Windberg

So, another post? Yes. The name of this blog involves both my life and math and recently there has been more life and less math in my days, but I've just come back from a sweet math meeting and I want to share with you again.

Happy Nikolaustag! Better late than never!

So, how have I been? Decent enough. Things are getting better for me in Regensburg, I am attending my classes, doing my homework, occasionally preparing at home, the usual, I am even studying bits of German and I like it. I haven't got used to the morning temperatures in my classrooms, though, so usually after my first class I am frozen to the bone, however wrapped-up I might be. Afterwards I usually go back to my room to drink some hot tea and snuggle up under the blankets.

Overall, I like some of my classes, I am making progress in my studies and I try to study some of the books I have long longed to read. Silverman and Cox have been on my bedside table for some time and I even get round to opening the books and marvelling at the wondrous elliptic curves.

Anyhow, Windberg. The annual meeting of the Regensburg Graduiertenkolleg "Curvatures, Cycles and Cohomology". The people there were so nice and lectures fun and accessible and yesterday we had a funny Cohomology evening: a two-hour panoramic view of the various cohomology theories one might encounter in geometry. And it was awesome. Though, we might have enjoyed it more had it not been at 10 pm. 

Windberg Monastery. Sort of looks like Špilberk in Brno.

So, that's it for now. I could not stay in Windberg for longer because of other travel plans but I loved it there and I am feeling postiive now that more math will come and things will get even better for me!

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And now for the people who are interested in how the evening went: I have not yet decided how to look at cohomology as I am mostly home schooled in this aspect, but for me it is both about invariants of some geometric spaces (for one, the well-behaved cohomology theories are usually homotopy invariant), encompassing geometry of the space (like Chern classes) and obstructions to some constructions (I think of Ext and splitting of short exact sequences, though) in terms of abelian groups. But many smarter people have thought longer about this. So, what have I got out of the talks? (All the misunderstandings are mine, remember, 10 pm.)

So as a prelude we already saw some chain complexes and simplicial homology in the morning and in the evening we started off with de Rham cohomology. For that, you need a smooth manifold and smooth differential forms with the exterior derivative. Surprisingly enough, the cohomology groups do not depend on the complex structure of the manifold as they are the same as singular cohomology with real coefficients. 

Then we moved on to sheaf cohomology, more categorical approach but for me also more appealing: one studies the geometrical objects by means of the functions defined on them. And this approach is absolutely natural because usually objects come with a nicely defined notion of functions on them. And functions describe the world (Tom Garrity). It also paves the way for all the abstract algebraic geometry. But the naive intuition of working with functions works, which is sweet.

The following bit was about Čech cohomology. For that I believe you play with real manifolds and glue functions on open coverings of your space. That is also a very useful concept as so many things are easily defined locally but one would like to piece the information together to get a global thingy. 

The most difficult or advanced topic of the night was crystalline cohomology. Toposes and cycle maps were mentioned (Thank you, Bernard). Anyway, it all comes from the celebrated Weil conjectures (counting points of varieties over finite fields), which would follow from constructing a suitable cohomology theory for varieties over finite fields. To be even more vague and imprecise and silly, you need to lift your curve defined over a finite field to something defined over a field of characteristic zero. Apparently, one can do this locally and then attempt to glue and then I was lost.

We wrapped up the evening with a discussion of how to actually work with cohomology. If we have a class in cohomology, how to decide whether it vanishes or not. The nonvanishing can be done finding a pairing in which the class does not vanish. So we played with compactly supported de Rham cohomology and bounded de Rham cohomology and paired the groups with homology, integrating things and checking well-definedness with Stokes and using the axiom of choice to get some nice limit from a bounded sequence which supposedly amounts to chosing a free ultrafilter on the integers. 

All in all a sweet evening!

Morning Windberg on the day of the cohomology evening. The following morning it was foggy and bleak.