Mathematics, an Attractive Science

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Mathematics, an Attractive Science
North-East Students Summer Training on Basic
Science NESST-BASE Bose Institute, Mayapuri,
Darjeeling
June 2, 2007

• Michel Waldschmidt
• Université P. et M. Curie Paris VI
• Centre International de Mathématiques Pures et
  Appliquées
• CIMPA

http//www.math.jussieu.fr/miw/
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• Lexplosion
• des
• Mathématiques

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http//smf.emath.fr/Publication/ExplosionDesMathe
matiques/ Presentation.html
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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 France Société de
Mathématiques Appliquées et Industrielles
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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 .
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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?

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Eugene Wigner

•  The unreasonable effectiveness
• of mathematics in the natural
• sciences 
• Communications in Pure and Applied
  Mathematics, vol. 13, No. I (February 1960)

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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.
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Weather forecast Probabilistic model for
the climate Stochastic partial differential
equations Statistics
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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.

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Knot theory in algebraic topology

• Classification of knots, search of invariants
• Surgical operations

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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.

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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

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Error Correcting Codes Data Transmission

• Telephone
• CD or DVD
• Image transmission
• Sending information through the Internet
• Radio control of satellites

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• Olympus Mons on Mars Planet
• Image from Mariner 2 in 1971.

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Sphere packing
The kissing number is 12
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Sphere Packing

• Kepler Problem maximal density of a packing of
  identical sphères
•   p / Ö 18 0.740 480 49
• Conjectured in 1611.
• Proved in 1999 by Thomas Hales.
• Connections with crystallography.

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Codes and Geometry

• 1949 Golay (specialist of radars) efficient code
  
• Eruptions on Io (Jupiters volcanic moon)
• 1963 John Leech uses Golays ideas for sphere
  packing in dimension 24 - classification of
  finite simple groups

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Data transmission

• French-German war of 1870, siege of Paris

Flying pigeons first crusade - siege of Tyr,
Sultan of Damascus
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Data transmission

• James C. Maxwell
• (1831-1879)
• Electromagnetism

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Cell Phones Information
Theory Transmission by Hertz waves Algorithmic,
combinatoric optimization, numerical treatment
of signals, error correcting codes. How to
distribute frequencies among users.
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Data Transmission
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Language Theory

• Alphabet - for instance 0,1
• Letters (or bits) 0 and 1
• Words (octets - example 0 1 0 1 0 1 0 0)

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ASCII

• American Standard Code for Information
  Interchange
• Letters octet
• A 01000001
• B 01000010

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Coding
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Error correcting codes
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Applications of error correcting codes

• Transmitions by satellites
• Compact discs
• Cellular phones

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Codes and Maths

• Algebra
• (discrete mathematics finite fields, linear
  algebra,)
• Geometry
• Probability and statistics

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Coding
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Coding
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• 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.

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Detecting one error

• Send twice the same message
• 2 code words on 422
• (1 useful letter of 2)

• Code words
• (two letters)
• 0 0
• 1 1
• Rate 1/2

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Correcting an error

• Send the same message three times
• 2 code words of 823
• (1 useful letter of 3)

• Code words
• (three letters)
• 0 0 0
• 1 1 1
• Rate 1/3

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• 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

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• Principle of coding correcting one error
• Two distinct code words have at least three
  distinct letters

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Detecting one error (again)

• Code words (three letters)
• 0 0 0
• 0 1 1
• 1 0 1
• 1 1 0
• Parity bit (x y z) with zxy.
• 42?22 code words of 823
• (2 useful letters of 3).
• Rate 2/3

2
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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.

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Correcting one error (again)

• Words of 7 letters
• Code words (1624 on 12827 )
• (a b c d e f g)
• with
• eabd
• facd
• gabc
• Rate 4/7

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How to compute e , f , g , from a , b , c , d.
eabd
d
a
b
facd
c
gabc
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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
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Listening to a CD

• On a CD as well as on a computer, each sound is
  coded by a sequence of 0s and 1s, grouped in
  octets
• Further octets are added which detect and correct
  small mistakes.

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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.

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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

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• 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.

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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.

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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

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Voyager (1979-81)

• Color photos of Jupiter and Saturn
• Golay code with 4096212 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.

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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 (71)?16 256).

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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

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• 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.

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• Simple strategy they agree that two of them
  abstain and the other guesses. Probability of
  winning 1/2.
• Is it possible to do better?

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

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• 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.

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• Wins!

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• Loses!

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• Winning

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• Losing

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• 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.

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• Are there better strategies?
• Answer NO!
• Are there other strategies giving the same
  probability 3/4?
• Answer YES!

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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).

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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).

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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

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• 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.

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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.

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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.

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• 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 0s
• and
• those with two or three 1s

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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 3
• Richard W. Hamming (1915-1998)

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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
  2n1.

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• 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

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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.

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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.

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• 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.

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• 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.

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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

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• (0,0,0)
• (0,0,1) (0,1,1)
• (1,0,1)

• (1,1,1)
• (1,1,0) (1,0,0)
• (0,1,0)

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• 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.

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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.

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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 128-16112 cases (one bets correctly, the
  6 others abstain)
• Probability of winning 112/1287/8

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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?

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• 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.

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• 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.

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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).

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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,

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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,

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Mathematics involved in optimization

• Algebra (linear and bilinear algebra,)
• Analysis (differential calculus, numerical
  analysis, )
• Probability theory.

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Optimal path

• How to go from O to F

B
A
c
a
d
C
b
y
x
F
O
D
z
t
E
af(x1,,xn)
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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.

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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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