1/√t)4
The mathematical
techniques for
distinguishing knotted
structures …
… are applied to
molecular biology
(RNA strands and
protein folding) and
theoretical physics
The future (#4): The Shape of the Universe
How can we work in more than
three dimensions ??


We can see three
dimensions—we can
look down from
above.
In four dimensions
we can only get
inside the space and
feel its slopes.
Opening Doors
on Pleasure and Opportunity

Those of us who love mathematics
love the opportunities it provides
and the pleasure it brings

Mathematics opens doors in
nearly every walk of life, and
every career you choose

Mathematicians become addicted
to struggling and solving problems
Marcus du Sautoy
“I love the buzz of discovering some
new eternal truth about the
mathematical world. The adrenaline
rush of creating a strange
symmetrical object never seen
before, with interesting new
properties, is addictive. Of course,
it’s wonderful if there is some killer
application of your discovery, but
that is rarely the motivation for
hiking to the extremes of the mathematical landscape.”
From his column in The Times, February 10, 2010
Vaughan Jones
“My most important result was the
index for subfactors. I was on a high
for months as the result revealed
itself. All I wanted to do was think
about the maths and any worldly
thing was an intrusion.”
“It is an amazing high that is a consequence of having
tried and tried with things falling apart all the time. Then
eventually you sense, then believe, then know, then are
able to prove, that it's actually for real.”
Hauke Groot
“I remember trying to solve a driveway design
problem. It was difficult because there were two
intersecting slopes. I struggled trying, and failing,
to remember my school trigonometry, looking
things up, trying different ideas.
Finally I got it. It was such a buzz. Partly because
I had spent so much time but finally did it.
And then I thought. Oh, there is another way to do this, so I worked on,
and came to the same solution but using another method, and then a
third one. It was so satisfying.
But I realised that I could not charge my client for the last three hours
work, because I already had the solution, I had just got carried away
with the mathematics.”
Mathematics Is
Full of Opportunities and
Unexpected Pleasures
Join us!
<b.barton@auckland.ac.nz>