Category: geometry

Forgetting can’t be that hard, can it?

Published February 24, 2017 by lievenlb

Geometers will tell you there are two ways to introduce affine schemes.

You can use structure sheaves. That is, compute all prime ideals of your ring and turn them into a space. Then, put a sheaf of rings on this space by localisation. You’ll get your ring back taking global sections.

Or, you might try the ‘functor of points’. That is, you take any other ring. Compute all ring-morphisms from yours to that one. You’ll recover your ring from Yoneda’s lemma.

And here’s the funny part.

Scheme-theorists claim there’s no differences between these two approaches. They are ‘equivalent’, as they prefer to say.

Do you believe them?

Let’s look at an example.

Take the ring of all polynomials with integer coefficients, $\mathbb{Z}[x]$.

Do you know all its prime ideals?

Sure, you’ll say.

There’s zero because it’s a domain. Then there are the ‘curves’. These are all prime numbers and all irreducible polynomials because it’s a UFD.

And then there are the ‘points’. They depend on a prime number $p$ and an irreducible polynomial which does not factor over $p$.

Not exactly rocket science, is it?

Okay, now let’s take them all together into a space.

Can you picture the intersection points of different curves? Let’s keep it simple. Take the curve given by a prime number $p$ and the one given by an irreducible polynomial $F(x)$. How do they intersect?

Easy! They are the factors of $F(x)$ modulo $p$.

Right, but can you picture this pattern for all prime numbers at once?

That depends on $F(x)$. David Mumford sketched the situation for $x^2+1$.

If $-1$ is a square modulo $p$, then $F(x)$ splits in two factors giving two points, such as along $5$. If not, $F(x)$ remains irreducible over $p$ and gives a thicker point like over $3$ or $7$. Except for the ‘odd’ case over $2$ where $F(x)$ is a square. Gauss knew already the situation for every prime.

But, what about arbitrary polynomials?

That’s a lot more difficult. Chebotarev knew how to get their Galois group from the factors at all primes.

So, you’ll need to solve deep problems in number theory before you can picture this space. The structure of the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ to name one.

I’m afraid nobody understands the space of all prime ideals of $\mathbb{Z}[x]$ completely, let alone its structure sheaf.

What about the other approach? Let’s try to understand the functor of points of $\mathbb{Z}[x]$.

Take any ring $R$. We need to figure out all ring-maps $\mathbb{Z}[x] \rightarrow R$. But, we know such a map once we know the image of $x$. That is, there are as many ring-maps as there are elements in the set $R$.

Forgetting all about addition and multiplication on $R$. It is just the forgetful functor from rings to sets.

And they claim this is equivalent to solving deep problems in number theory?

Forgetting can’t be that hard, can it?

2 Comments

Stirring a cup of coffee

Published December 23, 2016 by lievenlb

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.

Where are Grothendieck’s writings? (2)

Published December 19, 2016 by lievenlb

A couple of days ago, there was yet another article by Philippe Douroux on Grothendieck’s Lasserre writings “Inestimables mathématiques, avez-vous donc un prix?” in the French newspaper Liberation.

Not that there is much news to report.

I’ve posted on this before: Grothendieck’s gribouillis, Grothendieck’s gribouillis (2), and more recently Where are Grothendieck’s writings?

In that last post I claimed that the five metallic cases containing Grothendieck’s Lasserre notes were in a white building behind the police station of the sixth arrondissement of Paris.

I was wrong.

There’s a detail in Douroux’ articles I forgot to follow-up before.

Here’s the correct location:

What went wrong?

Here’s my ‘translation’ of part of chapter 46 of Douroux’ book “Alexandre Grothendieck, sur les traces du dernier genie des mathematiques”:

“On November 13th 2015, while the terrorist-attacks on the Bataclan and elsewhere were going on, a Mercedes break with on board Alexandre Jr. Grothendieck and Jean-Bernard, a librarian specialised in ancient writings, was approaching Paris from Lasserre. On board: 5 metallic cases, 2 red ones, 1 green and 2 blues.

At about 2 into the night they arrived at the ‘commissariat du Police’ of the 6th arrondissement. Jean-Bernard pushed open a heavy blue carriage porch, crossed the courtyard opened a second door and then a third one and delivered the cases.”

It all seemed to fit together: the ‘commissariat’ has a courtyard (but then, so do most buildings in the neighborhood) and has a blue carriage porch:

What went wrong?

I should have trusted Google-translate instead.

It translates the original text “…il garait sa voiture pres du commissariat…” more correctly into “…he parked his car near the police station…”. ‘Near’ as apposed to ‘at’…

We should have looked for a location close to the police station.

And, I should have looked up “Jean-Bernard, a librarian specialised in ancient writings”.

Who is Jean-Bernard?

In Douroux’ latest article there’s this sentence:

“Dès lors, on comprend mieux le travail de Jean-Bernard Gillot, libraire à Paris et expert en livres anciens et manuscrits scientifiques pour lequel les cinq malles contenant les écrits de Lasserre représentent l’affaire d’une vie.”

I’m not even going to make an attempt at translation, you know which tool to use if needed. Suffice it to say that the mysterious Jean-Bernard is no other than Jean-Bernard Gillot.

In 2005, Jean-Bernard Gillot took over the Librairie Alain Brieux, specialising in ancient scientific books and objects. Here’s a brief history of this antiques shop.

Relevant to our quest is that it is located 48, rue Jacob in Paris, just around the corner of the Police Station of the 6th arrondissement.

And, there is a beautiful heavy blue carriage porch, leading to an interior courtyard…

A quick look at the vast amount of scientific objects (such as these Napier’s bones) indicates that there must be adequate and ample storage space in the buildings behind the shop.

This is where the five metallic cases containing the Lasserre writings are at this moment.

What’s next?

We’re lightyears removed from Maltsiniotis’ optimistic vision, broadcast at the Grothendieck conference in Montpellier last year, that the BNF would acquire the totality of the writings and make them available to the mathematical community at large.

Apart from Maltsiniotis’ cursory inventory of (part of) the 93.000 pages, nobody knows what’s inside these five boxes, making it impossible to put a price tag on them.

Perhaps, the family should grant some bloggers access to the cases, in return for a series of (live)posts on what they’ll find inside…?!

One Comment