Okay, passed the second language exam. Now I only have the Generals left to look forward to for the remainder of this semester.

Compared to the French exam, the German one was a lot less intensive. But I am still glad I studied. (I spent the entirety of yesterday going through the paper I was to read for the exam to make sure there aren't any surprises.)

My German, well, is not very good. On the grammar side it is virtually non-existent. Of the four aspects (listening, speaking, writing, and reading), I can only do the very last. All I have is some basic conjugation rules (simple present and present passive forms) and a lot of vocabulary obtained by rote memorisation. So, I was, of course, ecstatic when the first thing Professor Gunning said was "Don't worry too much about the exact grammar, it is the math that is important": what a load off my shoulders.

I took my exam from a short paper by Max Landsberg called "Über die Fixpunkte kompacter Abbildungen" (On the fixed-point of compact maps), published in Mathematische Annalen 154, 427-431 (1964). It is a mix of topology and functional analysis, and is only 4.5 pages long (one of the criteria I used in the selection of the paper is that it be short; another is that it is related to analysis). It was quite fun going through the paper: I learned something about fixed-point theorems in the setting of uniform spaces (generalizations of metric spaces) and topological vector spaces, as well as about the techniques of nets (aka. Moore-Smith sequences: a sequence indexed not by an ordered set, but by a directed set; something useful in topology).

The exam itself was less than 10 minutes long (I'd say only slightly more than 5). He asked me to translate Satz 1, Korollar 1 und 2, Definition 1 und 2. Then he asked for Satz 2 and the corresponding Beweis for a change. After I gave satisfactory translation of all those, he told me I passed. I was rather glad about having only the material from the first two pages: there are a few phrases in pages 3 and 4 that I didn't quite understand during my practice runs, which might just be mathematical terms that doesn't have a good cognate in English.

All in all, an educating and fun experience.

Regarding a phrase that is really troubling me, on page 3, in the proof of theorem 3, there is the sentence

p sei das Minkowski-Funktional von W. Es wird ρ(x)=inf(1/p(φx),1) für jedes x ∈ W mit p(φx) ≠ 0 und ρ(x) = 1 für jedes x ∈ W mit p(φx)=0 gesetzt. Dann ist ρ eine reelle stetige Funktion (auf W) mit leicht erkennbaren Eigenschaften und die durch ...