The following content has been provided by the University of Erlangen-Nürnberg.

So welcome everybody to the course on the geometric anatomy of theoretical physics.

And you being advanced students, you know that theoretical physics is all about casting our concepts about the real world

into rigorous mathematical form, for better or worse.

But theoretical physics doesn't do that for its own sake, but it does so in order to fully explore the implications

of what our concepts about the real world are.

So to a certain extent, the spirit of theoretical physics can be cast into the words of Wittgenstein,

who said what we cannot speak about clearly, that we must pass over in silence.

Well, so indeed, if we have concepts about the real world and it's not possible to cast them into precise mathematical form,

that usually is an indicator that some aspects of these concepts have not been well understood.

So theoretical physics aims at casting these concepts into mathematical language.

But then mathematics is just that, it's just a language.

And if we want to extract physical conclusions from this formulation, we must interpret the language.

And that's not the purpose or the task of mathematics, that's the task of physicists.

And that is where it gets difficult again.

But again, mathematics is just a language.

And to again use Wittgenstein, he says the theorems of mathematics all say the same, namely nothing.

And so what does he mean by that?

Well, obviously, he doesn't mean that mathematics is useless.

He just refers to the fact that if we have a theorem of the type A if and only if B, A and B being propositions,

then obviously B says nothing else than A does.

And A says nothing else than B does.

It's a tautology.

Now, of course, mathematically, logically, it's a tautology.

But psychologically or for understanding of A, it may be very useful to have a reformulation of A in terms of B.

So with this understanding that the mathematics just gives us a language for what we want to do,

the idea of this course is to provide proper language for theoretical physics and certainly for the four courses you have heard so far.

So the aims of the course, the aim, the overarching aim is to provide proper mathematical language

for classical mechanics, electromagnetism, quantum mechanics, and statistical physics.

Now, so these subjects you studied in the modules TP1 to TP4 here at Erlangen or at many other universities.

And obviously, we're not going to revise all the mathematics that are needed for these four subjects,

but rather we will develop mathematics from a higher point of view that we can now approach that you have some prior knowledge of these subjects.

And so far, you have certainly used a fair amount of analysis in order to study these subjects.

Further, a fair amount of algebra, mainly linear algebra, but also nonlinear algebra at times, and geometry in one or the other form.

So very often in classical mechanics, you would appeal to geometric intuition in setting up a problem or finding the solution to a problem.

The same goes for electromagnetism. Algebra, you mainly used in quantum mechanics.

That's the modern way to do quantum mechanics is by means of linear algebra.

And in statistical physics, you use some geometry, maybe without really realizing it by looking at convex cones of states and so on.

Now, if you take these four fields and you try to be pedagogic about it, you draw circles around them.

So classical mechanics and electrodynamics certainly use a fair amount of analysis and geometry,

and so does general relativity for those of you who took that course or about to take it.

Well, quantum mechanics lies certainly well in the intersection of analysis and algebra, usually called functional analysis.

And well, algebra and geometry, let's do statistical physics justice, it more or less lies there.

Of course, this is not a pure picture. This is mainly where the intersections of these fields come to play.

And here in the middle, where these three fields intersect, we have differential geometry as a mathematical subject.

It's differential geometry that this course is mainly about.

So the mathematics of this course will be that part of the mathematics of these different theoretical subjects where these three fields intersect.

That's the rough idea of this course.

Now, the structure of this course, and indeed the development of differential geometry that we're about to start, starts well with sets.

So we'll talk about set theory, because any space in classical mechanics, the space of configurations,