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Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA http://www.math.jussieu.fr/~miw/ North-East Students’ Summer Training on Basic Science NESST-BASE Bose Institute, Mayapuri, Darjeeling June 2, 2007

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Page 1: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Mathematics, an Attractive Science

Michel WaldschmidtUniversité P. et M. Curie Paris VI

Centre International de Mathématiques Pures et Appliquées

CIMPA

http://www.math.jussieu.fr/~miw/

North-East Students’ Summer Training on Basic Science NESST-BASEBose Institute, Mayapuri, Darjeeling

June 2, 2007

Page 2: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

L’explosionL’explosion

des des

MathématiquesMathématiques

Page 3: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

http://smf.emath.fr/Publication/

ExplosionDesMathematiques/

Presentation.html

Page 4: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Explosion of Mathematics

• Weather forecast

• Cell phones

• Cryptography

• Control theory

• From DNA to knot theory

• Air transportation

• Internet: modelisation of traffic

• Communication without errors

• Reconstruction of surfaces for images

Société Mathématique de FranceSociété de Mathématiques Appliquées et Industrielles

Page 5: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Aim:

To illustrate with a few examples the usefulness of some mathematical theories which were developed only for theoretical purposes

Unexpected interactions between pure research and the real world .

Page 6: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Interactions between physics and mathematics

• Classical mechanics

• Non-Euclidean geometry: Bolyai, Lobachevsky, Poincaré, Einstein

• String theory

• Global theory of particles and their interactions: geometry in 11 dimensions?

Page 7: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Eugene Wigner:

« The unreasonable effectiveness

of mathematics in the natural

sciences »

Communications in Pure and Applied

Mathematics, vol. 13, No. I (February 1960)

Page 8: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Dynamical systems

Three body problems (Henri Poincaré)

Chaos theory (Edward Lorentz): the butterfly effect:Due to nonlinearities in weather processes, a butterfly flapping its wings in Tahiti can, in theory, produce a tornado in Kansas.

Page 9: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Weather forecast

Probabilistic model for the climate

Stochastic partial differential equations

Statistics

Page 10: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Weather forecast

• Mathematical models are required for describing and understanding the processes of meteorology, in order to analyze and understand the mechanisms of the climate.

• Some processes in meteorology are chaotic, but there is a hope to perform reliable climatic forecast.

Page 11: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Knot theory in algebraic topology

• Classification of knots, search of invariants

• Surgical operations

Page 12: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Knot theory and molecular biology

• The topology of DNA molecule has an action on its biological action.

• The surgical operations introduced in algebraic topology have biochemical equivalents which are realized by topoisomerases.

Page 13: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Finite fields and coding theory

• Solving algebraic equations with radicals: Finite fields theory Evariste Galois (1811-1832)

• Construction of regular polygons with rule and compass

• Group theory

Page 14: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Error Correcting Codes Data Transmission

• Telephone

• CD or DVD

• Image transmission

• Sending information through the Internet

• Radio control of satellites

Page 15: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Olympus Mons on Mars Planet

Image from Mariner 2 in 1971.

Page 16: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Sphere packing

The kissing number is 12

Page 17: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Sphere Packing

• Kepler Problem: maximal density of a packing of identical sphères :

  / 18= 0.740 480 49…

Conjectured in 1611.

Proved in 1999 by Thomas Hales.

• Connections with crystallography.

Page 18: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Codes and Geometry

• 1949 Golay (specialist of radars): efficient code

• Eruptions on Io (Jupiter’s volcanic moon)

• 1963 John Leech: uses Golay’s ideas for sphere packing in dimension 24 - classification of finite simple groups

Page 19: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Data transmission

French-German war of 1870, siege of Paris

Flying pigeons : first crusade - siege of Tyr, Sultan of Damascus

Page 20: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Data transmission

• James C. Maxwell

(1831-1879)

• Electromagnetism

Page 21: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Cell Phones

Information Theory

Transmission by Hertz waves

Algorithmic, combinatoric optimization, numerical treatment of signals, error correcting codes.

How to distribute frequencies among users.

Page 22: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Data Transmission

Transmission

Source Receiver

Page 23: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Language Theory

• Alphabet - for instance {0,1}

• Letters (or bits): 0 and 1

• Words (octets - example 0 1 0 1 0 1 0 0)

Page 24: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

ASCII

American Standard Code for Information Interchange

Letters octetA: 01000001B: 01000010… …

Page 25: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Coding

transmission

Source Coded Text

Coded Text

Receiver

Page 26: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Error correcting codes

Page 27: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Applications of error correcting codes

• Transmitions by satellites

• Compact discs

• Cellular phones

Page 28: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Codes and Maths• Algebra (discrete mathematics finite

fields, linear algebra,…)

• Geometry

• Probability and statistics

Page 29: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Coding

transmission

Source Coded Text

Coded Text

Receiver

Page 30: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Coding

transmission

Source Coded Text

Noise

CodedText

Receiver

Page 31: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Principle of coding theory : only certain words are permitted (code =

dictionary of allowed words).

The « useful » letters carry the information, the other ones (control bits) allow detecting errors.

Page 32: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Detecting one error

• Send twice the same message

2 code words on 4=22

(1 useful letter of 2)

Code words

(two letters)

0 0

1 1

Rate: 1/2

Page 33: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Correcting an error

• Send the same message three times

2 code words of 8=23

(1 useful letter of 3)

Code words

(three letters)

0 0 0

1 1 1

Rate: 1/3

Page 34: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

• Correct 0 0 1 as 0 0 0

• 0 1 0 as 0 0 0

• 1 0 0 as 0 0 0

and

• 1 1 0 as 1 1 1

• 1 0 1 as 1 1 1

• 0 1 1 as 1 1 1

Page 35: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Principle of coding correcting one error:

Two distinct code words have at least three distinct letters

Page 36: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Detecting one error (again)

Code words (three letters):0 0 00 1 11 0 11 1 0

Parity bit : (x y z) with z=x+y.4=22=2 code words of 8=23

(2 useful letters of 3).Rate: 2/3

2

Page 37: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Code words Non code words

0 0 0

0 1 1

1 0 1

1 1 0

0 0 1

0 1 0

1 0 0

1 1 1

Two distinct code words have at least two distinct letters.

Page 38: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Correcting one error (again)

Words of 7 letters

Code words: (16=24 on 128=27 )

(a b c d e f g)with

e=a+b+d f=a+c+d g=a+b+c

Rate: 4/7

Page 39: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

ab

d

c

e=a+b+d

g=a+b+c

f=a+c+d

How to compute e , f , g , from a , b , c , d.

Page 40: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

16 code words of 7 letters

0 0 0 0 0 0 0

0 0 0 1 1 1 0

0 0 1 0 0 1 1

0 0 1 1 1 0 1

0 1 0 0 1 0 1

0 1 0 1 0 1 1

0 1 1 0 1 1 0

0 1 1 1 0 0 0

1 0 0 0 1 1 1

1 0 0 1 0 0 1

1 0 1 0 1 0 0

1 0 1 1 0 1 0

1 1 0 0 0 1 0

1 1 0 1 1 0 0

1 1 1 0 0 0 1

1 1 1 1 1 1 1

Two distinct code words have at least three distinct letters

Page 41: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Listening to a CD

• On a CD as well as on a computer, each sound is coded by a sequence of 0’s and 1’s, grouped in octets

• Further octets are added which detect and correct small mistakes.

Page 42: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Coding the sound on a CD

Using a finite field with 256 elements, it is possible to correct 2 errors in each word of 32 octets with 4 control octets for 28 information octets.

Page 43: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

A CD of high quality may have more than 500 000 errors!

• 1 second of radio signal = 1 411 200 bits.• The mathematical theory of error correcting codes

provides more reliability and at the same time decreases the cost. It is used also for data transmission via the internet or satellites

Page 44: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Informations was sent to the earth using an error correcting code which corrected 7 bits on 32.

In each group of 32 bits, 26 are control bits and the 6 others contain the information.

Page 45: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Voyager 1 and 2 (1977)

• Journey: Cape Canaveral, Jupiter, Saturn, Uranus, Neptune.

• Sent information by means of a binary code which corrected 3 errors on words of length 24.

Page 46: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Mariner spacecraft 9 (1979)

• Sent black and white photographs of Mars

• Grid of 600 by 600, each pixel being assigned one of 64 brightness levels

• Reed-Muller code with 64 words of 32 letters, minimal distance 16, correcting 7 errors, rate 3/16

Page 47: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Voyager (1979-81)

• Color photos of Jupiter and Saturn• Golay code with 4096=212 words of 24

letters, minimal distance 8, corrects 3 errors, rate 1/2.

• 1998: lost of control of Soho satellite recovered thanks to double correction by turbo code.

Page 48: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

The binary code of Hamming and Shannon (1948)

It is a linear code (the sum of two code words is a code word) and the 16 balls of radius 1 with centers in the code words cover all the space of the 128 binary words of length 7

(each word has 7 neighbors (7+1)16= 256).

Page 49: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

The Hat Problem

• A team of three people in a room with black/white hats on their head (hat colors chosen at random). Each of them sees the color on the hat of the others but not on his own. They do not communicate.

• Everyone writes on a piece of paper the color he guesses for his own hat: black/white/abstain

Page 50: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

• The team wins if at least one of the three people does not abstain, and everyone who did not abstain guesses correctly the color of his hat.

Page 51: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

• Simple strategy: they agree that two of them abstain and the other guesses. Probability of winning: 1/2.

• Is it possible to do better?

Page 52: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

• Hint:

Improve the probability by using the available information: each member of the team knows the two other colors.

Page 53: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

• Better strategy: if a member sees two different colors, he abstains. If he sees the same color twice, he guesses that his hat has the other color.

Page 54: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Wins!

Page 55: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Loses!

Page 56: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Winning:

Page 57: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Losing:

Page 58: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

• The team wins exactly when the three hats do not have all the same color, that is in 6 cases of a total of 8

• Probability of winning: 3/4.

Page 59: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

• Are there better strategies?

Answer: NO!• Are there other strategies giving the same

probability 3/4?

Answer: YES!

Page 60: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Tails and Ends

• Throw a coin three consecutive times

• There are 8 possible sequences of results:

(0,0,0), (0,0,1), (0,1,0), (0,1,1),

(1,0,0), (1,0,1), (1,1,0), (1,1,1).

Page 61: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

If you bet (0,1,0), you have :

• All three correct results for (0,1,0).

• Exactly two correct results if the sequence is either (0,1,1), (0,0,0) or (1,1,0),

• Exactly one correct result if the sequence is either (0,0,1), (1,1,1) or (1,0,0),

• No correct result at all for (1,0,1).

Page 62: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Whatever the sequence is, among 8 possibilities,

each bet is winning in exactly 1 case

has exactly two correct results in 3 cases

has exactly one correct result in 3 cases

has no correct result at all in only 1 case

Page 63: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

• Goal: To be sure of having at least two correct results

• Clearly, one bet is not sufficient• Are two bets sufficient? Recall that there are 8 possible results,

and that each bet has at least two correct results in 4 cases.

Page 64: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Answer: YES, two bets suffice!

For instance bet

(0,0,0) and (1,1,1)

Whatever the result is, one of the two digits

0 and 1

occurs more than once.

Hence one (and only one) of the two bets

has at least two correct results.

Page 65: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Other solutions:

• Select any two bets with all three different digits, say

0 0 1 and 1 1 0

The result either is one of these, or else has just one common digit with one of these and two common digits with the other.

Page 66: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

• Come back with (0,0,0) and (1,1,1)The 8 sequences of three digits 0 and 1 split into two groups: those with two or three 0’sand those with two or three 1’s

Page 67: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Hamming Distance between two words:

= number of places where the two words

do not have the same letter

Examples:

(0,0,1) and (0,0,0) have distance 1

(1,0,1) and (1,1,0) have distance 2

(0,0,1) and (1,1,0) have distance 3Richard W. Hamming (1915-1998)

Page 68: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Hamming Distance

• Recall that the Hamming distance between two words is the number of places where letters differ.

• A code detects n errors iff the Hamming distance between two distinct code words is at least 2n.

• It corrects n errors iff the Hamming distance between two distinct code words is at least 2n+1.

Page 69: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

• The set of eight elements splits into two balls

• The centers are (0,0,0) and (1,1,1)

• Each of the two balls consists of elements at distance at most 1 from the center

Page 70: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Back to the Hat Problem

• Replace white by 0 and black by 1 hence the distribution of colors becomes a

word of three letters on the alphabet {0 , 1}• Consider the centers of the balls (0,0,0) and

(1,1,1).• The team bets that the distribution of colors

is not one of the two centers.

Page 71: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Assume the distribution of hats does not correspond to one of the centers

(0, 0, 0) and (1, 1, 1). Then:

• One color occurs exactly twice (the word has both digits 0 and 1).

• Exactly one member of the team sees twice the same color: this corresponds to 0 0 in case he sees two white hats, 1 1 in case he sees two black hats.

• Hence he knows the center of the ball: (0, 0, 0) in the first case, (1, 1, 1) in the second case.

• He bets the missing digit does not yield the center.

Page 72: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

• The two others see two different colors, hence they do not know the center of the ball. They abstain.

• Therefore the team win when the distribution of colors does not correspond to the centers of the balls.

• this is why the team win in 6 cases.

Page 73: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

• Now if the word corresponding to the distribution of the hats is one of the centers, all members of the team bet the wrong answer!

• They lose in 2 cases.

Page 74: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Another strategy:

• Select two words with mutual distance 3 = two words with three distinct letters, say (0,0,1) and (1,1,0)

• For each of them, consider the ball of elements at distance at most 1

Page 75: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

(0,0,0)

(0,0,1) (0,1,1)

(1,0,1)

(1,1,1)

(1,1,0) (1,0,0)

(0,1,0)

Page 76: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

• The team bets that the distribution of colors is not one of the two centers (0,0,1), (1,1,0) .

• A word not in the center has exactly one letter distinct from the center of its ball, and two letters different from the other center.

Page 77: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Assume the word corresponding to the distribution of the hats is not a center.

Then:

• Exactly one member of the team knows the center of the ball. He bets the distribution does not correspond to the center.

• The others do not know the center of the ball. They abstain.

• Hence the team win.

Page 78: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

The Hat Problem with 7 people

• The team bets that the distribution of the hats does not correspond to the 16 elements of the Hamming code

• Loses in 16 cases (they all fail)

• Wins in 128-16=112 cases (one bets correctly, the 6 others abstain)

• Probability of winning: 112/128=7/8

Page 79: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Tossing a coin 7 times

• There are 128 possible results

• Each bet is a word of 7 letters on the alphabet {0, 1}

• How many bets do you need if you want to guarantee at least 6 correct results?

Page 80: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

• Each of the 16 code words has 7 neighbors (at distance 1), hence the ball of which it is the center has 8 elements.

• Each of the 128 words is in exactly one of these balls.

Page 81: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

• Make 16 bets corresponding to the 16 code words: then, whatever the result is, exactly one of your bets will have at least 6 correct answers.

Page 82: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

The price of financial options

• Probability theory yields a modelisation of random processes. The prices of stocks traded on stock exchanges fluctuate like the Brownian motion.

• Optimal stochastic control involves ideas which previously occurred in physics and geometry (deformation of surfaces).

Page 83: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

How to control a complex world

• Managing distribution in an electricity network, studying the vibrations of a bridge, the flow of air around an airplane require tools from the mathematical theory of control (differential equations, partial derivatives equations) .

• The optimization of trajectories of satellites rely on optimal control, numerical analysis, scientific calculus,…

Page 84: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Optimization• Industry manufacturing, costs

reducing, decreasing production time, …

• Production of fabrics, shoes : minimizing waste, …

• Petroleum Industry : how to find the proper hydrocarbon mixtures, …

• Aero dynamism (planes, cars,…).• Aerospace industry : optimal

trajectory of an interplanetary spaceflight, …

Page 85: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Mathematics involved in optimization

• Algebra (linear and bilinear algebra,…)

• Analysis (differential calculus, numerical analysis, …)

• Probability theory.

Page 86: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Optimal path

How to go from O to F

AB

O

C

FD

E

ab

cd

x y

z t

a=f(x1,…,xn)

Page 87: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Trees and graphs

A company wants to find the best way (less expensive, fastest) for trucks which receive goods and deliver them at many different places.

Page 88: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

Applications of graph theory

• Electric circuits • How to rationalize the production

methods, to improve the organization of a company.

• How to manage the car traffic or the metro network.

• Informatics and algorithmic• Buildings and public works • Internet, cell phones

Page 89: Mathematics, an Attractive Science Michel Waldschmidt Université P. et M. Curie Paris VI Centre International de Mathématiques Pures et Appliquées CIMPA

http://smf.emath.fr/Publication/

ExplosionDesMathematiques/

Presentation.html