Articles – Konrad Voelkel's Blog – Mathemagically

Im vergangenen Semester hat die Fachschaft Mathematik Freiburg eine Evaluation von Tutoraten angeboten. Genau genommen haben wir den Tutoren angeboten, dass sie sich ausgedruckte Fragebögen in ihr Tutorat mitnehmen können, die sie dann ausgefüllt in einem Umschlag wieder zurückgeben. Wir haben das dann anonymisiert in ein Balkendiagramm verwandelt und den Tutoren per Email geschickt. So sollte jede/r in der Lage sein, den Tutoratsteilnehmern eine Feedback-Möglichkeit zu geben.

Da ich nun schon mehrmals darauf angesprochen wurde, möchte ich gerne den entsprechenden Fragebogen online stellen. Man kann so eine Evaluation relativ leicht selbst durchführen. Wenn man als Tutor die Antworten auszählt, so erkennt man noch an der Handschrift, wer was geschrieben hat, also sollte man das den Tutanden überlassen. Bei einer Gruppengröße von 10-20 Leuten geht das aber ganz fix und sollte kein Problem sein. Eine Computer-Auszählung ist eigentlich gar nicht so dringend nötig.

Hier gibt es das fertige PDF zum Ausdrucken

und hier gibt es den Quelltext als LaTeX-Dokument.

Das ist natürlich nur ein Vorschlag; ich bitte darum, das LaTeX-Dokument nach Gutdünken umzugestalten und so zu verwenden, wie man es für sinnvoll hält.

Wenn es jemanden interessiert, kann ich auch noch ein bisschen Arbeit darein stecken, unsere Erfassungsformulare (in HTML+CGI-Skript) sowie die Auswertungssoftware (Python+R) online zu stellen, aber das CGI-Skript sowie das R-Skript sind nicht von mir, daher dauert das ein bisschen. Man kann prinzipiell auch eine Erfassung am Rechner durchführen, dazu muss man im LaTeX-Dokument nur einen "Absenden"-Button hinzufügen und ein passendes CGI-Skript verwenden.
Eine sehr einfache Online-Lösung kann man sich auch mit Google Documents zusammenklicken (und sicherlich gibt es andere Anbieter mit ähnlichen Lösungen).

Zum Inhaltlichen: Wir hatten damals ein bisschen über den Fragebogen diskutiert und es gab verschiedene Meinungen. Im Großen und Ganzen wurden Freitext-Felder nicht so intensiv genutzt, wie man es sich vielleicht wünschen würde. Der Rest ist aber eher positiv angekommen. Um ehrliches Feedback zu bekommen, ist die Anonymität der ausfüllenden sehr wichtig. Es hilft auch, wenn die Teilnehmer wissen, dass sie am Ende profitieren, wenn man z.B. eine Evaluation nach 5 Wochen Tutorat durchführt, und noch einige Zeit vor sich hat. Man sollte nicht den Fehler machen, den Studenten zu viele Fragen zu stellen, denn dann bekommt man statt wohlüberlegte Antworten nur "passt schon" zurück (weniger ist mehr).

An manchen Universitäten gibt es bereits spezielle Evaluations-Büros (in Karlsruhe habe ich das gesehen), die sich generalstabsmäßig darum kümmern. Aber auch hierbei geht es oft nicht so sehr um einzelne Übungsgruppen, denn um ganze Massenveranstaltungen. Deshalb kann ich eine Evaluation solcher Kleingruppen-Übungen sehr empfehlen. Es hilft gerade Neulingen, die noch nicht so genau wissen, ob die Studenten wegen ihnen nichts kapieren, oder ob es an etwas anderem liegt

These days I read Akhil Mathew's post on Vaserstein's proof of the Quillen-Suslin theorem, once known as Serre's Problem. This inspired the following.

Serre asked whether algebraic vector bundles over affine space are all trivial or not. Quillen and Suslin proved independently that they are, in fact, all trivial. This is some kind of analogue to the topological situation, where all vector bundles over n-dimensional complex affine space (or even n-dimensional real affine space) are trivial.

In the topological case, one classical proof goes like this:
Denote by Ψ the functor of rank r vector bundles up to isomorphism on topological spaces. This functor is representable in the homotopy category, in particular Ψ(X) ≅ [X,Gr(r)], where Gr(r) is the infinite Grassmannian of r-planes in an arbitraily sized vector space and the parentheses [,] mean "homotopy classes of maps". To see this, embed any rank r vector bundle in a trivial one to get a map X->Gr(r), where different embeddings yield homotopic maps. Now observe that the pullback of the tautological bundle over Gr(r) along such a map gives a bundle isomorphic to the one you started with.

If X is contractible, i.e. homotopy equivalent to a point, [X,Gr(r)]=[pt,Gr(r)]={pt}, so there is nothing but one isomorphy-class of vector bundles on X, which must be the class of the trivial bundle. If X is affine, it is in particular contractible.

In the algebraic case, one might try to use this proof idea to proceed similarly. The first question arising here would be: which homotopy classes? Since the easiest example is affine space, one should choose a homotopy theory where affine space is contractible, which is the case for A¹-local simplicial homotopy theory, e.g. motivic homotopy theory.
The next question is now, whether we can represent the functor as in the topological setting.
This is certainly not the case: if G would be a classifying space (a representing object) for the functor Ψ, in the motivic homotopy theory, we'd have Ψ(X) ≅ [X,G], but the right-hand side is A¹-invariant by construction, while the left hand side is not! This means, [XxA¹,G]=[X,G] (along the projection morphism) but Ψ(XxA¹)≠Ψ(X) (this fails for example even in the case X=P¹).

One could stop here, given that there are nice proofs of the Quillen-Suslin theorem. But wait, Quillen-Suslin is only about affine space and we just tried to prove something about all schemes. Let's try to look at something weaker. What about (A¹-)contractible spaces?

There is a nice paper of Asok and Doran on vector bundles on contracible schemes which explains that there are lots of A¹-contractible schemes (over a field) with lots of vector bundles that are not trivial. These vector bundles are somewhat invisible, since they are indistinguishable by cohomology or K-theory: since the base is A¹-contractible and motivic cohomology as well as K-theory are representable in the A¹-homotopy category, cohomology and K-theory of these schemes are those of a point.

Well, it just seems the approach to Quillen-Suslin via A¹-homotopy theory is doomed. Now let me tell you that I think that is the case, but nevertheless, using the Quillen-Suslin theorem (and stronger results on the more general Bass-Quillen conjecture), Morel claims that the representability works like in the topological case, as long as we only look at affine schemes:

Let X be a smooth affine k-scheme, then Ψ(X) ≅ [X,Gr(r)].

For now, the proof didn't appear in any journal, but you can take a look at Morel's book-in-progess "A¹-algebraic topology over a field" here. The parts related to this discussion are mostly in section 7 and 8 and these sections were previously contained in an earlier paper draft called "A¹-homotopy classification of vector bundles over smooth affine schemes".

For some more info about the history and various approaches to Serre's problem I like to recommend Lam's excellent book "Serre's Problem on projective modules".

Here is my mind-map for first-course complex analysis. It contains some well-known theorems and "arrows" between them. There are some aspects which require an explanation:

An arrow (A --> B) roughly means "you could try to prove B by using A and maybe something else, although A might not be necessary to prove B". I tried to remove redundant arrows, so keep in mind you can prove most of the theorems from the Cauchy-Riemann differential equations and some calculus ...

Another hint on the interpretation of my mind-map:
"existence of anti-derivatives" in the context of the mind-map really means "holomorphic functions have anti-derivatives on every simply connected domain", and "homotopy-invariance of path integral" really means "the path integral over holomorphic functions is homotopy-invariant on every simply connected domain". While you can do complex analysis without homotopy theory (even without homology theory), I don't like that approach very much. At least by using homology, some statement are easier to state. With homotopy, they become easier to prove (assuming you are allowed to use algebraic topology).

I created the mind-map (on the occasion of my oral exams) to keep it as a reminder and to sort out which statements could be generalized to harmonic maps or even to analytic maps (that was fun!).

Here it is, and of course you can download it as a PDF or as a SVG (vector graphics) as well (click on the image to enlarge it):

The license is CC-BY-NC-SA (if you redistribute, put my name on it, don't make profit, share alike).

An incomplete list of theorems contained:

• Goursat's Lemma
• Cauchy's integral formula
• Liouville's Theorem
• Maximum Modulus Principle
• Open Mapping Theorem
• Identity Theorem
• Riemann's Removable Singularity Theorem
• Casorati-Weierstraß
• Residue Theorem
• Fundamental Theorem of Algebra
• Montel's Theorem
• Schwarz's Lemma
• Riemann Mapping Theorem

It's a nice exercise to try to prove these on your own.

Enjoy learning, repeating or staring at confusing arrows!

To prepare for my oral exams in algebraic geometry (covering Hartshorne's book "Algebraic Geometry" Chapter II and III) I sketched an overview diagram of morphism properties in the category of noetherian schemes. Maybe this is a good cheat sheet to keep with you while reading the book for the first or second time (ok, and I dropped a "Nisnevich" for no good reason, you can ignore it).

You can get a PDF version of the image or click on it to get a readable version.

I'm still in the process of writing down examples and counter-examples to these properties, maybe that list will be online some day (another kind of "counterexamples in algebraic geometry").

As always, I'm happy to hear any comments (did I miss an important arrow, did I get anything wrong) -- but I should stress that the diagram works in Hartshorne-world, not in EGA-terms (this kind of confusion cost me almost one entire day trying to prove wrong statements..)

UPDATE (2011-11-18): improved diagram (more information, less colour) and higher quality PNG file.