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Michi’s blog » archive for April, 2006

Reading Merkulov: Differential geometry for an algebraist (4 in a series)

  • April 24th, 2006
  • 2:52 pm

Suppose we have a presheaf \mathcal F of abelian groups over M and pick a point x. On the collection of all abelian groups defined over some neighbourhood of x (disjoint union) we put an equivalence relation which identifies f\in\mathcal F(U) and g\in\mathcal F(V) precisely if there is some open W in the intersection where f and g coincide. (or more precisely, their restrictions coincide). The set of equivalence classes turns out to be an Abelian group \mathcal F_x called the stalk of the presheaf \mathcal F at x.

So, with more fluff introduced, the stalk is all the elements in the presheaf that are defined above any neighbourhood of the point, and counted as the same if they seem to be.

For an open set U and a point x\in U there is a canonical group morphism \rho_x:\mathcal F(U)\to\mathcal F_x which sends an element f\in\mathcal F(U) to its equivalence class. This image is the germ of f at x.

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Weekly report: Settling in

  • April 22nd, 2006
  • 2:21 pm

My first week has passed. Today is saturday; and the move took place monday. So far, I’ve been running around doing bureaucracy and little else (I managed to leaf through the first 5 pages of Evens: Cohomology of Groups). Along the lines – I’ve received a summons to appear in front of the immigration authorities to explain my moving in, I’ve ran circles around the city trying to get someone to approve my swedish birth certificate et.c.

My apartment is small, neat and nice. It’s some 4×5 meters, with bed, bookcase, two tables, wardrobe, kitchenette, toilet with bath, balcony. And then all the things I brought with me – including a bookcase, three tables, computer, books-books-books, and much much more. I’ve gotten around to some interior decoration as well – putting up my swedish and my franconian flags on a wall. The endeffect is pretty – although I periodically have to remind myself that my putting up a swedish flag is no longer a sign of right-extremism but rather a sign of keeping in touch with home.

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Question for the mathematical audience

  • April 20th, 2006
  • 3:44 pm

I have now been staring at this particular sentence for way too long, and thus will start using any and all communication lines I can find to get assistance. Either I’m being way too stupid, or the author neglects to mention some salient detail.

Setup: \phi\colon G^\prime\to G is a group homomorphism, A\in kG-\operator{mod}, A^\prime\in kG^\prime-\operator{mod}. A can be given the structure of a kG^\prime-module by pulling back through \phi, i.e. we define g^\prime a:=\phi(g^\prime)a for g^\prime\in G^\prime and a\in A.

So far it’s all crystal clear for me. However, it then turns out that we’re highly interested in using a morphism f\in\operator{Hom}_G(A,A^\prime) and I cannot for the life of me find out how such beasts are guaranteed to exist. If it where f\in\operator{Hom}_{G^\prime}(A,A^\prime), I wouldn’t have any problems with it; but then the stuff I need/want to do with it don’t work out.

(133 words, 11 images)

Weekly posts – a new feature at this blog

  • April 15th, 2006
  • 7:27 pm

Now that my blog returns to its status of a PhDiary as I actually got a PhD position, I will introduce one flavour of regular postings. Instead of keeping in touch with people by mailing lists, livejournal, and everywhere else, there will be weekly postings here about life as a German PhD student.

So far, my entrance into German academics has had one feature above all else. Bureaucracy.

In order to even look at my contract, I needed to go, specifically therefore, to Jena, to fill out a questionnaire. This questionnaire is geared towards ascertaining that I am a good representant of the German state and its ideals. So, there are questions upon questions upon questions about my involvement with Stasi, my involvment with former DDR, whether I went to party schools, whether I’ve held party offices, et.c. et.c. Not to mention the centimeter-high stack of papers I got home to fill out on my own. With complete curriculum vitae from the age of 14. And Gods only know how many different obscure decisions to be made and forms to be filled in.

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Why I keep organizing congresses

  • April 5th, 2006
  • 9:07 am

Once upon a time, I wasn’t passionate about mathematics. Up to grade 6, I even disliked it quite a bit – it consisted of only mechanical plugging away of numbers, and training of multiplication tables that I had the feeling I already mastered.

Then something changed. Subtly at first – in grade 7, it started to gain texture, it got beyond the rote calculations ever so slightly. And so I started devouring the old popular mathematics texts my father kept in his bookcases. Soon, I stumbled across a new word – “integral calculus” – and of course asked my father to explain it. And thus it was that I, at the age of 13, got introduced to limits, derivatives and integrals.

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