THE ART OF RESEARCH 2005 Herzberg Lecture M. Ram Murty, FRSC Queen’s Research Chair Queen’s University

1. THE ART OF RESEARCH 2005 Herzberg Lecture M. Ram Murty, FRSC Queen’s Research Chair Queen’s University

2. What is research? • The art of research is really the art of asking questions. • In our search for understanding, the SOCRATIC method of questioning is the way.

3. QUESTION Socrates taught Plato that all ideas must be examined and fundamental questions must be asked for proper understanding.

4. Some basic questions seem to defy simple answers. • One can enquire into the nature of understanding itself. • But then, this would take us into philosophy.

5. What is 2 + 2 ? • The engineer takes out a calculator and finds the answer is 3.999. • The physicist runs an experiment and finds the answer is between 3.8 and 4.2. • The mathematician says he doesn’t know but can show that the answer exists. • The philosopher asks for the meaning of the question. • The accountant closes all doors and windows of the room and asks everyone, ‘What would you like the answer to be?’

6. Some Famous Questions • What is life? • What is time? • What is space? • What is light? • What is a number? • What is a knot?

7. The Eight-fold Way How to ask `good questions’? A good question is one that leads to new discoveries. We will present eight methods of generating `good questions’.

8. 1. SURVEY • The survey method consists of two steps. • The first is to gather facts. • The second is to organize them. • Arrangement of ideas leads to understanding. • What is missing is also revealed.

9. The Periodic Table • Dimitri Mendeleev organized the existing knowledge of the elements and was surprised to find a periodicity in the properties of the elements.

10. In the process of writing a student text in chemistry, Mendeleev decided to gather all the facts then known about the elements and organize them according to atomic weight.

11. The periodic table now sits as the presiding deity in all chemistry labs.

12. David Hilbert organized 23 problems at the ICM in 1900.

13. Hilbert Problems The 7th problem led to the development of transcendental number theory The 8th problem is the Riemann hypothesis. The 9th problem led to the development of reciprocity laws. The 10th problem led to the development of logic and diophantine set theory. The 11th problem led to the arithmetic theory of quadratic forms. The 12th problem led to class field theory.

14. Who wants to be a millionaire? • The Clay Mathematical Institute is offering $1 million (U.S.) for the solution of any of the following seven problems. • P=NP • The Riemann Hypothesis • The Birch and Swinnerton-Dyer conjecture • The Poincare conjecture • The Hodge Conjecture • Navier-Stokes equations • Yang-Mills Theory • www.claymath.org

15. 2. OBSERVATIONS • Careful observations lead to patterns and patterns lead to the question why?

16. The Michelson-Morley experiment showed that there was no need to postulate a medium for the transmission of light.

18. Archimedes

19. Archimedes and his bath • Archimedes goes to take a bath and notices water is displaced in proportion to his weight!

20. He was so happy with his discovery that he forgot he was taking a bath!!

22. 3. CONJECTURES • Careful observations lead to well-posed conjectures. • A conjecture acts like an inspiring muse. • Let us consider Fermat’s Last `Theorem.’

23. In 1637, Pierre de Fermat conjectured the following. Fermat’s Last Theorem

24. Fermat’s marginal note • Fermat was reading Bachet’s translation of the work of Diophantus. • 9+ 16 = 25 • + 144 = 169 • 64 + 225 = 289 • … …. . . He wrote his famous marginal note: To split a cube into a sum of two cubes or a fourth power into a sum of two fourth powers and in general an n-th power as a sum of two n-th powers is impossible. I have a truly marvellous proof of this but this margin is too narrow to contain it.

25. Srinivasa Ramanujan

26. Ramanujan was not averse to making extensive calculations on his slate.

27. Ramanujan made the following conjectures. • t is multiplicative: t( mn )=t(m)t (n) whenever m and n are coprime. • t satisfies a second order recurrence relation for prime powers. • |t(p)|< p11/2 • These are called the Ramanujan conjectures formulated by him in 1916 and finally resolved in 1974 by Pierre Deligne.

28. 4. RE-INTERPRETATION • This method tries to examine what is known from a new vantage point. • An excellent example is given by gravitation.

29. Newton’s theory of gravitation was inspired by Kepler’s careful observations.

30. Isaac Newton • Gravity is a force. • F=Gm1m2/r2

31. Albert Einstein • Gravity is curvature of space.

32. Gravity as curvature

33. Light and gravitational field

34. Bending of light due to gravity

35. Perihelion of Mercury

36. Black Holes • In 1938, Chandrasekhar predicted the existence of black holes as a consequence of relativity theory.

38. What is re-interpretation?

39. Unique Factorization Theorem • Every natural number can be written as a product of prime numbers uniquely. • For example, 12 = 2 X 2 X 3 etc.

40. Unique Factorization Revisited Euler

41. The Riemann Zeta Function

42. 5. ANALOGY • When two theories are analogous, we try to see if ideas in one theory have analogous counterparts in the other theory.

43. Zeta Function Analogies

44. The Langlands Program E. Hecke • This analogy signalled a new beginning in the theory of L-functions and representation theory. Harish-Chandra R. P. Langlands

45. The Doppler Effect • When a train approaches you the sound waves get compressed.

46. Police Radar • The police use the doppler effect to record speeding cars.

48. 6. TRANSFER • The idea here is to transfer an idea from one area of research to another. • A good example is given by the use of the doppler effect in weather prediction.

49. This is essentially the method of generalization. A simple example is given by the following observations. 13+23 = 9 = 32 13 +23+33 = 36 = 62 A general pattern? 13 + 23 + … + n3 = [n(n+1)/2]2 7. INDUCTION

50. GL(1): Riemann zeta function. GL(2): Ramanujan zeta function. Building on these two levels, Langlands formulated the general theory for GL(n). The Theory of L-functions