THE ART OF RESEARCH

2005 Herzberg Lecture

M. Ram Murty, FRSCQueen’s Research Chair

Queen’s University

What is research?What is research?

The art of research is really the art of The art of research is really the art of asking questions.asking questions.

In our search for understanding, the In our search for understanding, the SOCRATIC method of questioning is SOCRATIC method of questioning is the way.the way.

QUESTIONQUESTION

Socrates taught Plato that all ideasmust be examined and fundamental questions must be asked for properunderstanding.

Some basic questions seem to defy simple Some basic questions seem to defy simple answers.answers.

One can enquire into the nature of One can enquire into the nature of understanding itself. understanding itself.

But then, this would take us into philosophy.But then, this would take us into philosophy.

What is 2 + 2 ?What is 2 + 2 ?

The engineer takes out a calculator and finds the answer is The engineer takes out a calculator and finds the answer is 3.999.3.999.

The physicist runs an experiment and finds the answer is The physicist runs an experiment and finds the answer is between 3.8 and 4.2.between 3.8 and 4.2.

The mathematician says he doesn’t know but can show that The mathematician says he doesn’t know but can show that the answer exists.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 The accountant closes all doors and windows of the room and asks everyone, ‘What would you like the answer to and asks everyone, ‘What would you like the answer to be?’be?’

Some Famous QuestionsSome 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?

The Eight-fold WayThe Eight-fold Way

How to ask `good questions’?How to ask `good questions’?

A good question is one that leads to A good question is one that leads to new discoveries.new discoveries.

We will present eight methods of We will present eight methods of generating `good questions’. generating `good questions’.

1. SURVEY1. SURVEY

The survey method consists of two The survey method consists of two steps.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 Arrangement of ideas leads to understanding.understanding.

What is missing is also revealed.What is missing is also revealed.

The Periodic TableThe Periodic Table

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

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

The periodic table now sits as The periodic table now sits as the presiding deity in all the presiding deity in all

chemistry labs.chemistry labs.

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

Hilbert ProblemsHilbert Problems

The 7The 7thth problem led to the problem led to the development of development of transcendental number theorytranscendental number theory

The 8The 8th th problem is the Riemann problem is the Riemann hypothesis.hypothesis.

The 9The 9th th problem led to the problem led to the development of reciprocity development of reciprocity laws.laws.

The 10The 10thth problem led to the problem led to the development of logic and development of logic and diophantine set theory.diophantine set theory.

The 11The 11thth problem led to the problem led to the arithmetic theory of quadratic arithmetic theory of quadratic forms.forms.

The 12The 12thth problem led to class problem led to class field theory. field theory.

Who wants to be a Who wants to be a millionaire?millionaire?

The Clay Mathematical Institute is offering $1 The Clay Mathematical Institute is offering $1 million (U.S.) for the solution of any of the million (U.S.) for the solution of any of the following seven problems.following seven problems.

P=NPP=NP The Riemann HypothesisThe Riemann Hypothesis The Birch and Swinnerton-Dyer conjectureThe Birch and Swinnerton-Dyer conjecture The Poincare conjectureThe Poincare conjecture The Hodge ConjectureThe Hodge Conjecture Navier-Stokes equationsNavier-Stokes equations Yang-Mills TheoryYang-Mills Theory www.claymath.orgwww.claymath.org

2. OBSERVATIONS2. OBSERVATIONS

Careful Careful observations lead observations lead to patterns and to patterns and patterns lead to patterns lead to the question why?the question why?

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

ArchimedesArchimedes

Archimedes and his bathArchimedes and his bath

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

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

3. CONJECTURES3. CONJECTURES

Careful observations lead to well-Careful observations lead to well-posed conjectures.posed conjectures.

A conjecture acts like an inspiring A conjecture acts like an inspiring muse.muse.

Let us consider Fermat’s Last Let us consider Fermat’s Last `Theorem.’`Theorem.’

Fermat’s Last TheoremFermat’s Last Theorem In 1637, Pierre de In 1637, Pierre de

Fermat conjectured Fermat conjectured the following. the following.

Fermat’s marginal noteFermat’s marginal note

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

He wrote his famous marginal note:To split a cube into a sum of two cubesor a fourth power into a sum of two fourthpowers and in general an n-th power as a sumof two n-th powers is impossible. I have a truly marvellous proof of this but this margin is too narrow to contain it.

……

Srinivasa RamanujanSrinivasa Ramanujan

Ramanujan was not averse to Ramanujan was not averse to making extensive calculations making extensive calculations

on his slate.on his slate.

Ramanujan made the following Ramanujan made the following conjectures.conjectures.

is multiplicative: is multiplicative: mnmnm)m) (n) (n) whenever m and n are coprime.whenever m and n are coprime.

satisfies a second order recurrence satisfies a second order recurrence relation for prime powers.relation for prime powers.

p)|< pp)|< p11/211/2

These are called the Ramanujan These are called the Ramanujan conjectures formulated by him in 1916 and conjectures formulated by him in 1916 and finally resolved in 1974 by Pierre Deligne. finally resolved in 1974 by Pierre Deligne.

4. RE-INTERPRETATION4. RE-INTERPRETATION

This method tries to examine what is This method tries to examine what is known from a new vantage point.known from a new vantage point.

An excellent example is given by An excellent example is given by gravitation.gravitation.

Newton’s theory of gravitation Newton’s theory of gravitation was inspired by Kepler’s was inspired by Kepler’s

careful observations.careful observations.

Isaac NewtonIsaac Newton

Gravity is a force.Gravity is a force.

F=GmF=Gm11mm22/r/r22

Albert EinsteinAlbert Einstein

Gravity is Gravity is curvature of space.curvature of space.

Gravity as curvatureGravity as curvature

Light and gravitational fieldLight and gravitational field

Bending of light due to Bending of light due to gravitygravity

Perihelion of MercuryPerihelion of Mercury

Black HolesBlack Holes

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

What is re-interpretation?What is re-interpretation?

Unique Factorization Unique Factorization TheoremTheorem

Every natural number can be written Every natural number can be written as a product of prime numbers as a product of prime numbers uniquely.uniquely.

For example, 12 = 2 X 2 X 3 etc.For example, 12 = 2 X 2 X 3 etc.

Unique Factorization Unique Factorization RevisitedRevisited

Euler

The Riemann Zeta FunctionThe Riemann Zeta Function

5. ANALOGY5. ANALOGY

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

Zeta Function AnalogiesZeta Function Analogies

The Langlands ProgramThe Langlands Program

This analogy This analogy signalled a new signalled a new beginning in the beginning in the theory of L-theory of L-functions and functions and representation representation theory.theory.

E. Hecke

Harish-ChandraHarish-Chandra

R. P. LanglandsR. P. Langlands

The Doppler EffectThe Doppler Effect

When a train When a train approaches you approaches you the sound waves the sound waves get compressed.get compressed.

Police RadarPolice Radar

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

6. TRANSFER6. TRANSFER

The idea here is to The idea here is to transfer an idea transfer an idea from one area of from one area of research to another.research to another.

A good example is A good example is given by the use of given by the use of the doppler effect in the doppler effect in weather prediction.weather prediction.

7. INDUCTION7. INDUCTION

This is essentially This is essentially the method of the method of generalization. generalization.

A simple example A simple example is given by the is given by the following following observations.observations.

1133+2+23 3 = 9 = 3 = 9 = 322

113 3 +2+233+3+33 3 = 36 = 6= 36 = 622

A general pattern?A general pattern?

113 3 + 2+ 23 3 + … + n+ … + n3 3 ==

[n(n+1)/2][n(n+1)/2]22

The Theory of L-functionsThe Theory of L-functions

GL(1): Riemann zeta GL(1): Riemann zeta function.function.

GL(2): Ramanujan GL(2): Ramanujan zeta function.zeta function.

Building on these Building on these two levels, two levels, Langlands Langlands formulated the formulated the general theory for general theory for GL(n).GL(n).

8. CONVERSE8. CONVERSE

Whenever A implies B we may ask if Whenever A implies B we may ask if B implies A. 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.

ElectromagnetismElectromagnetism

An electric current An electric current creates a magnetic creates a magnetic field.field.

One may ask if the One may ask if the converse is true.converse is true.

Does a magnetic field Does a magnetic field create an electric create an electric current?current?

Converse TheoryConverse Theory

We have seen that the Riemann zeta function and We have seen that the Riemann zeta function and Ramanujan’s Delta series have similar properties.Ramanujan’s Delta series have similar properties.

We also learned that Langlands showed that these We also learned that Langlands showed that these zeta functions arise from automorphic zeta functions arise from automorphic representations.representations.

The question of whether all such objects arise from The question of whether all such objects arise from automorphic representations is called converse automorphic representations is called converse theory.theory.

Langlands proved a 2-dimensional reciprocity law.Langlands proved a 2-dimensional reciprocity law.

New DirectionsNew Directions

Feynman diagramsFeynman diagrams Knot theoryKnot theory Zeta functionsZeta functions Multiple zeta Multiple zeta

valuesvalues

NUMBER THEORY NUMBER THEORY AND PHYSICSAND PHYSICS

SUMMARYSUMMARY

SURVEYSURVEY OBSERVATIONSOBSERVATIONS CONJECTURESCONJECTURES RE-INTERPRETATIONRE-INTERPRETATION ANALOGYANALOGY TRANSFERTRANSFER INDUCTIONINDUCTION CONVERSECONVERSE