Transcript The origins of probability theory

S.R. Srinivasa Varadhan
• Born in Madras (now Chennai), India,
January 2, 1940.
• B.Sc. Presidency College, 1959
• Ph.D. Indian Institute of Statistics, 1963
• From 1963 at the Courant Institute of
Mathematical Sciences at New York
University
Scientific Work
• Varadhan is a probabilist, often working
on problems from mathematical physics
with tools from partial differential
equations
• Areas: Martingale problems and
diffusion theory, large particle systems,
hydrodynamical limits, random walks in
random media, quantum field theory....
Large Deviations
...... but the citation particularly
emphasizes his work on large
deviations
Probability theory
From games of chance to large
deviations
Games of chance
• Geronimo Cardano (1501-1576): “De ludo
alea” (published 1663)
• Blaise Pascal (1623-1662) and Pierre de
Fermat (1601-1656), letter correspondence
1654
• First textbook: Christian Huygens: “De
ratiociniis in Alea Ludo”, 1657
Limit laws
• The Law of Large Numbers: Jakob
Bernoulli (1654-1705), “Ars
Conjectandi”, published 1713
• The Central Limit Theorem: Abraham
De Moivre (1667-1754), “The Doctrine
of Chances” (2nd edition), 1738
The Law of Large Numbers
• Coin tossing:
The Central Limit Theorem
• Bell shaped curves - mean and variance
Large Deviations
• Harald Cramér (1893-1985)
Importance of Large
Deviations:
• Insurance companies: Probability of a
“bad year”
• Constructions: Impact on waves on oil
drilling platforms in the North Sea
• Networks: Probability that a network will
break down due to overload during peak
hours
Varadhan’s Contribution:
• Turned Large Deviation theory into an
extremely smooth, powerful and
efficient tool in many areas of
mathematics and related fields. His
Large Deviation Principle succinctly
sums up what is needed to apply the
technique.
Subtle effects
• Large deviation results are much more
subtle than classical limit laws — the
nature of each individual experiment
becomes important
Techniques
• Varadhan’s work is a tour de force
combining techniques from probability
theory, nonlinear analysis, partial
differential equations and functional
analysis.
Applications
• The Large Deviation Principle has
applications in statistics, insurance,
finance, statistical physics,
hydrodynamics, partial differential
equations.....
• And, of course, in many parts of
probability theory
Other contributions by
Varadhan:
• Martingale problems (with D.W.
Stroock)
• Hydrodynamical limits
• Random walks in random media
• Etc., etc.
A worthy winner
• Varadhan is very highly regarded in the
probability community, not only for his
scientific results, but also for his style.
He is friendly, accessible, but with high
standards. His emphasis is always on
ideas and general methods, rather than
technicalities.