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Michel Waldschmidt Université P. et M. Curie Paris VI Société Mathématique de France

October 15, 2005. Khemarak University, Phnom Penh. Explosion of Mathematics. Michel Waldschmidt Université P. et M. Curie Paris VI Société Mathématique de France. http://www.math.jussieu.fr/~miw. L’explosion des Mathématiques.

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Michel Waldschmidt Université P. et M. Curie Paris VI Société Mathématique de France

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  1. October 15, 2005 Khemarak University, Phnom Penh Explosion of Mathematics Michel Waldschmidt Université P. et M. Curie Paris VI Société Mathématique de France http://www.math.jussieu.fr/~miw

  2. L’explosion des Mathématiques

  3. http://smf.emath.fr/Publication/ExplosionDesMathematiques/ Presentation.html

  4. 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 Explosion of Mathematics Société Mathématique de France Société de Mathématiques Appliquées et Industrielles

  5. 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 .

  6. 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?

  7. Eugene Wigner: « The unreasonable effectiveness of mathematics in the natural sciences » Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960)

  8. 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.

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

  10. 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.

  11. Knot theory in algebraic topology • Classification of knots, search of invariants • Surgical operations

  12. 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.

  13. 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

  14. Error Correcting Codes Data Transmission • Telephone • CD or DVD • Image transmission • Sending information through the Internet • Radio control of satellites

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

  16. Sphere packing The kissing number is 12

  17. Sphere Packing • Kepler Problem:maximal density of a packing of identical sphères: p / Ö 18=0.740 480 49… Conjectured in1611. Proved in1999 by Thomas Hales. • Connections with crystallography.

  18. 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

  19. Data transmission French-German war of 1870, siege of Paris Flying pigeons : first crusade - siege of Tyr, Sultan of Damascus

  20. Data transmission • James C. Maxwell (1831-1879) • Electromagnetism

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

  22. Data Transmission

  23. Language Theory • Alphabet - for instance {0,1} • Letters (or bits): 0 and 1 • Words (octets - example 0 1 0 1 0 1 0 0)

  24. ASCII American Standard Code for Information Interchange Letters octet A: 01000001 B: 01000010 … …

  25. Coding

  26. Cryptography

  27. Encryption for security

  28. Applications of cryptography • Credit cards • Web security • Imaging • Encrypted television, • Telecommunications

  29. Mathematics in cryptography • Algebra • Arithmetic, number theory • Geometry

  30. History Encryption using alphabetical transpositions and substitutions (Julius Caesar). 1586, Blaise de Vigenère (key: «table of Vigenère») 1850, Charles Babbage (frequency of occurrences of lettres)

  31. Interpretation of hieroglyphs • Jean-François Champollion (1790-1832) • Rosette stone (1799)

  32. Any secure encyphering method is supposed to be known by the ennemy The security of the system depends only on the choice of keys. Auguste Kerckhoffs «La  cryptographie militaire», Journal des sciences militaires, vol. IX, pp. 5–38, Janvier 1883, pp. 161–191, Février 1883 .

  33. 1917, Gilbert Vernam (disposable mask) Example:the red phone Kremlin/White House 1940, Claude Shannon proves that the only secure private key systems are those with a key at least as long as the message to be sent.

  34. Enigma

  35. Alan Turing Deciphering coded messages (Enigma) Computer science

  36. Colossus Max Newman, the first programmable electronic computer (Bletchley Park before 1945)

  37. Théorie de l’Information Claude Shannon A mathematical theory of communication Bell System Technical Journal, 1948.

  38. Claude E. Shannon, "Communication Theory of Secrecy Systems ", Bell System Technical Journal , vol.28-4, page 656--715, 1949. .

  39. DES: Data Encryption Standard In 1970, NBS (National Board of Standards) put out a call in the Federal Register for an encryption algorithm • with a high level of security which does not depend on the confidentiality of the algorithm but only on secret keys • using secret keys which are not too large • fast, strong, cheap • easy to implement DES was approved in 1978 by NBS

  40. Algorithm DES:combinations, substitutions and permutations between the text and the key • The text is split in blocks of 64 bits • The blocks are permuted • They are cut in two parts, right and left • Repetition 16 times of permutations and substitutions • One joins the left and right parts and performs the inverse permutations.

  41. Diffie-Hellman:cryptography with public key • W. Diffie and M.E. Hellman, New directions in cryptography, IEEE Transactions on Information Theory, 22 (1976), 644-654

  42. RSA (Rivest, Shamir, Adleman - 1978)

  43. R.L. Rivest, A. Shamir, and L.M. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Communications of the ACM (2) 21 (1978), 120-126

  44. Public key Easy to multiply two numbers, even if they are large. If you know only the product, it is difficult to find the two numbers.

  45. Example p=1113954325148827987925490175477024844070922844843 q=1917481702524504439375786268230862180696934189293 pq=2135987035920910082395022704999628797051095341826417406442524165008583957746445088405009430865999

  46. Quizz du malfaiteur Apprenez les maths pour devenir chef du Gang http://www.parodie.com/monetique/hacking.htm http://news.voila.fr/news/fr.misc.cryptologie

  47. Primality tests • Given an integer, decide whether it is the product of two smaller numbers or not. • 8051 is composite 8051=8397, 83 are 97 prime Today’s limit : more than 1000 digits

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