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Mathematics (and High-Tech Applications) for the New Millennium

Mathematics (and High-Tech Applications) for the New Millennium. PETER E. TRAPA Department of Mathematics University of Utah. E8. E8 was a worldwide media event in March, 2007:. E8. E8 was a worldwide media event in March, 2007:

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Mathematics (and High-Tech Applications) for the New Millennium

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  1. Mathematics (and High-Tech Applications) for the New Millennium PETER E. TRAPA Department of MathematicsUniversity of Utah

  2. E8 • E8 was a worldwide media event in March, 2007:

  3. E8 • E8 was a worldwide media event in March, 2007: • New York Times (front page), Le Monde, London Timee, Los Angeles Times, ….

  4. E8 • E8 was a worldwide media event in March, 2007: • New York Times (front page), Le Monde, London Timee, Los Angeles Times, …. • Nature, Science, Scientific American, Economist, ….

  5. E8 • E8 was a worldwide media event in March, 2007: • New York Times (front page), Le Monde, London Timee, Los Angeles Times, …. • Nature, Science, Scientific American, Economist, …. • Al Arabiya TV (Dubai), Good Morning America, Fox News,…

  6. E8 • E8 was a worldwide media event in March, 2007: • New York Times (front page), Le Monde, London Timee, Los Angeles Times, …. • Nature, Science, Scientific American, Economist, …. • Al Arabiya TV (Dubai), Good Morning America, Fox News,… • Yahoo news (top 5 news, top emailed news story for several days).

  7. E8 • E8 was a worldwide media event in March, 2007: • New York Times (front page), Le Monde, London Timee, Los Angeles Times, …. • Nature, Science, Scientific American, Economist, …. • Al Arabiya TV (Dubai), Good Morning America, Fox News,… • Yahoo news (top 5 news, top emailed news story for several days). • AP and other wire services.

  8. E8 • E8 was a worldwide media event in March, 2007: • New York Times (front page), Le Monde, London Timee, Los Angeles Times, …. • Nature, Science, Scientific American, Economist, …. • Al Arabiya TV (Dubai), Good Morning America, Fox News,… • Yahoo news (top 5 news, top emailed news story for several days). • AP and other wire services.

  9. E8 • E8 was a worldwide media event in March, 2007: • New York Times (front page), Le Monde, London Timee, Los Angeles Times, …. • Nature, Science, Scientific American, Economist, …. • Al Arabiya TV (Dubai), Good Morning America, Fox News,… • Yahoo news (top 5 news, top emailed news story for several days). • AP and other wire services. • Statement read on the floor of Congress by Rep. McNerney (California).

  10. Department of Mathematics at the U • Rough statistics: • 45 tenured or tenure-track faculty • 20 postdoctoral instructors • 80 graduate students

  11. Department of Mathematics at the U • Facilities: • new renovation • new computer labs • new tutoring center

  12. Department of Mathematics at the U • National Recognition: • Ranked 34th in recent U.S. News and World Report survey. • Top 12 among public institutions. • Very high levels of support from the National Science Foundation and the National Security Agency.

  13. Mathematics at the U Applied Mathematics Pure Mathematics

  14. Outline 1. A pure math example. (Number Theory) 2. An applied math example. (Symmetry) 3. A problem in the design of communication networks. 4. A unifying framework (Langlands Program). 5. Solution of network problem. (Applications) 6. E8 and future applications.

  15. A Pure Math Example

  16. A Pure Math Example 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 , 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54,55, 56, 57, 58 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, …

  17. Prime Numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 , 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, …

  18. Typical Pure Math Question On average, how are the prime numbers spaced?

  19. Typical Pure Math Question For example, can one estimate how many prime numbers are there between 1,000,000,000,000,000,000 and 2,000,000,000,000,000,000? Can one say how accurate this estimate is?

  20. Bernhard Riemann How are the prime numbers spaced? There is a very precise conjectural answer proposed about 150 years ago (The Riemann Hypothesis). But even today no one has any idea of how to prove it.

  21. How are the prime numbers spaced? The Riemann Hypothesis Bernhard Riemann This is such a fundamental problem that the Clay Mathematics Institute in Cambridge has offered a $1M prize for its solution. Jim Carlson Clay Institute

  22. Outline 1. A pure math example. (Number Theory) 2. An applied math example. (Symmetry) 3. A problem in the design of communication networks. 4. A unifying framework (The Langlands Program). 5. Solution of network problem. (Appliucations)

  23. Applied Math Example: The Vibrating String

  24. Applied Example: The Vibrating String • Joseph Fourier (around 1820): • The motion of the string can be suitably decomposed as a superposition of harmonics. • This idea is fundamental to all signal processing.

  25. Typical Applied Math Questions • In sending (or compressing) the information of a vibrating string, what are the “most important” harmonics to keep?

  26. Jared Tanner Warnock Professor Typical Applied Math Questions • In sending (or compressing) the information of a vibrating string, what are the “most important” harmonics to keep? • If some harmonics are lost how can the original signal be reconstructed?

  27. Relation with Symmetry The harmonics are exactly the oscillating functions that “wind” neatly around a circle:

  28. Relation with Symmetry Can ask the same question for other symmetric objects. For example: What are the “harmonics” that wrap neatly around a sphere? These “harmonics” are exactly what is needed to understand the hydrogen atom.

  29. Some Generalities Trickle-down: the pure mathematics of yesterday often becomes the primary tool of today’s applied mathematics. In contrast to other basic sciences, the time scale of this kind of trickle-down is relatively long.

  30. Some Generalities Trickle-down: the pure mathematics of yesterday often becomes the primary tools of today’s applied mathematics. Pure mathematical reasoning and analysis is portable. John Warnock Math BS ’62, MS ’64 EE PhD `69

  31. Jared Tanner Warnock Professor Some Generalities Trickle-down: the pure mathematics of yesterday often becomes the primary tools of today’s applied mathematics. Pure mathematical reasoning and analysis is portable. John Warnock Math BS ’62, MS ’64 EE PhD `69

  32. Outline 1. A pure math example. (Number Theory) 2. An applied math example. (Symmetry) 3. A problem in the design of communication networks. 4. A unifying framework (The Langlands Program). 5. Solution of network problem. (Appliucations)

  33. Networks • A network can be represented by a set of nodes • and a collection of edges connecting distinct • nodes.

  34. Networks • A network can be represented by a set of nodes • and a collection of edges connecting distinct • nodes.

  35. Example: High School Friendships

  36. Example: A Website

  37. Example: A Website

  38. Example: High School Dating

  39. The Expander Problem • Construct a network where: • Information spreads quickly • The number of edges is relatively small

  40. Fast Spread of Information (but too many edges)

  41. Few edges (but slow information transfer)

  42. The Expander Constant • Assign to each network N a constant e(N) • (the expander constant) that incorporates • the cost of edges and the benefit of fast • information transfer.

  43. The Expander Constant • Assign to each network N a constant e(N) (the expander constant) that incorporates the cost of edges and the benefit of fast information transfer. • With an appropriate definition, there is an • optimal expander constant. That is, e(N) • is bounded.

  44. The Expander Constant • Assign to each network N a constant e(N) (the expander constant) that incorporates the cost of edges and the benefit of fast information transfer. • With an appropriate definition, there is an optimal expander constant. That is, e(N) is bounded. • e(N)≤ 11/2

  45. Obvious Challenge • Design a family of networks with expander • coefficients as close as possible to the • optimal value 11/2. • Turns out to be very difficult.

  46. Outline 1. A pure math example. (Number Theory) 2. An applied math example. (Symmetry) 3. A problem in the design of communication networks. 4. A unifying framework (Langlands Program). 5. Solution of network problem. (Applications)

  47. The Unifying Vision of Langlands Symmetry (Harmonic Analysis) The Langlands Program (1960’s) Number Theory Robert Langlands

  48. The Unifying Vision of Langlands Symmetry (Harmonic Analysis) The Langlands Program (1960’s) Number Theory Robert Langlands

  49. Institute for Advanced Study

  50. Institute for Advanced Study

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