Where is mathematics taking me? An exciting ride into the future.

1. Where is mathematicstaking 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

2. Is there a future? (#1)

3. Apophis

4. 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.

5. Is there a future? (#2)

6. 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.

7. 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.

8. Is there a future (#3): Our Hearts

9. 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

10. Calcium transfer This occurs in your heart, throat, gut, or wherever muscles contract. Not to mention in neurons in your brain!

11. 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 ?

12. Air Crew Scheduling Let us say we have nine flights between four airports as follows: 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 DEL HYD BOM MAA

13. 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 4 6 1 7 2 5 8 9 3

14. 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 4 6 1 7 2 5 8 9 3

15. 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. Each column represents a CREW SCHEDULE. AI 1 AI 2 AI 3 AI 4 AI 5 AI 6 AI 7 AI 8 AI 9 PATHWAY 4 4 6 6 1 7 2 2 5 8 8 9 3

16. Air Crew Scheduling 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.

17. Air Crew Scheduling We chose columns 3, 7, and 12. 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.

18. 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.

19. 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.

20. 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.

21. 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.

22. 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.

23. The fish’smovementis, in fact,the mostefficientpossible toreach thatposition.

24. The future (#2): Google How does Google order all the pages it finds when you ask it to search forsomething? If I search for “HyderabadSchools” Google findsover 2 million pages. How does it decide which webpages to show on the first screen?

25. Why does “Oakridge” get shown on the first screen, and the “Hyderabad Parents Association” not appear until screen 41?

26. 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 pagesare linked to each other. In this diagram I have drawnthose links as arrows. For example, B links to twoother pages, A and C.

27. 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?

28. It is possible to multiplymatrices. If you multiplythis matrix by itself, thenyou get another matrixthat tells you whether youcan get from one page toanother in TWO steps. What do you notice ?

29. 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 two steps. For example, if youstart on page C, thenthe probability thatyou will end up onpage B is 0.44 or 44% A B C D E A B C D E

30. 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 !

31. 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.

32. So Google just ranks the pages in orderfrom the most likely to the least likely. In this case it would rank them: B, A, C, E, D.

33. 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.

34. 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 onthe Pacific Coast of NorthAmerica were being loggedpeople became concernedabout the Spotted Owl.

35. 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 IterativeEquations to show howthey progressed, makingfield observations for thecoefficients.

36. 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 theseequations so thateach equation useseach variable (withzero coefficients).

37. 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 writethese equationsin a matrix form,the elements of thematrix are thecoefficients.

38. 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).

39. 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

41. The future (#4): The Shape of the Universe

42. 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.

43. Opening Doorson 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

44. Marcus du Sautoy “I love the buzz of discovering somenew eternal truth about themathematical world. The adrenalinerush of creating a strangesymmetrical object never seenbefore, with interesting newproperties, is addictive. Of course,it’s wonderful if there is some killerapplication of your discovery, butthat is rarely the motivation forhiking to the extremes of the mathematical landscape.” From his column in The Times, February 10, 2010

45. Vaughan Jones “My most important result was theindex for subfactors. I was on a highfor months as the result revealeditself. All I wanted to do was thinkabout the maths and any worldlything 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.”

46. HaukeGroot “I remember trying to solve a driveway designproblem. It was difficult because there were twointersecting slopes. I struggled trying, and failing,to remember my school trigonometry, lookingthings up, trying different ideas. Finally I got it. It was such a buzz. Partly becauseI 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.”

47. Mathematics IsFull of Opportunities and Unexpected Pleasures Join us! <b.barton@auckland.ac.nz>