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My friend here in Goettingen, Daniel Pape, has told me about his very short paper in which he proves an interesting lemma about Von Neumann dimension. AFAIU it was earlier proved by Gabor Elek in a bit less general context. The lemma concerns discrete groups and it has immediate interesting applications for amenable groups. Read the rest of this entry »

This is a first part of notes from a lecture given by Alex Hoffnung on 4.02.2009 in Goettingen. You can learn from it what (do I think) a categorification and decategorification is, and see various examples of it. Next part will deal with categorification of Hecke algebras, which was the main point of Alex’ talk. However, it’s not coming until I teach myself about how to get an algebraic group out of a Dynkin diagram, because otherwise it’s “monkey see, monkey do”. Also, thanks to Peter Arndt here in Goettingen, for clarifying me a point about pullbacks of groupoids. Read the rest of this entry »

Due to a bonfire I’m unable to write anything coherent. Also, I’m quite dissapointed by my yesterday post (I spent ~2 hours on it and described only ≤ half of one lecture (out of ~4)). So I’ll only list the topics I wanted to blog about: Read the rest of this entry »

First thing about GGT I want to remark is very positive: it’s straightforward; one doesn’t need to read heavy books with definitions to acquaint with actual problems. After this first day I have a feeling that GGT (or rather GGT represented by people on this specific school/conference) is very close to what Lieven Le Bruyn described here as a healthy branch of mathematics.

Whole day was full of lectures: apart from coffe breakes there was only one long (2 hours) break for a lunch. First lecture was at 9:00 (which is way to early for me, I tried to sleep on almost every coffe break) and last question session was at 19:20. Actually, it’s a bit to much for me, as some lecturers are handing out some exercises and there is virtually no time to solve them. But to much is better then to little :-).

There is parallel conference taking place here on the topic of differential equations, so there’s some additional fun from talking with all these guys who know how to solve PDEs and stuff. God I wish I had this easiness with deltas and epsilons. :-)

Here are some notes from today lectures. Read the rest of this entry »

My exams are over – I start my summer holidays. First spot is Topics in Geometric Group Thoery School/Conference. I thought I’m not really into GGT, but while preparing for this event I changed my mind a little bit. However, I came here mostly because people who came here are great if one wants to talk about mathematics (of any sort).

Today we just came to Bedlewo Math Conference Center, all the real stuff is starting tomorrow (it’s after midnight here, so actually today), but we had some time to drink few beers and talk about mathematics. I learned about at least two very cool things.

As a bonus, read about my very very weird mathematical dream :-).

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I wrote an introductory note about elliptic curves and modular functions. At the begining it was supposed to be just an introduction to an article about very cool proof of Siegel. However, in the meantime I broke my finger (quite severely, bone is in many pieces…) and it’s still hard for me to use keyboard.

Accordingly, first few paragraphs aren’t very relevant, as they describe my admiration of Siegel’s proof.

I use Unicode for mathematical notation. Enjoy :-)

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I got back from Belgium, from Ulrike Tillmann lectures. I’m quite glad I went there, I’m still going trough my notes and type them (in polish, it’s far easier for me, because I have ~15 pages of raw notes; I’ll make however an english review of what happened in few days) into a computer.

In this article I’ll be putting links to polish notes from consecutive lectures as I complete writing them.

They are a little bit chaotic, but I’ll answer any questions with pleasure. Also, I’ll put here any comments that I’ll receive by mail from my friends.

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I continue to read Sheldon Katz’s book “Enumerative Geometry and String Theory”. Here’s question I posted on sci.physics. If you have any answer, comment or ANYTHING, please (rather: I BEG YOU, CRYING ON THE FLOOR!!!) share them. It’s really hard to find someone who is mathematician and who’s interested enough in physics to answer even such simple questions… In my department there is only one more guy I didn’t yet ask (about what follows or some other physical questions) , but I don’t trust him – he was a physicist in his youth. Read the rest of this entry »

So I’ve passed my homological algebra exam. Today morning, I unexpectedly realized that I’m going to have a mathematics exam (yes… with the great pain I admit that HA is mathematics :-) and I actually don’t know any proof. This is because HA is so tedious – normally, I feel very uncomfortable if I don’t check at least some details by myself, prove few things by myself, etc. This time, I didn’t check any details. I wanted to several times, but looking at them (I was using a book by Gelfand & Manin) almost always discouraged me.

It’s not that there was nothing that interested me in itself – for example, I really wanted to know why when one passes from category of chains over abelian category to category localized in quasiisomorphisms (see below) then one gets all homotopic morphisms identified. In aforementioned book there are few definitions and diagrams -“nothing is happening”, but I don’t get illuminated even after longer moment of looking at them.

Correspondingly, my knowledge of derived/triangulated categories, simplicial objects and derived functors of nonadditive functors (these were the main topics of a course) is a little bit like a knowledge of history – I know essential facts, I even see some causal connections between them, but I miss something – a spirit of mathematics?.

I’d love to see some comments on how you are/were learning homological algebra. Especially, I’d love to know how (for example) Terrence Tao learned HA (as everybody probably knows, there is non-zero chance of meeting him around wordpress.com, so who knows :-). My General Theory of Geniuses(;-) predicts that he (or maybe He?:-) spent as much time as I did (or less, of course) on the given topic, but his unconsciousness checked all the details and communicated them to his consciousness, one day or another (actually, my GTG says that my unconsiousness also checked all the details, but it saved them for itself, damn bastard.)

Below follows a survey of what I’ve learned on this HA course, part I: derived categories, with some additional thoughts on the subject. I believe you may find it worth reading if you already know some basics of HA (nothing more than classical derived functors) and you’d like to read some informal introduction, to have some view on a matter before studying it for real. Comments are appreciated. Read the rest of this entry »

Alon Levy points that equation I’m interested in,

x2+7=2n,

is called Ramanujan equation. (You must know who Ramanujan was.) He also gives a full solution (which I haven’t verified yet) and a link to alternative solution. I’ve checked in Alan Baker’s Concise introduction to the theory of numbers that indeed it was one of Ramanujan’s many conjectures that above equation has solutions only for n=3,4,5,7 and 15. It was proved by T. Nagell in 1948. (Precise references are given here.)

However, I still have an impression that something funny is going on. Read the rest of this entry »