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What use has a mathematician for symmetry?
Mogens Flensted-Jensen
SEST
Friday 2 December 2011
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The general opinion about mathematicians
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The general opinion about studentsamong mathematicians
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Main Theorem:
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Main Theorem:
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Mathematics is Modelling
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But: Simple calculations can lead to complicated numbers
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Mathematics is Teaching
Teaching of mathematics to non-mathematicians:
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Mathematics is Research
• Doing mathematical research is a kind of art:• You must understand (to a certain extend)
the known mathematical world (theory)• You must see some “interesting” unexplored
region• You must begin to explore such a region• You design or discover the right “map” of the
region (i.e. formulate a hypothesis) – This is the “art” part
• You must prove it rigorously – This is where you need craftsmanship and ingenuity
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In mathematics we talk about “beauty”
when the “art” of designing the “map” gives a result, which is
• Build on easy accessible concepts• Easy to conceive and understand the
structure and the content• Has not been understood before• Is difficult to prove rigorously
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Symmetric Spaces
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Mercer Oak, near Institute for Advanced Study
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My topic: Harmonic Analysis
The classical theory• On R:
– Fourier Integrals (xexp(λx))
• On T=R/Z:– Fourier Series (t exp(2πnt))
• On R and T:– Fourier Inversion Formula– Plancherel Formula
• On R:– Paley-Wiener Theorem
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Modern Highlight 1: Harish-ChandraPlancherel Formula for G
• Discrete series• Asymptotic
expansions• Spherical functions
Key Paper:
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Modern Highlight 2: Helgason Geometric Analysis on G/K
• Spherical functions and Paley-Wiener theorems
• Poisson transform: Helgason conjecture
Key paper:
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Symmetric Spacesinmathematical terms
A Symmetric Space is an affine manifold for which the geodesic reflection in any point is an affine isomorphism
U/K
G/K
G/H
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My simple idea for the construction of the discrete spectrum for G/H (1980):
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Mittag-Leffler Institute,Djursholm, Stockholm, Sweden 1970-71 and 1995
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Plancherel Formula for G/H
MLI November 1995
Henrik Schlichtkrull
And
Erik van den Ban
Paley-Wiener Theorem for G/H
MLI November 1995
Patrick Delorme
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I did not talk much about symmetry and mathematics
Anyway
Thank You for your patience.
Mogens