1. What use has a mathematician for symmetry?
Mogens Flensted-Jensen
SEST
Friday 2 December 2011

2. The general opinion about mathematicians

3. The general opinion about students among mathematicians

4. Main Theorem:

5. Main Theorem:

6. Mathematics is Modelling

7. But: Simple calculations can lead to complicated numbers

8. Mathematics is Teaching
Teaching of mathematics to non-mathematicians:

9. 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

10. 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

11. Symmetric Spaces

12. Mercer Oak, near Institute for Advanced Study

13. My topic: Harmonic Analysis
The classical theory
On R:
Fourier Integrals (xexp(λx))
On T=R/Z:
Fourier Series (t exp(2πnt))
On R and T:
Fourier Inversion Formula
Plancherel Formula
Paley-Wiener Theorem

14. Modern Highlight 1: Harish-Chandra Plancherel Formula for G
Discrete series
Asymptotic expansions
Spherical functions
Key Paper:

15. Modern Highlight 2: Helgason Geometric Analysis on G/K
Spherical functions and Paley-Wiener theorems
Poisson transform: Helgason conjecture
Key paper:

16. Symmetric Spaces in mathematical terms
U/K
G/K
A Symmetric Space is
an affine manifold for
which the geodesic
reflection in any point is
an affine isomorphism
G/H

17. My simple idea for the construction of
the discrete spectrum for G/H (1980):

18. Mittag-Leffler Institute,Djursholm, Stockholm, Sweden
and 1995

19. 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

20. I did not talk much about symmetry and mathematics Anyway
Thank You for your patience.
Mogens