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

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 (xexp(λx)) On T=R/Z: Fourier Series (t exp(2πnt)) On R and T: Fourier Inversion Formula Plancherel Formula Paley-Wiener Theorem

Modern Highlight 1: Harish-Chandra Plancherel 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 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

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