what use has a mathematician for symmetry?
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What use has a mathematician for symmetry?. Mogens Flensted-Jensen SEST Friday 2 December 2011. The general opinion about mathematicians. The general opinion about students among mathematicians. Main Theorem:. Main Theorem:. Mathematics is Modelling. - PowerPoint PPT PresentationTRANSCRIPT
What use has a mathematician for symmetry?
Mogens Flensted-Jensen
SEST
Friday 2 December 2011
The general opinion about mathematicians
The general opinion about studentsamong mathematicians
Main Theorem:
Main Theorem:
Mathematics is Modelling
But: Simple calculations can lead to complicated numbers
Mathematics is Teaching
Teaching of mathematics to non-mathematicians:
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
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
Symmetric Spaces
Mercer Oak, near Institute for Advanced Study
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
Modern Highlight 1: Harish-ChandraPlancherel Formula for G
• Discrete series• Asymptotic
expansions• Spherical functions
Key Paper:
Modern Highlight 2: Helgason Geometric Analysis on G/K
• Spherical functions and Paley-Wiener theorems
• Poisson transform: Helgason conjecture
Key paper:
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
My simple idea for the construction of the discrete spectrum for G/H (1980):
Mittag-Leffler Institute,Djursholm, Stockholm, Sweden 1970-71 and 1995
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
I did not talk much about symmetry and mathematics
Anyway
Thank You for your patience.
Mogens