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Good morning ladies and gentlemen. So as you can see from the program, I'll give two lectures

today and the overall title is representations of league groups and the subtitle is, well,

four lectures aimed at physicists. And my background is not physics but mathematics

but I've always admired physics and the many interesting problems that come out of physics.

Now we do have an extra hour at the end of the day so there we can discuss as much as we want.

Some of the questions that you may have in my lectures, maybe look at some of the exercises

I put on the overview of the list of contents of my lectures and then I have my last lectures

on Thursday and Friday. Now I have to start rather slow and so I apologize to those people

who know already a lot of representation theory and group theory but I'll start slow so we have

a common background, a common notation basically to establish that. Now let me just start by

recalling that in 1905 Einstein introduced the special theory of relativity. So he introduced

special relativity in the sense of giving us the relevant symmetries of space-time.

So we can think of this as the study of the special relativity symmetry group of space-time.

So this is where he introduced the symmetries of space-time.

Then in 1913, exactly 100 years ago, Niels Bohr introduced his atomic model at that time

which was the first attempt at giving a model that could begin to predict and explain some of the

experiments that one saw in say the emission of energy quanta from atoms. So here we had the idea

of spectrum. In fact he could explain the size of the so-called, for example the Bellmore lines

in the hydrogen spectrum could be explained and so there was the idea of spectrum.

And also in the air there was the idea of symmetry. And my lectures actually aim at going a little

bit deeper into these two concepts. The concept of symmetry, which is going to be the representation

theory from a mathematical point of view, and the idea of spectrum which in some sense has two

meanings. It has the meaning of the spectrum that we observe in spectral analysis and spectral

experiments but also has a well-defined mathematical notion. It's a well-defined mathematical notion

that I'll also try to explain in these lectures. So it's in the spectrum that we see the quantum

effects coming up and somehow the aim of the lectures is to marry these two concepts, the concept

of symmetry and group and the concept of spectrum and see how these two in the mathematical theory

that I'll explain interact. I should also say that of course there are some famous lectures by

Felix Klein, the so-called Erlangen Lectures where he, oh also around a hundred years ago,

was proposing the point of view that study a geometry via its symmetries. So again, symmetries.

So he had in mind what is Euclidean geometry. Well this is just studying the objects and the

quantities that are invariant under the Euclidean group of motion. What is spherical geometry?

That's studying the quantities that are invariant under the rotation group of the sphere. Same for

hyperbolic geometry. And so he very much liked this point of view that to study a geometry is the

same as studying its group of symmetries. And we can steal this idea and say study anything

via its symmetries. And this anything, this could be a physical system, could be a mass or a particle

or it could be an electron, it could be some mesons or baryons. And the idea would be to study these

by first understanding very well the symmetries involved and how they manifest themselves in the

description of the system in the model of the system. And actually the same idea goes through in

mathematics. Every time you see a mathematical object you want to study its symmetries. And what's

a symmetry? Well a symmetry can be thought of as transformations preserving the essential structure.

So of course it depends on what is essential to you. So up here in special relativity the essential

thing is what happens when you change the observer from one observer to another, say another one

that's moving in a straight line with a fixed velocity relative to the first one. What happens to

your system that you're looking at? That's what special relativity is doing for you. And so that

in that case would be the symmetries. What are the symmetries of say a particle in a force field?

Well if you have symmetries that preserve the energy, that preserve the Hamiltonian, that would be

a good thing to think about as a symmetry. And I'll be precise about exactly what symmetries we

work on and which will also help us in understanding how we can actually get to representation theory