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