Introduction

In 2015, a course (International Winter School on Gravity and Light), on the mathematical foundations of general relativity (manifolds, differential geometry, etc.) was held by Frederic P. Schuller (and others) at the Johannes Kepler University, Linz, Austria, 2015.

There are two main playlists on YouTube:

• Central lecture course;
• Tutorials.

And, somewhat complementary, some unofficial notes (.pdf) by Richie Dadhley.

Finally, it seems that a considerable amount of time was dedicated to workshops (10h over two weeks, “where more advanced problems are solved in teams”, by comparison with the tutorials, 12h, “to practice the basics”), but unfortunately, nothing has been found pertaining to their content¹.

Tutorial PDFs & missing tutorial videos

There were official websites for the course, which are as of today (December 2022) unavailable; two of them are archived to some degree on archive.org:

• www.gravity-and-light.org;
• gravity-and-light.herokuapp.com.

I’ve taken the liberty to import from the latter all available tutorial .pdf files, and three tutorial videos (7, 8, 9/10) missing in the official YouTube playlist. We’re still missing the 14th and 17th video tutorials, which don’t seem available anywhere. I obviously don’t own any of this material, and am merely reuploading them to increase their visibility (archive.org pages aren’t directly google-able as far as I can tell).

Here are the missing videos, and here’s a playlist containing all the available tutorials (14 and 17 are still missing then).

More courses by prof. Schuller

Frederic P. Schuller has two more courses on his YouTube channel:

• Lectures on Geometrical Anatomy of Theoretical Physics;
• Lectures on Quantum Theory.

The former seems to be a slower-paced version of the Heraeus course.

There’s a set of notes (cached .pdf) for the former, by written by Simon Rea. And there’s also one (cached .pdf) for the latter, originally written by Simon Rea, and later completed by Richie Dadhley.

Both contain elements of the course (e.g. some proofs) that have been relegated to the otherwise unavailable problem sheets.

Note: I’ve reworked the proof of the invertibility of the Fourier operator, in a hopefully clearer fashion (QM18). From the same lecture, here’s a proof for Parseval’s theorem.

Clarity

All three courses are remarkable, by comparison with similar material, for their mathematical precision (physicists can be mathematically “sloppy”), and for their clarity. By clarity, I mean that arguments are well-articulated (clear), and don’t assume too much from the students.

Bear in mind that clarity doesn’t mean easy: the subjects are technical, require a certain amount of work and background knowledge. Besides a (finite) linear algebra course, I guess a real analysis course (there are others; M. Penn’s courses tend to be well-articulated as well) is a great way to get a rigorous understanding of limits, convergence and related topics, as well as a feel for their generalization in more abstract settings (mainly metric/topological spaces).

Anyway, here’s a typical example of a well-articulated argument, from the lecture 4 of the quantum theory course (c. 43:17). Again, this assumes that you’re not completely oblivious to topological arguments, nor to “basic” mathematical reasoning².

We’ve defined the orthogonal complement \(M^\perp\) of \(M\) in \(\mathcal{H}\), where \(M\) is a (not necessarily closed) linear subspace of a (separable) Hilbert space \((\mathcal{H}, +, ., <\cdot,\cdot>)\), as:

\[\begin{aligned} M^\perp := \{ \psi\in\mathcal{H}\ |\ (\forall\mu\in M),\ <\mu,\psi> = 0 \} \end{aligned}\]

There’s a property that \(M^\perp\) is then a closed linear subspace (the closedness implies, by a previous result, completeness, and thus that \(M^\perp\) forms a sub-Hilbert space). In Mathematical Methods in Quantum Mechanics by Gerald Teschl, this result is concisely justified en-passant in section \(1.3\):

By continuity of the scalar product it follows that \(M^\perp\) is a closed linear subspace […].

That’s it.

On the other hand, F. Schuller will quickly explain why it’s a linear subspace in the first place (similar results have been demonstrated more thoroughly earlier, hence why he’s lighter on details here), and then follows to prove the closed property more thoroughly (I’ll be omitting details here actually, so refer to the video for more): first by observing that \(M^\perp\) is actually:

\[\begin{aligned} M^\perp = \bigcup_{\mu\in M} \text{preimg}_{\omega_\mu}(\{0_{\mathbb{C}}\}); \qquad \text{with }\quad \omega_\mu := \begin{pmatrix} \mathcal{H} & \rightarrow & \mathbb{C} \\ \psi & \mapsto & <\mu,\psi> \\ \end{pmatrix} \end{aligned}\]

Then by recalling some notions and results from topology, including that a closed set is, by definition, one which has an open complement, and that an arbitrary (i.e. potentially infinite) intersection of closed set is closed, from which we could conclude, were we able to prove for an arbitrary \(\mu\in M\), that \(preimg_{\omega_\mu}(\{0_{\mathbb{C}}\})\) is closed. Which is then equivalent to prove that its complement is open; said complement can be reformulated as:

\[\begin{aligned} \mathcal{H}\backslash \text{preimg}_{\omega_\mu}(\{0_{\mathbb{C}}\}) = \{ \psi\in\mathcal{H}\ |\ <\mu,\psi> \neq 0_{\mathbb{C}} \} = \{ \psi\in\mathcal{H}\ |\ <\mu,\psi> = \omega_\mu(\psi) \in \mathbb{C}\backslash\{0_\mathbb{C}\} \} = \text{preimg}_{\omega_\mu}(\mathbb{C}\backslash\{0_\mathbb{C}\}) \end{aligned}\]

But, single-element sets are closed with respect to the standard topology on \(\mathbb{C}\)³, so their complements are open. In particular, \(\{0_\mathbb{C}\}\) is closed, and thus \(\{\mathbb{C}\backslash\{0_\mathbb{C}\}\}\) is opened.

Finally, because \(\omega_\mu\) is but an application of the inner-product, by continuity of the inner-product (established/mentioned in an earlier lecture, and used many times already) \(\omega_\mu\) is continous, and thus (recall that topologically-wise, a function is continuous if the pre-images of open sets are open) \(\mathcal{H}\backslash preimg_{\omega_\mu}(\{0_{\mathbb{C}}\}) \) must be open, and thus \(preimg_{\omega_\mu}(\{0_{\mathbb{C}}\})\) is closed, implying that \(M^\perp\) is closed. \(\Box\)

Note: This kind of situation is common in upper mathematical material: courses are typically taught to advanced students, which are expected to be able to fill the blanks with little effort, allowing teachers to be more concise.

Note: Teschl’s version is not to be dismissed either: Schuller’s is a great introductory material, but Teschl’s conciseness gives you plenty of room for strengthening your understanding.

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