Mathreader's Blog

I heard the name Coxeter before, guessing somewhat vaguely that he was a geometer contributing to group theory as well. And of course I knew the book on regular polytopes. But I did not know anything about him. Since last September when I started my new postdoc, the name has been mentioned every day, but I was still ignorant about him. One day I wondered into the library, just for browsing (enjoying the existence of the bookshelves and printed books before they disappear) and I found this: Siobhan Roberts King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry.

This is one of the best mathematical biographies, if not the best one. It was a page turner for me. Well, I’m interested in the topic as my current research is related to his works. I have nostalgic feel about Cambridge. I like Escher’s drawings. His last conference trip was to Budapest in 2002.  But apart from the fact that I am obviously biased, this book conveys a decent amount of mathematical knowledge (e.g. regular polyhedra, good introduction to Coxeter diagrams, etc). There is also an attempt to explain what a group is. Obviously I cannot judge how successful that is.

You can also learn about the whole Bourbaki phenomenon, and finally understand that why is it so that about the same topic one book is a horrible read while the other one is pure entertainment.

The endnotes are overwhelming, almost a fifth of the book. This shows that there was a great deal of research behind this book, but also hints that there could be a lighter edition of this book. Of course, the bibliography, appendices and the publication list is essential.

The description of Coxeter’s life is so detailed that sometimes I felt like a voyeur, seeing too much of his private life. I don’t know, maybe that makes the whole book such a good read.

In sum: I never was that much interested in geometry, but after reading this I would like to read his books on Geometry.

As lecture notes I had to write the following short survey paper. This is basically a collection of book reviews. Books from the last two decades that somehow provide new perspectives on the ancient questions like What is Mathematics? Created or discovered? etc.

What is Mathematics? – a Short Introduction to the Contemporary Thinking about Math

This book tells the story of the symmetry groups from the beginnings up to the suspectedly finished classification of finite simple groups and stops at the surprising (to say the least)  and unexplained connections between group theory, number theory and physics. One of the finest pieces of living math.

BTW, in my opinion, the existence of Monster and Moonshine settles the Math as Discovery vs. Creation debate.

I read this book twice. First, some 3 years ago. I think I did that read somewhat superficially, concentrating more on the persons involved and their stories, but not on the mathematical side. For the second time, I read it more slowly constantly checking up the mathematical notions/theorems appearing in the story. This needs some extra work as an annotated bibliography is missing. Apart from that this is an excellent popular math book. The author is an insider, and he could find the proper balance between the amount of math and story (clearly not enough math for a mathematician but the difficult notions are carefully simplified for the ones outside group theory/math).

Author’s page

I found the following quote from John von Neumann in a book as a motto (I don’t remember which one, maybe David Joyner’s Adventures in Group Theory, but need to check):

Young man, in mathematics you don’t understand things. You just get used to them.

I liked this very much. My girlfriend (she’s a math teacher and also doing a PhD on teaching math) liked it too. So we were happy to see that quote…

…Then few months later the quote came up again in a discussion, and I realized that we had two different interpretations of it. Her take:

• This is how (rather badly) math is taught nowadays. We teach mechanistic algorithms and the pupils do not need to know why do the recipes work as long as they can carry out the calculation, they get good marks.

My take:

• As you dig deeper into math or go higher into the thin air of extreme abstraction (choose your favourite metaphor) the mathematical objects are no longer directly grounded in our everyday 3D physical bodily experience. You can understand number 3, just 3 sheeps or 3 fingers, easy, but how about generalised reciprocities in number theory? No matter how abstract the objects are, if you meet them regularly, they become good friends and you don’t care about their ontological status, they become part of your world.

So who is right? What did the old master mean?

Little investigation reveals that it was said to a young physicist who complained when having difficulties understanding a mathematical method… Still, both explanations seem possible.

Here is a shot of the math library corner of the boat where I live. Some of these books will be reviewed here in near future, but not only these. This selection is clearly biased towards group theory, algebraic geometry, computation. These I need for my work.

Strange book. It is somewhere between a popular science book and a textbook. Maybe the best use is for the  not number-theorist mathematicians to understand what is going on there.

It is an easy read, at least for the first 2 parts. I understood the new concept quickly and the structures that I did know beforehand are also explained very nicely and quickly. Sometimes their strategy is that say something firm, easy to understand/remember then to admit that it is a lie, the situation is a bit more complicated. It seems to be a useful technique.

So what is achieved in the first 2 parts? Group theory with permutation and matrix representations, elliptic functions, Galois groups, characters. Then it all comes together in reciprocity laws, using Galois representations for understanding the set of solutions of equations… but at this point, in Part 3 it becomes quite steep. New concepts are coming in a pace that full understanding is not possible any more.  Understanding Wiles’ proof after reading a thin book is not feasible under normal circumstances. Most probably  that is not the aim here, it is rather wetting the appetite for learning more, and that is done well.

An index of symbols would be very useful. The authors keep referring back to previous definitions, reflect on notations, so the omission of this is quite surprising.

The text contains some philosophical reflection on math at some random points, and these are quite good, though I was quite surprised to hear that some people think that mathematics is finished. I never met those people.

The preface for the paperback edition is great fun. It is a pity that that sense of humour is not pursued in the main text.

publisher’s page

This is not just a book to read and forget, but will be the base for my research for the following years. I’m lucky enough to know the authors and with my colleagues we had a reading club where we had been reading the final drafts. We did not get far in that but it was quite rewarding. This is first class cutting edge math (with a list of open problems).

Now I have a real copy of the book in my hands, so real reading can finally begin. As a trained philosopher I start with the preface and the introduction only.  First it gives the history of semigroup theory. This may not be the full story but a well rounded, coherent one. One can argue that you don’t need history in math. False. You need the right amount, just to give the perspective, the framework to understand theorems and their applications. I’ve been working in (applied semigroup theory for the last 7 years, I knew some results, but I was lacking the vision of the field before reading this introduction.

Not surprisingly the intro summarizes the book’s main thesis (more on this later), and you also get insights how mathematics is done plus you get a good laugh (Les Monts d’Auvergne).

Clearly I am biased I believe (and working on it) that semigroup theory will have important applications in other branches of science (e.g. biology).

I’ll write many times about this book as reading goes…

Publisher’s page:

The q-theory of Finite Semigroups

A popular science book on the Riemann conjecture. Not the only one. This one is a good read for anyone interested, maybe a bit too much psychology (I mean the recurring case of Louis de Branges), but a very accurate picture of the mindset of today’s practising mathematicians.

So far not enough reason for a review, but this book has a feature which is extraordinary. Six little chapters at the end of the book, called toolkits, written for those who want to understand the math as well, not just the story. Starting with exponents and logarithms ending with eigenvalues. The aim is to explain the main mathematical ideas with minimum terminology/overhead/formalism. It’s like ‘Hamlet in 2 paragraphs’, ’1page War and Peace Abridged’. A toolkit is like the moment when the pupils suddenly see the light when the teacher, after an hour long explanation, in despair tries to rephrase the whole idea in a snappy metaphor.

I see an opportunity here, for math-toolkit-literature. The recipe is this, write a serious math book. Then ask a journalist to listen to you in cafes for a few afternoon, while you  try to explain the content of the book. Then ask him to write a few pages long summary. If the journalist is on the level of Kral Sabbagh, then you can include the toolkit into your book, to make it a bestseller in academic bookshops.

Karl Sabbagh

There is quite some trouble in trying to classify the genre of this book. The closest match would be the ‘popular math’ category, but it violates the basic rule of popular science books: it is full with equations. Actually there are pages containing only math. Diving into Fourier-series without any hesitation. Is it a calculus book then? No, it is also about emotions, friendship, love. Not even a psychological one.

It is about how math interacts with life on every level, both personal matters and applied math. Continuity versus sudden changes, infinity versus limits. Yes, it is more like a philosophical treatise on the very fundamental ideas of calculus. Philosophical treatise, not in the dry sense, but more like Plato’s dialogues. Though again, not quite.  By the way, thumbs up for promoting the idea of real physical mail correspondence. I do not mean that it is the future, just saying that it had some values that email communication certainly does not have. Think about the dedication and time needed to sit down and write a letter.

As you can see, this book is rather odd, but nevertheless it is a great book. People say that it can be read with ignoring the mathematical bits, I certainly doubt that.

The book itself, as a physical entity, is a gem. Very nice binding, hardcover, high quality yellowish shade paper.

Strongly recommended! If you don’t happen to know the ‘Monk and the Mountain Riddle’, you will have a good laugh at the simplest solution.

Well, this is my research area: finite transformation semigroups. An introduction on this topic was long time due. We have a few introductory books on semigroup theory, and obviously transformation semigroups are mentioned but not as the main focus. This is surprising knowing the fact transformations have the same role in semigroup theory as permutations in group theory.

Motivations to read it:

1. The topic is so fundamental, that being a semigroupist this is a must check.
2. Though the field is young, it is already quite wide, so there is chance that we missed a paper somewhere containing some trick which could save us a month’s work in research.

The book itself is a very nice Springer hardcover (what a privilege to have a book like this!).

The usual Latex typeset with figures done by XYpic (that’s my guess, pretty sure though).

I don’t know the authors, never seen any publications before, so my expectations are really based on what is on the cover. So there are high expectations for this book, so I’ll start. We’ll see…

Book info:

Classical Finite Transformation Semigroups, An Introduction

Series: Algebra and Applications , Vol. 9 Ganyushkin, Olexandr, Mazorchuk, Volodymyr

2008, Approx. 340 p. 4 illus., Hardcover, ISBN: 978-1-84800-280-7.