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

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

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

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

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

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

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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’.

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

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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. Dimitri Mendeleev organized the existing knowledge of the elements and was surprised to find a periodicity in the properties of the elements.

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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. 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.

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

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

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

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

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

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

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Archimedes

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

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

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

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

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Fermat’s marginal note Fermat was reading Bachet’s translation of the work of Diophantus. Fermat was reading Bachet’s translation of the work of Diophantus. 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.

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Srinivasa Ramanujan

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

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Ramanujan made the following conjectures. is multiplicative: mn m) (n) whenever m and n are coprime. is multiplicative: mn m) (n) whenever m and n are coprime. satisfies a second order recurrence relation for prime powers. satisfies a second order recurrence relation for prime powers. p)|< p 11/2 p)|< p 11/2 These are called the Ramanujan conjectures formulated by him in 1916 and finally resolved in 1974 by Pierre Deligne. These are called the Ramanujan conjectures formulated by him in 1916 and finally resolved in 1974 by Pierre Deligne.

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

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

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Isaac Newton Gravity is a force. Gravity is a force. F=Gm 1 m 2 /r 2 F=Gm 1 m 2 /r 2

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Albert Einstein Gravity is curvature of space. Gravity is curvature of space.

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Gravity as curvature

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Light and gravitational field

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Bending of light due to gravity

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Perihelion of Mercury

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

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What is re-interpretation?

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

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Unique Factorization Revisited Euler

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The Riemann Zeta Function

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

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Zeta Function Analogies

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

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

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

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

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7. INDUCTION This is essentially the method of generalization. This is essentially the method of generalization. A simple example is given by the following observations. A simple example is given by the following observations. 1 3 +2 3 = 9 = 3 2 1 3 +2 3 = 9 = 3 2 1 3 +2 3 +3 3 = 36 = 6 2 1 3 +2 3 +3 3 = 36 = 6 2 A general pattern? A general pattern? 1 3 + 2 3 + … + n 3 = 1 3 + 2 3 + … + n 3 = [n(n+1)/2] 2 [n(n+1)/2] 2

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

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8. CONVERSE Whenever A implies B we may ask if B implies A. Whenever A implies B we may ask if B implies A. This is called the converse question. This is called the converse question. A good example occurs in physics. A good example occurs in physics.

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Electromagnetism An electric current creates a magnetic field. An electric current creates a magnetic field. One may ask if the converse is true. One may ask if the converse is true. Does a magnetic field create an electric current? Does a magnetic field create an electric current?

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Converse Theory We have seen that the Riemann zeta function and Ramanujan’s Delta series have similar properties. We have seen that the Riemann zeta function and Ramanujan’s Delta series have similar properties. We also learned that Langlands showed that these zeta functions arise from automorphic representations. We also learned that Langlands showed that these zeta functions arise from automorphic representations. The question of whether all such objects arise from automorphic representations is called converse theory. The question of whether all such objects arise from automorphic representations is called converse theory. Langlands proved a 2-dimensional reciprocity law. Langlands proved a 2-dimensional reciprocity law.

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New Directions Feynman diagrams Feynman diagrams Knot theory Knot theory Zeta functions Zeta functions Multiple zeta values Multiple zeta values NUMBER THEORY AND PHYSICS NUMBER THEORY AND PHYSICS

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SUMMARY SURVEY SURVEY OBSERVATIONS OBSERVATIONS CONJECTURES CONJECTURES RE-INTERPRETATION RE-INTERPRETATION ANALOGY ANALOGY TRANSFER TRANSFER INDUCTION INDUCTION CONVERSE CONVERSE

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