# Stirring a cup of coffee

Please allow for a couple of end-of-semester bluesy ramblings. I just finished grading the final test of the last of five courses I lectured this semester.

Most of them went, I believe, rather well.

As always, it was fun to teach an introductory group theory course to second year physics students.

Personally, I did enjoy our Lie theory course the most, given for a mixed public of both mathematics and physics students. We did the spin-group $SU(2)$ and its connection with $SO_3(\mathbb{R})$ in gruesome detail, introduced the other classical groups, and proved complete reducibility of representations. The funnier part was applying this to the $U(1) \times SU(2) \times SU(3)$-representation of the standard model and its extension to the $SU(5)$ GUT.

Ok, but with a sad undertone, was the second year course on representations of finite groups. Sad, because it was the last time I’m allowed to teach it. My younger colleagues decided there’s no place for RT on the new curriculum.

Soit.

The final lecture is often an eye-opener, or at least, I hope it is/was.

Here’s the idea: someone whispers in your ear that there might be a simple group of order $60$. Armed with only the Sylow-theorems and what we did in this course we will determine all its conjugacy classes, its full character table, and finish proving that this mysterious group is none other than $A_5$.

Right now I’m just a tad disappointed only a handful of students came close to solving the same problem for order $168$ this afternoon.

Clearly, I gave them ample extra information: the group only has elements of order $1,2,3,4$ and $7$ and the centralizer of one order $2$ element is the dihedral group of order $8$. They had to determine the number of distinct irreducible representations, that is, the number of conjugacy classes. Try it yourself (Solution at the end of this post).

For months I felt completely deflated on Tuesday nights, for I had to teach the remaining two courses on that day.

There’s this first year Linear Algebra course. After teaching for over 30 years it was a first timer for me, and probably for the better. I guess 15 years ago I would have been arrogant enough to insist that the only way to teach linear algebra properly was to do representations of quivers…

Now, I realise that linear algebra is perhaps the only algebra course the majority of math-students will need in their further career, so it is best to tune its contents to the desires of the other colleagues: inproducts, determinants as volumes, Markov-processes and the like.

There are thousands of linear algebra textbooks, the one feature they all seem to lack is conciseness. What kept me going throughout this course was trying to come up with the shortest proofs ever for standard results. No doubt, next year the course will grow on me.

Then, there was a master course on algebraic geometry (which was supposed to be on scheme theory, moduli problems such as the classification of fat points (as in the car crash post, etale topology and the like) which had a bumpy start because class was less prepared on varieties and morphisms than I had hoped for.

Still, judging on the quality of the papers students are beginning to hand in (today I received one doing serious stuff with stacks) we managed to cover a lot of material in the end.

I’m determined to teach that first course on algebraic geometry myself next year.

Which brought me wondering about the ideal content of such a course.

Half a decade ago I wrote a couple of posts such as Mumford’s treasure map, Grothendieck’s functor of points, Manin’s geometric axis and the like, which are still quite readable.

In the functor of points-post I referred to a comment thread Algebraic geometry without prime ideals at the Secret Blogging Seminar.

As I had to oversee a test this afternoon, I printed out all comments (a full 29 pages!) and had a good time reading them. At the time I favoured the POV advocated by David Ben-Zvi and Jim Borger (functor of points instead of locally ringed schemes).

Clearly they are right, but then so was I when I thought the ‘right’ way to teach linear algebra was via quiver-representations…

We’ll see what I’ll try out next year.

You may have wondered about the title of this post. It’s derived from a paper Raf Bocklandt (of the Korteweg-de Vries Institute in Amsterdam) arXived some days ago: Reflections in a cup of coffee, which is an extended version of a Brouwer-lecture he gave. Raf has this to say about the Brouwer fixed-point theorem.

“The theorem is usually explained in worldly terms by looking at a cup of coffee. In this setting it states that no matter how you stir your cup, there will always be a point in the liquid that did not change position and if you try to move that part by further stirring you will inevitably move some other part back into its original position. Legend even has it that Brouwer came up with the idea while stirring in a real cup, but whether this is true we’ll never know. What is true however is that Brouwers refections on the topic had a profound impact on mathematics and would lead to lots of new developments in geometry.”

I wish you all a pleasant end of 2016 and a much better 2017.

As to the 168-solution: Sylow says there are 8 7-Sylows giving 48 elements of order 7. The centralizer of each of them must be $C_7$ (given the restriction on the order of elements) so two conjugacy classes of them. Similarly each conjugacy class of an order 3 element must contain 56 elements. There is one conjugacy class of an order 2 element having 21 elements (because the centralizer is $D_4$) giving also a conjugacy class of an order 4 element consisting of 42 elements. Together with the identity these add up to 168 so there are 6 irreducible representations.

# NaNoWriMo (3)

In 2001, Eugenia Cheng gave an interesting after-dinner talk Mathematics and Lego: the untold story. In it she compared math research to fooling around with lego. A quote:

“Lego: the universal toy. Enjoyed by people of all ages all over the place. The idea is simple and brilliant. Start with some basic blocks that can be joined together. Add creativity, imagination and a bit of ingenuity. Build anything.

Mathematics is exactly the same. We start with some basic building blocks and ways of joining them together. And then we use creativity, and, yes, imagination and certainly ingenuity, and try to build anything.”

She then goes on to explain category theory, higher dimensional topology, and the process of generalisation in mathematics, whole the time using lego as an analogy. But, she doesn’t get into the mathematics of lego, perhaps because the talk was aimed at students and researchers of all levels and all disciplines.

There are plenty of sites promoting lego in the teaching of elementary mathematics, here’s just one link-list-page: “27 Fantastic LEGO Math Learning Activities for All Ages”. I’m afraid ‘all ages’ here means: under 10…

Can one do better?

Everyone knows how to play with lego, which shapes you can build, and which shapes are simply impossible.

Can one tap into this subconscious geometric understanding to explain more advanced ideas such as symmetry, topological spaces, sheaves, categories, perhaps even topos theory… ?

Let’s continue our

[section_title text=”imaginary iterview”]

Question: What will be the opening scene of your book?

Alice posts a question on Lego-stackexchenge. She wants help to get hold of all imaginary lego shapes, including shapes impossible to construct in three-dimensional space, such as gluing two shapes over some internal common sub-shape, or Escher like constructions, and so on.

Question: And does she get help?

At first she only gets snide remarks, style: “brush off your French and wade through SGA4”.

Then, she’s advised to buy a large notebook and jot down whatever she can tell about shapes that one can construct.

If you think about this, you’ll soon figure out that you can only add new bricks along the upper or lower bricks of the shape. You may call these the boundary of the shape, and soon you’ll be doing topology, and forming coproducts.

These ‘legal’ lego shapes form what some of us would call a category, with a morphism from $A$ to $B$ for each different way one can embed shape $A$ into $B$.

Of course, one shouldn’t use this terminology, but rather speak of different instruction-manuals to get $B$ out of $A$ (the morphisms), stapling two sets of instructions together (the compositions), and the empty instruction-sheet (the identity morphism).

Question: But can one get to the essence of categorical results in this way?

Take Yoneda’s lemma. In the case of lego shapes it says that you know a shape once you know all morphisms into it from whatever shape.

For any coloured brick you’re given the number of ways this brick sits in that shape, so you know all the shape’s bricks. Then you may try for combination of two bricks, and so on. It sure looks like you’re going to be able to reconstruct the shape from all this info, but this quickly get rather messy.

But then, someone tells you the key argument in Yoneda’s proof: you only have to look for the shape to which the identity morphism is assigned. Bingo!

Question: Wasn’t your Alice interested in the ‘illegal’ or imaginary shapes?

Once you get to Yoneda, the rest follows routinely. You define presheaves on this category, figure out that you get a whole bunch of undesirable things, bring in Grothendieck topologies to be the policing agency weeding out that mess, and keep only the sheaves, which are exactly the desired imaginary shapes.

Question: Your book’s title is ‘Primes and other imaginary shapes’. How do you get from Lego shapes to prime numbers?

By the standard Gödelian trick: assign a prime number to each primitive coloured brick, and to a shape the product of the brick-primes.

That number is a sort of code of the shape. Shapes sharing the same code are made up from the same set of bricks.

Take the set of all strictly positive natural numbers partially ordered by divisibility, then this code is a functor from Lego shapes to numbers. If we extend this to imaginary shapes, we’ll rapidly end up at Connes’ arithmetic site, supernatural numbers, adeles and the recent realisation that the set of all prime numbers does have a geometric shape, but one with infinitely many dimensions.

Not sure yet how to include all of this, but hey, early days.

Question: So, shall we continue this interview at a later date?

No way, I’d better start writing.

# NaNoWriMo (2)

Two more days to go in the NaNoWriMo 2016 challenge. Alas, it was clear from the outset that I would fail, bad.

I didn’t have a sound battle plan. Hell, I didn’t even have a a clue which book to write…

But then, I may treat myself to a SloWriMo over the Christmas break.

For I’ve used this month to sketch the vaguest possible outlines of an imaginary book.

[section_title text=”An imaginary interview”]

Question: What is the title of your book?

I don’t know for sure, but my working title is Primes and other imaginary shapes.

Question: What will the cover-illustration look like?

At the moment I would settle for something like this:

Question: Does your book have an epigraph?

That’s an easy one. Whenever this works out, I’ll use for the opening quote:

[quote name=”David Spivak in ‘Presheaf, the cobbler'”]God willing, I will get through SGA 4 and Lurie’s book on Higher Topos Theory.
[/quote]

Question: Any particular reason?

Sure. That’s my ambition for this book, but perhaps I’ll save Lurie’s stuff for the sequel.

Question: As you know, Emily Riehl has a textbook out: Category Theory in Context. Here’s a recent tweet of hers:

Whence the question: does your book have a protagonist?

Well, I hope someone gave Emily the obvious reply: Yoneda! As you know, category theory is a whole bunch of definitions, resulting in one hell of a lemma.

But to your question, yes there’ll be a main character and her name is Alice.

I know, i know, an outrageous cliché, but at least I can guarantee there’ll be no surprise appearances of Bob.

These days, Alices don’t fall in rabbit holes, or crawl through looking-glasses. They just go online and encounter weird and wondrous creatures. I need her to be old enough to set up a Facebook and other social accounts.

My mental image of Alice is that of the archetypical STEM-girl

In her younger years she was a lot like Lewis Carroll’s Alice. In ten years time she’ll be a copy-cat Alice Butler, the heroine of Scarlett Thomas’ novel PopCo.

Question: What will be the opening scene of your book?

Alice will post a question on Lego-stackexchenge, and yes, to my surprise such a site really exists

(to be continued, perhaps)

# NaNoWriMo (1)

Some weeks ago I did register to be a participant of NaNoWriMo 2016. It’s a belated new-year’s resolution.

When PS (pseudonymous sister), always eager to fill a 10 second silence at family dinners, asked

(PS) And Lieven, what are your resolutions for 2016?

she didn’t really expect an answer (for decades my generic reply has been: “I’m not into that kinda nonsense”)

(Me) I want to write a bit …

stunned silence

(PS) … Oh … good … you mean for work, more papers perhaps?

(Me) Not really, I hope to write a book for a larger audience.

(PS) Really? … Ok … fine … (appropriate silence) … Now, POB (pseudonymous other brother), what are your plans for 2016?

If you don’t know what NaNoWriMo is all about: the idea is to write a “book” (more like a ‘shitty first draft’ of half a book) consisting of 50.000 words in November.

We’re 5 days into the challenge, and I haven’t written a single word…

Part of the problem is that I’m in the French mountains, and believe me, there always more urgent or fun things to do here than to find a place of my own and start writing.

A more fundamental problem is that I cannot choose between possible book-projects.

Here’s one I will definitely not pursue:

[section_title text=”The Grothendieck heist”]

“A group of hackers uses a weapon of Math destruction to convince Parisian police that a terrorist attack is imminent in the 6th arrondissement. By a cunning strategy they are then able to enter the police station and get to the white building behind it to obtain some of Grothendieck’s writings.

A few weeks later three lengthy articles hit the arXiv, claiming to contain a proof of the Riemann hypothesis, by partially dismantling topos theory.

Bi-annual conferences are organised around the globe aimed at understanding this weird new theory, etc. etc. (you get the general idea).

The papers are believed to have resulted from the Grothendieck heist. But then, similar raids are carried out in Princeton and in Cambridge UK and a sinister plan emerges… “

Funny as it may be to (ab)use a story to comment on the current state of affairs in mathematics, I’m not known to be the world’s most entertaining story teller, so I’d better leave the subject of math-thrillers to others.

Here’s another book-idea:

[section_title text=”The Bourbaki travel guide”]

The idea is to hunt down places in Paris and in the rest of France which were important to Bourbaki, from his birth in 1934 until his death in 1968.

This includes institutions (IHP, ENS, …), residences, cafes they frequented, venues of Bourbaki meetings, references in La Tribu notices, etc.

This should lead to some nice Parisian walks (in and around the fifth arrondissement) and a couple of car-journeys through la France profonde.

Of course, also some of the posts I wrote on possible solutions of the riddles contained in Bourbaki’s wedding announcement and the avis de deces will be included.

Here the advantage is that I have already a good part of the raw material. Of course it still has to be followed up by in-situ research, unless I want to turn it into a ‘virtual math traveler ’s guide’ so that anyone can check out the places on G-maps rather than having to go to France.

I’m still undecided about this project. Is there a potential readership for this? Is it worth the effort? Can’t it wait until I retire and will spend even more time in France?

Here’s yet another idea:

[section_title text=”Mr. Yoneda takes the Tokyo-subway”]

This is just a working title, others are “the shape of prime numbers”, or “schemes for hipsters”, or “toposes for fashionistas”, or …

This should be a work-out of the sga4hipsters meme. Is it really possible to explain schemes, stacks, Grothendieck topologies and toposes to a ‘general’ public?

At the moment I’m re-reading Eugenia Cheng’s “Cakes, custard and category theory”. As much as I admire her fluent writing style it is difficult for me to believe that someone who didn’t knew the basics of it before would get an adequate understanding of category theory after reading it.

It is often frustrating how few of mathematics there is in most popular maths books. Can’t one do better? Or is it just inherent in the format? Can one write a Cheng-style book replacing the recipes by more mathematics?

The main problem here is to find good ‘real-life’ analogies for standard mathematical concepts such as topological spaces, categories, sheaves etc.

The tentative working title is based on a trial text I wrote trying to explain Yoneda’s lemma by taking a subway-network as an example of a category. I’m thinking along similar lines to explain topological spaces via urban-wide wifi-networks, and so on.

But al this is just the beginning. I’ll consider this a success only if I can get as far as explaining the analogy between prime numbers and knots via etale fundamental groups…

If doable, I have no doubt it will be time well invested. My main problem here is finding an appropriate ‘voice’.

At first I wanted to go along with the hipster-gimmick and even learned some the appropriate lingo (you know, deck, fin, liquid etc.) but I don’t think it will work for me, and besides it would restrict the potential readers.

Then, I thought of writing it as a series of children’s stories. It might be fun to try to explain SGA4 to a (as yet virtual) grandchild. A bit like David Spivak’s short but funny text “Presheaf the cobbler”.

Once again, all suggestions or advice are welcome, either as a comment here or privately over email.

Perhaps, I’ll keep you informed while stumbling along NaNoWriMo.

At least I wrote 1000 words today…