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So my two talks, this will be a continuation of Pavel Shvera lecture series on group-valued moment maps,

or actually also on moment maps and moduli of flat connections.

Well, actually I would like to take a somewhat different angle from what you heard in Pavel's talks.

But to make a point of contact, let me recall the basic definition of the Hamiltonian g-spaces that you saw in the Pavel's talks.

So let's recall at least one definition.

So a Hamiltonian g-space, well, it's a triple, a two-form on M, and a map mu from M to the dual space of the Lie algebra of the group G.

Such that the following conditions are satisfied.

I guess by now you know those conditions better than I do, but let me still reproduce them here.

So M omega is a symplectic manifold.

So then there is a moment map condition, it is saying that the contraction of omega with a fundamental vector field

generated by the G action is precisely the exterior differential of this function, the pairing of mu with x,

where x are elements of the Lie algebra.

And finally, the map mu is equivalent for the quadrant action.

So that's basically the definition where we start, and maybe one word about this definition.

It certainly, well, at first maybe you don't see it, but it's a very beautiful and very, very successful definition.

In the following sense, it has many beautiful or miraculous or many surprising consequences.

Some of them you already seen in this course, but let me just make for you a short list,

not exhaustive list, of what are the miracles or what are the beauties that come from this definition.

So one thing that you already seen in this course and also which was mentioned in the Tudor's course is the theory of reduction.

Right, starting from Hamiltonian G-spaces, from various versions of Hamiltonian G-spaces,

you can go and reduce, simplify various problems and mechanics,

and also go and construct very interesting examples of symplectic manifolds using the theory of symplectic quotients,

or using the theory in quasi-Humiltonian quotients in the group-valued moment map theory.

But then this is just one of several very, very interesting things which come out from this definition.

So another big story is the theory of convexity.

Under various conditions or various assumptions on the group G,

there are exciting and surprising convexity properties on the image of the moment map.

This is perhaps would be a topic for the whole lecture series if you wanted to do it.

So another very interesting topic is localization.

So we'll see a lot of it today and tomorrow, so perhaps right now I don't give you a preview and I'll start on it in some minutes from now.

One more topic I wanted to mention, it's not a complete list, but just to say.

So this is so-called Q bracket with R equals zero.

So that's the famous quantization commutes with reduction principle.

So this perhaps makes the closest or one of the closest contacts to physics or to the quantization theory.

Even though here the quantization is usually meant in terms, well, it depends in terms of the index theory.

So maybe it's not the most directly how physicists would view quantization, but at least that's a good approximation.

So these two things, those four things and many others, they come from this definition.

And all of them are kind of small or big miracles.

Everything here is very, very beautiful.

That's perhaps one interesting feature of the moment maps theory.

Maybe from my personal experience, I started my career in physics, but then I was attracted on the mathematics side.

And one of the big attractions, maybe the main attraction was the moment maps theory.

It's so, so beautiful. You cannot resist it.

Now, in this lecture in the lecture series of Pavel, you already seen quite a lot of reduction.

Of course, reduction is an enormous subject. There are many, many things.

But you have seen many things about reduction there.

So for those these last two lectures, I chose to speak about localization.

I'll give you some kind of introduction into the localization part of the story.

So that will be the topic of these two lectures.