Today, Jacek Brodzki was giving a lecture on K-theory of C*-algebras (as a part of a course on KK-theory). I wonder whether I’m not wasting my time attending this course (lectures are alright, but I don’t understand much), but at least I was able to hear this:

(Before proving Bott periodicity theorem) “Unfortunately, Raoul Bott died last year. However, his results are…” – There was a pause, I coudn’t resist and said – “still correct”.

Also on the same lecture:

(While proving Bott Periodicity Theorem): “This is the key insight, which can be traced back to Atiyah’s original proof for topological K-theory: a*b => a*b” Someone from the public: “Very deep indeed. You mean a*b => b*a?” “Well, if you insist. It will take us much more work, but hell with that. Yeah, a*b => b*a”.

This comes from xkcd comic which I’m now going through (because – and I want to emphasize causality – I should learn for Hodge Theory exam). Olek Zablocki, one of cleverer mathematics students I personally know (here’s how clever he is globally), told me once that he thinks it’d be cool to be a carpenter. This topic is very broad, and I haven’t time right now to blog (I have to read whole xkcd archive learn for my exams.)

It’s more than a month since I wrote last post and unfortunately it seems that it’ll take some time ’till I’ll be able to blog again. For last couple of weeks I was back in my home, in Szczecin, and I was supposed to be learning hard. I did not, and accordinly I have to learn hard now (of course, I don’t. I’m f****** wasting my time but I just cannot focus on things when I am forced to).

I already passed PDE exam, now I’m trying to focus on “Complex manifold topology and Hodge theory” and there is still plenty to do about Homological algebra II…….

link: Scott Aaronson wrote a superb article on quantum factoring algorithm.

Oh yes, and the onions… I like them very much, I add them to much of what I eat and accordingly if I’m going to buy myself some onions I don’t buy 3 onions but 5kg so that I have a supply for two months. I put them in refrigerator and it never happens that these onions rot or anything like that.

And yesterday, after I opened my refrigerator for the first time in about 4 weeks (I was back in Szczecin for this long) I discovered that 3 of my precious onions produced green shoots! In refrigerator. For 1.5 month. I was stunned! Almost touched! I just can’t eat them now – this great force of life makes me shed a tear. I moved them out of refrigerator onto windowsill and on monday I’ll buy them some soil and a flowerpot.

Alon Levy points that equation I’m interested in,


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 »

(I don’t think that being educated dummy is something bad – in fact, I consider myself to be one).

Around month ago, I gave here, at Warsaw University, a very introductory talk about Gromov-Witten invariants. It was aimed at students who hadn’t heard about this topic but who knew standard things from university course: cohomology, vector bundles, Chern classes, Poincare duality, etc. There were also some local wise men and they didn’t say I totally screw it up.

It was based on first 9 chapters of a Sheldon Katz’s fantastic book: “Enumerative geometry and string theory” . This is suberbly written, it starts at the level of highschool students and it ends… well I don’t actually know where it ends, because I still have some chapters to read, but it surely introduces Gromov-Witten invariants.

Accordingly, the book has actually one disadvantage: Katz tries to explain really all, even things like what topology is. So, in case You know at least very roughly things I mentioned above, You may want to read notes I prepared.

Everything is in “algebraic geoemtry framework” – so that it’s meaningful to speak about for example cubics in P2. When I write Pn, I mean CPn. I use unicode for mathematical notation.

Any kind of feedback is welcomed!

BTW, I just found out that Sheldon is actually a name, not surname. When I was writing the notes, I didn’t check it and so I write “Sheldon’s book” instead of “Katz’s book”. Sorry for that.

BTW2, There’s a new edition of a book “J-holomorphic curves and symplectic topology” by Dusa McDuff and Dietmar Salamon. It has about 5 000 000 pages (I know what I’ve seen. However, Amazon says it has 669). First edition had 200 and I could at least dream of understanding it someday. Life’s so damn cruel.

BTW3, WOW!!! Maxim Kontsevich comes to Warsaw next week!!!! Believe it or not, but in the last post I chose Kontsevich as a random Fields medalist! :-).

Read the rest of this entry »

The way I see it: there’s a kind of a soul in human body that wants to study hard, learn complex theories, understand why Maxim Kontsevich got a Fields Medal, etc. And there is also resistance of matter – world around that don’t want me to do all these things. Resistance of matter has its agents everywhere.

Read the rest of this entry »

Recently, I’ve been wondering about natural solutions of equation

x2 + 7 = 2n

While I’m still working on it, let’s find all natural solutions of similar equation

x2 + 1 = 2n

using the same methods of simple algebraic number theory as before. This time my post is rather short.

Read the rest of this entry »

Today my teacher told me about a following math problem: find all natural numbers n and x such that equality

x2 + 7 = 2n

holds (I will call this equality (1)). Everyone who competed in various mathematical olimpiads have seen lots of similar problems. Why this one is particularly interesting? My teacher was told about it by one of his highschool pupils who just came back from a “Zwardon math camp” (link in polish) – a place where best-in-mathematics polish highschool students come to train before International Mathematical Olympiad. All this very clever guys tried to work out the solution but they failed.

So, it might be just that there is no “elementary” solution. However, my teacher told me about what he came up with using some simple algebraic number theory (though, there is no solution yet – I hope somebody reading this can provide me with complete solution, this or another way).

Read the rest of this entry »

As I wrote earlier, I’m reading “The road to reality” (R2R) by Roger Penrose (RP). I want to share some thoughts about what I have already read:
1. The greatness of Euclid (and also Gauss, to some extent)
2. Uses of negative numbers in physics
Read the rest of this entry »

Few days ago I started to read “The road to reality” by sir Roger Penrose. I didn’t get far yet, but I like one thing about it very much: it’s definetely more than popular science (example: precise definitions of connection and curvature) but it is not academic textbook. The purpose seems to be
giving ideas of contemporary physics and mathematics in as precise as possible_without_pain way. I will write more soon.