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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
L’explosionL’explosion
des des
MathématiquesMathématiques
http://smf.emath.fr/Publication/
ExplosionDesMathematiques/
Presentation.html
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
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 .
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?
Eugene Wigner:
« The unreasonable effectiveness
of mathematics in the natural
sciences »
Communications in Pure and Applied
Mathematics, vol. 13, No. I (February 1960)
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.
Weather forecast
Probabilistic model for the climate
Stochastic partial differential equations
Statistics
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.
Knot theory in algebraic topology
• Classification of knots, search of invariants
• Surgical operations
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.
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
Error Correcting Codes Data Transmission
• Telephone
• CD or DVD
• Image transmission
• Sending information through the Internet
• Radio control of satellites
Olympus Mons on Mars Planet
Image from Mariner 2 in 1971.
Sphere packing
The kissing number is 12
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.
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
Data transmission
French-German war of 1870, siege of Paris
Flying pigeons : first crusade - siege of Tyr, Sultan of Damascus
Data transmission
• James C. Maxwell
(1831-1879)
• Electromagnetism
Cell Phones
Information Theory
Transmission by Hertz waves
Algorithmic, combinatoric optimization, numerical treatment of signals, error correcting codes.
How to distribute frequencies among users.
Data Transmission
Transmission
Source Receiver
Language Theory
• Alphabet - for instance {0,1}
• Letters (or bits): 0 and 1
• Words (octets - example 0 1 0 1 0 1 0 0)
ASCII
American Standard Code for Information Interchange
Letters octetA: 01000001B: 01000010… …
Coding
transmission
Source Coded Text
Coded Text
Receiver
Error correcting codes
Applications of error correcting codes
• Transmitions by satellites
• Compact discs
• Cellular phones
Codes and Maths• Algebra (discrete mathematics finite
fields, linear algebra,…)
• Geometry
• Probability and statistics
Coding
transmission
Source Coded Text
Coded Text
Receiver
Coding
transmission
Source Coded Text
Noise
CodedText
Receiver
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.
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
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
• 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
Principle of coding correcting one error:
Two distinct code words have at least three distinct letters
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
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.
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
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.
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
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.
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.
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
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.
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.
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
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.
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).
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
• 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.
• Simple strategy: they agree that two of them abstain and the other guesses. Probability of winning: 1/2.
• Is it possible to do better?
• Hint:
Improve the probability by using the available information: each member of the team knows the two other colors.
• 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.
Wins!
Loses!
Winning:
Losing:
• 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.
• Are there better strategies?
Answer: NO!• Are there other strategies giving the same
probability 3/4?
Answer: YES!
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).
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).
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
• 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.
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.
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.
• 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
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)
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.
• 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
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.
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.
• 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.
• 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.
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
(0,0,0)
(0,0,1) (0,1,1)
(1,0,1)
(1,1,1)
(1,1,0) (1,0,0)
(0,1,0)
• 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.
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.
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
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?
• 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.
• 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.
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).
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,…
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, …
Mathematics involved in optimization
• Algebra (linear and bilinear algebra,…)
• Analysis (differential calculus, numerical analysis, …)
• Probability theory.
Optimal path
How to go from O to F
AB
O
C
FD
E
ab
cd
x y
z t
a=f(x1,…,xn)
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.
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
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ExplosionDesMathematiques/
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