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

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slide1

THE ART OF RESEARCH

2005 Herzberg Lecture

M. Ram Murty, FRSC

Queen’s Research Chair

Queen’s University

what is research
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.
slide3
QUESTION

Socrates taught Plato that all ideas

must be examined and fundamental

questions must be asked for proper

understanding.

slide4
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.
what is 2 2
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?’
some famous questions
Some Famous Questions
  • What is life?
  • What is time?
  • What is space?
  • What is light?
  • What is a number?
  • What is a knot?
the eight fold way
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’.

1 survey
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.
the periodic table
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.
slide10
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.
hilbert problems
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.

who wants to be a millionaire
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
2 observations
2. OBSERVATIONS
  • Careful observations lead to patterns and patterns lead to the question why?
slide16
The Michelson-Morley experiment showed that there was no need to postulate a medium for the transmission of light.
archimedes and his bath
Archimedes and his bath
  • Archimedes goes to take a bath and notices water is displaced in proportion to his weight!
3 conjectures
3. CONJECTURES
  • Careful observations lead to well-posed conjectures.
  • A conjecture acts like an inspiring muse.
  • Let us consider Fermat’s Last `Theorem.’
fermat s marginal note
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.

ramanujan made the following conjectures
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.
4 re interpretation
4. RE-INTERPRETATION
  • This method tries to examine what is known from a new vantage point.
  • An excellent example is given by gravitation.
isaac newton
Isaac Newton
  • Gravity is a force.
  • F=Gm1m2/r2
albert einstein
Albert Einstein
  • Gravity is curvature of space.
black holes
Black Holes
  • In 1938, Chandrasekhar predicted the existence of black holes as a consequence of relativity theory.
unique factorization theorem
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.
5 analogy
5. ANALOGY
  • When two theories are analogous, we try to see if ideas in one theory have analogous counterparts in the other theory.
the langlands program
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

the doppler effect
The Doppler Effect
  • When a train approaches you the sound waves get compressed.
police radar
Police Radar
  • The police use the doppler effect to record speeding cars.
6 transfer
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.
7 induction
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
the theory of l functions
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
8 converse
8. CONVERSE
  • Whenever A implies B we may ask if B implies A.
  • This is called the converse question.
  • A good example occurs in physics.
electromagnetism
Electromagnetism
  • An electric current creates a magnetic field.
  • One may ask if the converse is true.
  • Does a magnetic field create an electric current?
converse theory
Converse Theory
  • 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.
  • The question of whether all such objects arise from automorphic representations is called converse theory.
  • Langlands proved a 2-dimensional reciprocity law.
new directions
Feynman diagrams

Knot theory

Zeta functions

Multiple zeta values

NUMBER THEORY AND PHYSICS

New Directions
summary
SUMMARY
  • SURVEY
  • OBSERVATIONS
  • CONJECTURES
  • RE-INTERPRETATION
  • ANALOGY
  • TRANSFER
  • INDUCTION
  • CONVERSE