Every september I coorganize with my friends a “math camp”. The idea is as follows: just before summer holidays we choose a topic we want to learn, we choose a suitable book and everyone of us chooses ~1 of chapters. Then, during 3 months of swimming/skiing/hiking, we become “experts” on chosen chapters. Finally, in september, we meet in Wiselka – small town at the polish seaside (we self-made a big blackboard out of stiff black linen which is very easy to travel with) – and everyone explains their chapters to all the others.

This year we have a serious problem with choosing a topic/book – we have three very interesting propositions and choosing only one is very difficult.

I’ll write here briefly about our previous camps (in case somebody was interested in proved ideas for such a camp) and about propositions for this year’s camp (in case somebody wanted to comment on that, which would be appreciated).


First camp was devoted to John Milnor‘s “Topology from differential point of view“. I was a bit frustrated before this camp (it was after 1st year of studying math) – during a whole year I had been learning very tedious things: calculus, general topology, set theory, basic algebra etc. This camp, with this specific book, was exactly what I needed. This book is superbly written, there are practically no tedious part of it (yet it is strictly mathematical text, not a “popular mathematics”) and it finishes with results that were very impressing to me, despite my lack of mathematical sophistication.

Some main points of this book are Sard’s theorem (with very easy to understand proof), properties of a degree of a smooth map, Euler characteristic through vector fields and computation of some homotopy groups of spheres through cobordism theory. We had two “adults” with us and they supplemented us with some additional topics (like, say, Hopf fibration and different points of view on Euler characteristic). Generally I think It’s very good to have some “adults” on a camp when one isn’t mathematically sophisticated enought – I remember that I felt much more surely when they were there to correct any misunderstandings.

There were plenty of drawings on a blackboard, reasonings were nontrivial and intuitive at the same time – I was relieved that there indeed is mathematics that I strived for for one long year. Milnor became my personal hero (we were also told at this camp what did Milnor prove about 7-dimensional spheres and it seemed just awesome).


However, at the time I suspected that maybe it’s not really Milnor who is the best, but rather all good mathematicians, in particular all Fields medallists, write very nice-to-read books. Accordingly, on the next camp (it took place in february, because first camp was such a huge fun and we wanted to repeat it as soon as possible) we decided to give a chance another Fields medallist, Jean Pierre Serre, and his book “A course in Arithmetics“.

Well, in a way, this camp was also a success, for example because I met there Olek Zablocki, by now my good friend, whose speed in mathematical reasoning was at that time astonishing to me. Meeting him then was a very cool experience. Also, we did some crazy swimming in the see when the temperature was far below the zero, there was heavy wind blowing and snow was everywhere around. That was really cool. We made a bet about who will go further into the see and one who won was a girl! You wouldn’t guess this if you saw her…

However, Serre is not a good hero in this story. Serre is a bad guy. Say whatever you want (“Serre is a beauty lover (easthetics lover)” , for polish readers: “Serre to pięknoduch”, quote is by A.S. Bialynicki-Birula, my favourite lecturer in Warsaw), but I was so dissapointed by Serre. I suspect that it’s very different when one listens to his lectures, but his book is very hard to read for beginners. At any given time, I just didn’t know what we’re doing and, even worse, where we’re actually heading. No comments of any kind on why what we’re doing is important or interesting. Only this year I really have learned why some of these things may be absorbing.

I plan to read his book once more in a near future to see my reaction. Maybe now I’ll say that it’s beautiful and easthetically pleasing BUT don’t take this book as a book for a camp for if you’re not sophisticated enough (maybe it’s a good book for an event where “adults” do most of a talking, but it’s not my kind of a camp.)


Accordingly, following september we quite well knew what we’re going to do: Milnor’s “Morse theory“. Actually we’ve done only first two chapters: the one on finite dimensional Morse theory and the “Rapid course on differential geometry” one. Again: Milnor is simply the best. The book is quite much more sophisticated than his “Topology from…” but basic truths about it remain the same: superbly written, provides a reader with geometrical intuition, non-tedious in every aspect. Sorry, but I just so feel like writing it once more: Milnor is the best.

My friend, Jarek Kedra (who was an “adult” on most of our camps) wrote once a paper with Milnor’s wife, Dusa McDuff. Accordingly, he once went to Stony Brook (that’s where Milnors reside) to talk with her. When, just before, he told me about his plans, I asked him to collect Milnor’s autograph on some of his book or a t-shirt. Unfortunately, Jarek just laughted at me. Nevertheless, Jarek is also sort of the best, as he actually shook hands with Milnor. Wow. :-)

(In case Jarek is reading: you’re the best nevertheless, without you and two other “adults” our camps wouldn’t be the same :-)

(Yeah, definetely. If not you and Gal, who would go and swim naked in a sea? :-)


After that september I started my two-years period of studying in Warsaw (which I’m finishing just now) and so it was hard for me to organize a winter camp. For the following september I quite knew what I’m willing to do – read two other parts of Milnor’s “Morse theory”). In Warsaw I made friends with some great young people and almost all of them agreed to come to Wiselka. Accordingly, I think this last camp was the coolest one. During the year before the camp, most of us attended advanced differential geometry course (which I especially enjoyed, as I’ve already knew some of it from last camp) and “Morse theory” allowed us to see very interesting applications of it. Highlights of a book are, among the others, investigations of a topology of Lie groups (including a version of Bott periodicity), results on homotopy groups of spheres (inluding Freudenthal Suspension Theorem and some theorems relating curvature to topology (“if curvature is everywhere < 0 then the space is homeomorphic to affine space”)

True anecdote: on this camp there were also some younger people which decided to have a camp on different topic (they said that what we’re doing is to hard for them – I think they were wrong). They weren’t as focused on mathematics as we were (on the camp we talk mainly about mathematics, even on the beach). On one occasion one girl from a younger group told us something like: “I can’t understand you. Normal young people should talk about sex and you talk about mathematics.” :-)

(Maybe it’s because there’s only one girl among us :-)


Sideremark: one so feels that book by Milnor and Stasheff “Characteristic classes” and Milnor’s “Lectures on an h-cobordism theorem” are not really wholy written by Milnor (the latter is a lecture notes written by two of his students). They lack clarity of Master’s works.


So I come to the subject of this year’s camp. One proposition, quite natural perhaps in the light of above, is to learn a bit about singularities in topology. More specifically, Jarek Kedra’s proposition is to learn about fundamental group of a completion of a (perhaps singular) curve in complex projective plane (from Ichiro Shimada’s paper) (one reason it’s interesting is a theorem by Denis Auroux which states that every symplectic 4-manifold can be covered by complex projective plane and this (branched) covering can be more-or-less understood in terms of a set of singular points of this covering (which is a curve) and monodromy around this curve (which has to do with fundamental group). Shimada’s notes are a bit to little for a camp and we would add some chapters from Milnor’s “Singular points of complex hypersurfaces“) I don’t know this book but it’s Milnor so it must be good (and Jarek Kedra says these two papers could nicely complete each other).

The second proposition is more or less mine: to learn basics of TQFT (Topological Quantum Field Theory). I motivate it in two ways: first, it’s connected to physics, and physics is interesting. We could perhaps learn how it’s actually connected to physics (maybe ask somebody to give us one or two lectures on these connections which nobody of us understands at all?). Second, it’s connected to mathematics, apparently quite strongly: I was on Ulrike Tillman’s lectures devoted to the proof of Mumford Conjecture, and speaking freely in a language of various field theories was apparently a prerequisite (and TQFT is perhaps the simplest of field theories, so learning it seriously would be perhaps a good starting points for further studies). Our text would be probably Frank Quinn’s “Lectures on Axiomatic Quantum Field Theory” (from this book). Jarek Kedra said that he really likes this idea and proposed some further texts to choose from: V. Turaev’s “Quantum invariants of knots and 3-manifolds” and M.Atiyah’s “The geometry and physics of knots”.

(Field theories in mathematics were coinvented by Sir Michael Atiyah. The only mathematical book which I wholy read by myself is Atiyah/McDonald “Commutative Algebra”. It is almost as good as Milnor’s books.)


And finally, the Introduction to Noncommutative Geometry camp. Also sort of my proposition, but all agreed that it could be a good idea (NGC is strongly represented among Warsaw mathamaticians and we’re all “aware” of the existence of NCG)

However, my proposition at least partially is provoked by this NCG blog, on which some mystical statements are being told. And I like this sort of mysticism. I like the idea of investigating properties of space of Penrose tilings – I can’t see any method that would allow it, and Alain Connes claims that NCG makes it possible.

The problem is: none of us can say what it’s really about and it’s hard for us to estimate whether we’ll like it or not, based only on texts we don’t really understand. We could ask one of aforementioned Warsaw mathematicians to give us an advice, but, well, at least I don’t know how to talk with them – they simply know to much (Anecdote: “You want to learn Mathematics? Go ask Tomek Maszczyk what is 2+2″). NCG is a very vast subject and it’s hard for us to ask specific question. And asking nonspecific question provokes answer which fails to be understandable :-).

So, I wrote one of the authors of above blog, Masoud Khalkhali (actually, it’s his idea to write about a math camp on a blog):

Every september me and my friends organize a “math camp”. The idea is as follows: around june (i.e. now) we choose what we want to learn, we
choose a specific text (less than 200 pages long), everyone (~10 people) chooses a part of it and becomes “an expert” on it. Finally we meet and talk what we’ve learned.

Partly because of your blog (I put there few comments as sirix), we consider making sort of “Introduction to NCG camp”. However, we don’t know whether we could learn anything meaningful (because of the vastness of topics of NCG). We know more or less some homological algebra (first few chapters of Weibel, not everybody knows derived categories), algebraic topology (~Hatcher), differential geometry/topology (one of previous camps was on Milnor’s “Morse theory”, we learned also Milnor’s “Characteristic classes”), functional analysis (all the standard topics “up to” spectral theory of bounded self-adjoint operators). Most of us don’t know any physics.

For example, our other idea for this year’s camp is Frank Quinn’s “Lectures on axiomatic TQFT” (table of contents).

Could you provide us with an idea of what we could do on our camp if we were willing to learn some NCG?

Here’s what he answered me:

Dear Lukasz Grabowski,
Having an NCG camp is an exciting idea! Your background sure is enough to start off. I can think of three, or 4 things right now: A chapter
from Alain Connes’ 1994 book (available online on his website at http://www.alainconnes.org). The survey paper (joint with Marcolli) A walk in the noncommutative garden

I have also put two survey articles on NCG on the archive.

These might give you an idea and hope it is of any help. If you need more consulting , surely I will be more than happy to discuss things with
you. Also discussing such things in the blog is a good idea since I am sure will be beneficial to other users as well.
Masoud (khalkhali)

I looked on above papers, and I like especially one of them: “Very basic NCG”. I think it would be very good to teach us some language. But we have this rule that on every camp there must be some interesting and nontrivial theorem. So we’d perhaps have to take something from Alain Connes’ book. That’s sort of risky: for a layman like me Connes has one serious advantage: he’s, like Milnor, a Fields’ medallist; and he has one serious disadvantage: he’s, like Serre, French ;-). So the question is whether his book is more Milnorish or Serreish in flavour :-).

So, as you see, we have a serious problem here. All three are tempting but we can choose just one. Feel free to comment on that.

I’ll let you know about any progress in decision making process.

One more thing: If you’re by any chance a math student from, say, Germany or Czech Republic then we could perhaps make a “joint camp” this time or in future – Leave a comment or mail me if you like the idea (my mail is on “About” page).