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Also blogs at NeverEndingBooks.
MapMathInterbellum

European mathematics in 1927

Here’s a map of the (major) mathematical centers in Europe (in 1927), made for the Rockefeller Foundation.

Support by the Rockefeller foundation was important for European Mathematics between the two world wars. They supported the erection of the Mathematical Institute in Goettingen between 1926-1929 and creation of the Institut Henri Poincare in Paris at about the same time.

Careers of people such as Stefan Banach, Bartel van der Waerden and Andre Weil benefitted hugely from becoming fellows of the Rockefeller-funded International Educational Board in the 20ties.

The map itself shows that there were three major centers at the time: Goettingen, Paris and Rome (followed by Berlin and Oxford, at a distance).

Also the distribution by topics (the pie-charts per university) is interesting: predominantly Analysis (red) with a fair share of Geometry (yellow), Number Theory (green) and Applied Mathematics (blue). Philosophy (black) was even more important than Algebra (orange) which existed only in Goettingen (Noether, van der Waerden) and Berlin.

I’d love to see a similar map for 2014…

A larger version of the map can be found here.

There’s a corresponding map for the USA here.

ENS

the birthday of Grothendieck topologies

This is the story of the day the notion of ‘neighbourhood’ changed forever (at least in the geometric sense).

For ages a neighbourhood of a point was understood to be an open set of the topology containing that point. But on that day, it was demonstrated that the topology of choice of algebraic geometry, the Zariski topology, needed a drastic upgrade.

This ultimately led to the totally new notion of Grothendieck topologies, which aren’t topological spaces at all.

Formally, the definition of Grothendieck topologies was cooked up in the fall of 1961 when Grothendieck visited Zariski, Mike Artin and David Mumford in Harvard.

The following spring, Mike Artin ran a seminar resulting in his lecture notes on, yes, Grothendieck topologies.

But, paradigm shifts like this need a spark, ‘une bougie d’allumage’, and that moment of insight happened quite a few years earlier.

It was a sunny spring monday afternoon at the Ecole Normal Superieure. Jean-Pierre Serre was giving the first lecture in the 1958 Seminaire Claude Chevalley which that year had Chow rings as its topic.

That day, april 21st 1958, Serre was lecturing on algebraic fibre bundles:

He had run into a problem.

If a Lie group $G$ acts freely on a manifold $M$, then the set of $G$-orbits $M/G$ is again a manifold and the quotient map $\pi : M \rightarrow M/G$ is a principal $G$-fibre bundle meaning that for sufficiently small open sets $U$ of $M/G$ we have diffeomorphisms

$\pi^{-1}(U) \simeq U \times G$

that is, locally (but not globally) $M$ is just a product manifold of $G$ with another manifold and the $G$-orbits are all of the form $\{ u \} \times G$.

The corresponding situation in algebraic geometry would be this: a nice, say reductive, algebraic group $G$ acting freely on a nice, say smooth, algebraic variety $X$. In this case one can form again an orbit space $X/G$ which is again a (smooth) algebraic variety but the natural quotient map $\pi : X \rightarrow X/G$ rarely has this local product property…

The reason being that the Zariski topology on $X/G$ is way too coarse, it doesn’t have enough open sets to enforce this local product property.

(For algebraists: let $A$ be an Azumaya algebra of rank $n^2$ over $\mathbb{C}[X]$, then the representation variety $\mathbf{rep}_n(A)$ is a principal $\mathbf{PGL}_n$-bundle over $X$ but is only local trivial in the Zariski topology when $A$ is a trivial Azumaya algebra, that is, $End_{\mathbb{C}[X]}(P)$ for a rank $n$ projective module $P$ over $\mathbb{C}[X]$.)

But, Serre had come up with a solution.

He was going to study fibre bundles which were locally ‘isotrivial’, meaning that they had the required local product property but only after extending them over an unamified cover $Y \rightarrow X$ (what we now call, an etale cover) and he was able to clasify such fibre bundles by a laborious way (which we now call the first etale cohomology group).

The story goes that Grothendieck, sitting in the public, immediately saw that these etale extensions were the correct generalization of the usual (Zariski) localizations and that he could develop a cohomology theory out of them in all dimensions.

According to Colin McLarty Serre was ‘absolutely unconvinced’, since he felt he had ‘brutally forced’ the bundles to yield the $H^1$’s.

We will never known what Serre actually wrote on the blackboard on april 21st 1958.

The above scanned image tells it is an expanded version of the original talk, written up several months later after the ICM-talk by Grothendieck in Edinburgh.

By that time, Grothendieck had shown Serre that his method indeed gives cohomology in all dimensions,and convinced him that this etale cohomology was likely to be the “true cohomology needed to prove the Weil conjectures”.

Prep-notes dump

Here are the scans of my rough prep-notes for some of the later seminar-talks. These notes still contain mistakes, most of them were corrected during the talks. So, please, read these notes with both mercy and caution!

Hurwitz formula imples ABC : The proof of Smirnov’s argument, but modified so that one doesn’t require an $\epsilon$-term. This is known to be impossible in the number-theory case, but a possible explanation might be that not all of the Smirnov-maps $q~:~\mathsf{Spec}(\mathbb{Z}) \rightarrow \mathbb{P}^1_{\mathbb{F}_1}$ are actually covers.

Frobenius lifts and representation rings : Faithfully flat descent allows us to view torsion-free $\mathbb{Z}$-rings with a family of commuting Frobenius lifts (aka $\lambda$-rings) as algebras over the field with one element $\mathbb{F}_1$. We give several examples including the two structures on $\mathbb{Z}[x]$ and Adams operations as Frobenius lifts on representation rings $R(G)$ of finite groups. We give an example that this extra structure may separate groups having the same character table. In general this is not the case, the magic Google search term is ‘Brauer pairs’.

Big Witt vectors and Burnside rings : Because the big Witt vectors functor $W(-)$ is adjoint to the tensor-functor $- \otimes_{\mathbb{F}_1} \mathbb{Z}$ we can view the geometrical object associated to $W(A)$ as the $\mathbb{F}_1$-scheme determined by the arithmetical scheme with coordinate ring $A$. We describe the construction of $\Lambda(A)$ and describe the relation between $W(\mathbb{Z})$ and the (completion of the) Burnside ring of the infinite cyclic group.

Density theorems and the Galois-site of $\mathbb{F}_1$ : We recall standard density theorems (Frobenius, Chebotarev) in number theory and use them in combination with the Kronecker-Weber theorem to prove the result due to James Borger and Bart de Smit on the etale site of $\mathsf{Spec}(\mathbb{F}_1)$.

New geometry coming from $\mathbb{F}_1$ : This is a more speculative talk trying to determine what new features come up when we view an arithmetic scheme over $\mathbb{F}_1$. It touches on the geometric meaning of dual-coalgebras, the Habiro-structure sheaf and Habiro-topology associated to $\mathbb{P}^1_{\mathbb{Z}}$ and tries to extend these notions to more general settings. These scans are unintentionally made mysterious by the fact that the bottom part is blacked out (due to the fact they got really wet and dried horribly). In case you want more info, contact me.

$\mathbb{F}_1$ and noncommutative geometry

why noncommutative geometry?

Some motivate noncommutative geometry as follows : assume you have a space (or variety) $X$ on which a group $G$ acts wildly so that the ‘orbit-space’ $X/G$ does not exists or has bad topological properties. Let $A$ be the ring of continuous functions on $X$ (or the coordinate ring $\mathcal{O}(X)$), then every $g \in G$ acts as an automorphism $\alpha_g$ on $A$.

Traditionally one associates the orbit-space (when possible) to the commutative fixed-point algebra $A^G$. However, when this algebra is too small to give information on the $G$-orbits in $X$ one can still associate a noncommutative algebra to the situation, the crossed product algebra $A \ast G$ which as a vectorspace is merely $A \otimes \mathbb{C} G$ but with multiplication induced by $(a \otimes g) (b \otimes h) = a \alpha_g(b) \otimes g h$. Some argue that ringtheoretical invariants of $A \ast G$ give some insight into the horrible orbit-space $X/G$.

relevant to $\mathbb{F}_1$-geometry?

We’ve defined an algebra $A$ over $\mathbb{F}_1$ to be a torsion-free $\mathbb{Z}$-ring having a commuting family of endomorphisms $\psi^n~:~A \rightarrow A$ having the property that for every prime number $p$ the endomorphism $\Psi^p$ is a lift of the Frobenius map on $A/pA$. This gives an action by endomorphisms of the multiplicative monoid $\mathbb{N}_{\times}$ on $A$.

We’ve interpreted this additional structure as descent-data from $\mathbb{Z}$ to $\mathbb{F}_1$. Now, in the case of Galois-descent between two fields $k \subset K$ with $Gal(K/k)=G$, the $k$-algebra corresponding to a $K$-algebra $A$ with descent-data $G \rightarrow Aut(A)$ is, of course, the fixed-point algebra $A^G$.

Of course, in the $\mathbb{F}_1$-setting it makes no sense to look at the fixed-point ring $A^{\mathbb{N}_{\times}}$, but we can still consider the corresponding noncommutative ring

$A \ast \mathbb{N}_{\times}$

which as a $\mathbb{Z}$-module is the tensor-product $A \otimes_{\mathbb{Z}} \mathbb{Z} [\mathbb{N}_{\times}]$ where $\mathbb{Z} [\mathbb{N}_{\times}]$ is the monoid-algebra of the commutative monoid $\mathbb{N}_{\times}$. As above, the multiplication is induced by the rule (using the variables $X_n = 1 \otimes n$)

$(a X_n) (b X_m) = a \Psi^n(b) X_{mn}$

If you are a lowly ringtheorist this is already daunting enough because the fact that the crossing is made with endos rather than autos kills most of the desired properties of your noncommutative ring (for example Noetherianness). But, if your a $C^{\ast}$-algebraist then you want to complicate matters even more as you need variables $X_n^{\ast}$ corresponding to the $X_n$ satisfying suitable properties. If this is possible, we will denote the noncommutative algebra generated by $A$, the $X_n$ and the $X_n^*$ by $A \circ \mathbb{N}_{\times}$.

the giant mashup-algebra $\mathbb{Z}[\mathbb{Q}/\mathbb{Z}] \circ \mathbb{N}_{\times}$, aka BC

Lots of papers are written trying to get novel insights into the BC-algebra by looking at its adelic-, motivic-, semi-hemi-demi-, p-adic-, $\mathbb{F}_1$-gadgety or whatever-comes-next interpretation. It is the archetypical example of the above construction.

Let’s define it by generators and relations using its ‘integral’ incarnation. Generators are $e(r)$, one for each $r \in \mathbb{Q}/\mathbb{Z}$ and elements $\tilde{\mu}_n$ and $\mu_n^*$ for $n \in \mathbb{N}_+$. The relations are

$e(r) e(s) = e(r+s)~\forall r,s \in \mathbb{Q}/\mathbb{Z}$

$\tilde{\mu}_n \tilde{\mu}_m = \tilde{\mu}_{nm}~\forall n,m \in \mathbb{N}_+$

$\mu_n^* \mu_m^* = \mu^*_{nm}~\forall n,m \in \mathbb{N}_+$

$\mu_n^* \tilde{\mu}_n = n~\quad \text{and} \quad \tilde{\mu}_n \mu^*_m = \mu^*_m \tilde{\mu}_n~\quad~\text{whenever} \quad (m,n)=1$

$\mu^*_n e(r) = e(nr) \mu^*_n~\forall r \in \mathbb{Q}/\mathbb{Z}, n \in \mathbb{N}_+$

$e(r) \tilde{\mu}_n = \tilde{\mu}_n e(nr)~\forall r \in \mathbb{Q}/\mathbb{Z}, n \in \mathbb{N}_+$

$\tilde{\mu}_n e(r) \mu^*_n = \sum_{ns=r} e(s)~\forall r \in \mathbb{Q}/\mathbb{Z}, n \in \mathbb{N}_+$

The first relations imply that the $\mathbb{Z}$-ring generated by the $e(r)$ is the integral group-ring $\mathbb{Z}[\mathbb{Q}/\mathbb{Z}]$. Taking $e(r) \mapsto e^{2 \pi i r}$ we see that this ring is isomorphic to the integral group-ring $\mathbb{Z}[\pmb{\mu}_{\infty}]$ of the multiplicative group of all roots of unity.

$\mathbb{Z}[\pmb{\mu}_{\infty}]$ is a $\lambda$-ring (actually, our best shot at the algebraic closure $\overline{\mathbb{F}}_1$) with endomorphisms $\Psi^n(e^{2 \pi i r}) = e^{2 \pi i nr}$ (which correspond to the endomorphisms $e(r) \mapsto e(nr)$ in $\mathbb{Z}[\mathbb{Q}/\mathbb{Z}]$).

Hence, we see that the subring generated by the $e(r)$ and the $\mu_n^*$ is actually isomorphic to the noncommutative crossed product $\mathbb{Z}[\pmb{\mu}_{\infty}] \ast \mathbb{N}_{\times}$ constructed before. The full BC-algebra is then what we have denoted $\mathbb{Z}[\pmb{\mu}_{\infty}] \circ \mathbb{N}_{\times}$.

More information on the (classical) BC-algebra can be found in these neverendingbook-posts : as a giant mash-up of arithmetical information and its relation to the Riemann zeta-function.

In view of the Borger-de Smit result characterizing the etale site of $\mathsf{Spec}(\mathbb{F}_1)$ it is perhaps interesting to consider the multi-variate BC-algebras $\mathbb{Z}[\pmb{\mu}_{\infty}] \otimes \cdots \otimes \mathbb{Z}[\pmb{\mu}_{\infty}] \circ \mathbb{N}_{\times}$ defined in the now obvious way.

More food-for-thought : take your favorite torsion free $\mathbb{Z}$-ring $A$ and construct your own BC-lookalike algebra $W(A) \circ \mathbb{N}_{\times}$ making clever use of the Adams operations $\Psi^n$ and the ‘Verschiebung’-operations on the ring of big Witt vectors $W(A)$.

Aliens and reality

October 21st : Dear Diary,

today’s seminar was fun, though a bit unconventional. My goal was to explain faithfully flat descent, but at the last moment i had this urge to let students discover the main idea themselves (in the easiest of examples) by means of a thought experiment :

“I am an alien (laughter…), and a very stubborn alien at that. There’s just one field, the complex numbers $\mathbb{C}$, and all rings are $\mathbb{C}$-algebras. I’ve heard strange rumours that you humans believe in a geometry ‘hidden under $\mathbb{C}$’, something called real manifolds. What then is an algebra over this obviously virtual ‘real’ field?”

Their first hurdle was to convey the concept of complex conjugation as the alien(me) was unwilling to decompose a complex number $c$ into two ‘ghost components’ $a+bi$. Still i had to concede that i knew about addition and multiplication, i had a $1$ and a square root of $-1$, which for some reason they preferred to call $i$.

‘Oh, but then you know about $\mathbb{Z}[i]$! You just add a number of times $1$’s with $i$’s.’

‘Why are you humans so obsessed with counting? We do not count! We can’t! We have neither fingers nor toes!’

Admittedly a fairly drastic intervention, but i had to keep them on the path leading to Galois descent… After a while we agreed on a map (they called it conjugation) sending sums to sums, products to products and taking a root of unity to its inverse.

Next, they asked me to be a bit flexible and allow for ‘generalized’ fields such as the one consisting of all elements fixed under conjugation! Clearly, the alien refused : ‘We’re not going on that slippery road called generalization, we’ve seen the havock caused by it in human-mathematics.’

It took them a while to realize they would never be able to sell me an $\mathbb{R}$-algebra $A$, but perhaps they could try to sell me the complex algebra $B= A \otimes_{\mathbb{R}} \mathbb{C}$?

Alien : ‘But, how do i recognize one of your algebras among mine? Is there a test to detect them?’

Humans : ‘Yes, they have a map (which we know to be the map $a \otimes c \mapsto a \otimes \overline{c}$, but you cannot see it) sending sums to sums, products to products that extends the conjugation on $\mathbb{C}$.’

Alien : ‘But if i take a basis for any of my algebras and apply conjugation to all its coordinates, then surely all my algebras have this property, not?’

Humans : ‘No, such maps are good for sums, but not always for products. For example, take $\mathbb{C}[x]/(x^2-c)$ for $c$ a complex-number not fixed under conjugation.’

Alien : ‘Point taken. But then, your algebras are just a subclass of my algebras, right?’

Humans : ‘No! An algebra can have several of such additional maps. For example, take $B = \mathbb{C} \times \mathbb{C}$ then there is one sending $(a,b)$ to $(\overline{a},\overline{b})$ and another sending it to $(\overline{b},\overline{a})$. (because we know there are two distinct real algebras $\mathbb{R} \times \mathbb{R}$ and $\mathbb{C}$ of dimension two, tensoring both to $\mathbb{C} \times \mathbb{C}$.)’

By now, the alien and humans agreed on a dictionary : what to humans is the $\mathbb{R}$-algebra $A$ is to the alien the complex algebra $B=A \otimes \mathbb{C}$ together with a map $\gamma_B : B \rightarrow B$ sending sums to sums, products to products and extending conjugation on $\mathbb{C}$ (this extra structure, the map $\gamma_B$, is called the ‘descent data’).

A human-observed $\mathbb{R}$-algebra morphism $\phi : A \rightarrow A’$ is to the alien the $\mathbb{C}$-algebra morphism $\Phi = \phi \otimes id_{\mathbb{C}} : B \rightarrow B’$ which commutes with the extra structures, that is, $\Phi \circ \gamma_B = \gamma_{B’} \circ \Phi$.

Phrased differently (the alien didn’t want to hear any of this) : there is an equivalence of categories between the category $\mathbb{R}-\mathsf{algebras}$ of commutative $\mathbb{R}$-algebras and the category $\gamma-\mathsf{algebras}$ consisting of complex commutative algebras $B$ together with a ringmorphism $\gamma_B$ extending complex conjugation and with morphisms $\mathbb{C}$-algebra morphisms compatible with the $\gamma$-structure.

Further, what to humans is the base-extension (or tensor) functor

$- \otimes_{\mathbb{R}} \mathbb{C}~:~\mathbb{R}-\mathsf{algebras} \rightarrow \mathbb{C}-\mathsf{algebras}$

is (modulo the above equivalence) to the alien merely the forgetful functor

$\mathsf{Forget}~:~\gamma-\mathsf{algebras} \rightarrow \mathbb{C}-\mathsf{algebras}$

stripping off the descent-data.

After the break (yes, it took us that long to get here) we used this idea to properly define obviously non-existing rings living ‘under $\mathbb{Z}$’, or if you like silly terminology, algebras over the field with one element $\mathbb{F}_1$.

Alien : ‘Ha-ha-ha, a field with one element? Surely you’re joking Mr. Human’

Note to self : Dare to waste time like this in a seminar.