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Category: geometry

Car crashes in scheme theory

What do you get when two cars crash head on at full speed?

A heap of twisted metal.

What do you get when two tiny cars crash head on at full speed?

A smaller heap of twisted metal.

In the limit, what do you get when two point cars crash head on at full speed?

A point of twisted metal?

No, you get a point car with a better GPS system!

Huh?

All a point can see of a function is its own value. Call this a basic, grade one GPS.

When two points collide you’ll get a grade two GPS: not only does it give you its value, but also the tendency of the function in a tiny neighbourhood of the point.

It doesn’t matter whether the two point-cars collide along a line, or in the plane or in 24-dimensional space, you’ll always get the same grade two GPS.

carcrash

Things become more interesting when three or more point-cars crash together.

For three point-cars colliding in the plane a crash-scene investigator can tell from the resulting grade three GPS whether they crashed along a curve, or more randomly.

Again, if this crash happened in a higher dimensional space, you’d still have only hese two types of grade three GPS’s.

And if 4 point-cars pile up?

Then, there are 4 possible types of grade four GP-systems. For crashes of 5 point cars there are 9 possible types, and so on.

Always a finite number?

Hah, no! Seven or more point-cars can pile up in infinitely many ways.

How on earth would you prove such a thing?

A bit of classical geometry, and an extra bit of GIT.

If you’d try to describe all possible 7-car configurations up to isomorphisms, one of the subproblems you run into is to classify all possible intersections of two quadrics in $\mathbb{P}^3$. These usually give you an elliptic curve, and there is a 1-parameter family of those.

Wow!

It only gets better.

Let’s say a grade n GPS knows $n$ behavioural facts about functions, among which its value, in some locally closed neighborhood of the point.

Does such a GPS necessarily comes from the collision of n point cars in some high dimensional space?

Or put differently, can you unravel any GPS into n distinct point cars, just before they crash?

The way you ask this I suspect you’re going to tell me it’s not always possible. But, I haven’t the faintest clue on how to approach such a problem.

You’re right, it’s not always possible.

There exist grade 8 GPS’s which you can’t get from smashing 8 point cars in a 4 or higher dimensional space.

And you’ll need some scheme theory to prove this.

It’s pretty easy to show that the component containing n distinct point-cars in the scheme of all possible n-point configurations has dimension $n^2$. Follows because the automorphism group of n distinct point-cars is just the group of all permutations of the n points.

All grade n GPS’s made from massive car collisions are points in that component.
But then, the tangent space in such a point must have dimension at least $n^2$.

And then, all one has to do is to engineer a grade n GPS with a smaller dimensional tangent space.

A Berkeley group of 9 geometrical engineers did just that: Jonah Blasiak, Dustin Cartwright, David Eisenbud, Daniel Erman, Mark Haiman, Bjorn Poonen, Bernd Sturmfels, Mauricio Velasco, and Bianca Viray.

Here’s their simplest example:

$\mathbb{C}[a, b, c, d]/(a^2, ab, b^2, c^2, cd, d^2, ad − bc)$

Details can be found here:

B. Poonen : The moduli space of commutative algebras of finite rank.

B. Poonen : Isomorphism types of commutative algebras of finite rank over an algebraically closed field.

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Toposes alive and kicking at IHES

After 50 years, vivid interest in topos theory seems to have returned to one of the most prestigious research institutes, the IHES. Last november, there was the meeting Topos a l’IHES.

At the meeting, Celine Loozen filmed a documentary which is supposed to have as its title “Unifying Worlds”. Its very classy trailer is now on YouTube (via +David Roberts).

How did topos theory, a topic considered by most to be far too abstract to be useful to main stream mathematics, suddenly return in such force?

It always helps when a couple of world-class mathematicians become interest in the topic, for their own particular reasons. Clearly, the topic gathers considerable momentum if these people are all permanent members of the IHES.

A lot of geometric information is contained in the category of all sheaves on the geometric object. Topos theory offers a way to construct ‘geometries’ out of nothing, that is, out of arbitrary categories.

Take your favourite category $\mathbf{C}$, then “presheaves” on $\mathbf{C}$ are defined to be contravariant functors $\mathbf{C} \rightarrow \mathbf{Sets}$. For any Grothendieck topology on $\mathbf{C}$ one can then restrict to the sub-category of “sheaves” for this topology, and that’s your typical topos.

Alain Connes got interested in topos theory because he observed that even for the most trivial of categories, such as the monoid category with just one object and endomorphisms the multiplicative semigroup $\mathbb{N}_{\geq 1}^{\times}$, and taking the coarsest of all Grothendieck topologies, one gets interesting objects of baffling complexity.

One of the ‘invariants’ one can associate to a topos is its collection of “points”. Together with Katia Consani, Connes computed in Geometry of the Arithmetic Site that the collection of points of this simple presheaf topos is exactly the set of adele classes $\mathbb{Q}^{\ast}_+ \backslash \mathbb{A}^f_{\mathbb{Q}} / \hat{\mathbb{Z}}^{\ast}$.

Here’s what Connes himself said about this revelation (followed by an attempted translation):

——————————————————

(50.36)

And,in this example, we saw the wonderful notion of a topos, developed by Grothendieck.

It was sufficient for me to open SGA4, a book written at the beginning of the 60ties or the late fifties.

It was sufficient for me to open SGA4 to see that all the things that I needed were there, say, how to construct a cohomology on this site, how to develop things, how to see that the category of sheaves of Abelian groups is an Abelian category, having sufficient injective objects, and so on … all those things were there.

This is really remarkable, because what does it mean?

It means that the average mathematician says: “topos = a generalised topological space and I will never need to use such things. Well, there is the etale cohomology and I can use it to make sense of simply connected spaces and, bon, there’s the chrystaline cohomology, which is already a bit more complicated, but I will never need it, so I can safely ignore it.”

And (s)he puts the notion of a topos in a certain category of things which are generalisations of things, developed only to be generalisations…

But in fact, reality is completely different!

In our work with Katia Consani we saw not only that there is this epicyclic topos, but in fact, this epicyclic topos lies over a site, which we call the arithmetic site, which itself is of a delirious simplicity.

It relies only on the natural numbers, viewed multiplicatively.

That is, one takes a small category consisting of just one object, having this monoid as its endomorphisms, and one considers the corresponding topos.

This appears well … infantile, but nevertheless, this object conceils many wonderful things.

And we would have never discovered those things, if we hadn’t had the general notion of what a topos is, of what a point of a topos is, in terms of flat functors, etc. etc.

(52.27)

——————————————————-

Pierre Cartier has a very wide interest in mathematical theories, the wilder the better: Witt rings, motifs, cosmic Galois groups, toposes…

He must have been one of the first people to speak about toposes at the Bourbaki seminar. In february 1978 he gave the talk Logique, categories et faisceaux, d’apres F. Lawvere et M. Tierney (and dedicated to Grothendieck’s 50th birthday).

He also gave the opening lecture of the Topos a l’IHES conference.

In this fragment of an interview with Stephane Dugowson and Anatole Khelif in 2014 he plays down his own role in the development of topos theory, compared to his contributions in other fields, such as motifs.

——————————————————-

(46:24)

Well, I didn’t invest much time in topos theory.

Except, I once gave a talk at the Bourbaki seminar on the use of topos theory in logic, such as the independence of the axiom of choice, that is, on the idea of forcing.

But, it was just this talk, I didn’t do anything original in it.

Then there is nonstandard analysis, where one can formulate certain things in terms of topos theory. When I got interested in nonstandard analysis, I had this possible application of topos theory in mind.

At the moment when you have a nonstandard model of the integers or more generally of set theory, then one has two models of set theory, that is two different toposes, and then one obviously tries to compare them.

In that sense, I was completely aware of the fact that everything I was doing could be expressed in the language of toposes,or at least in the philosophy of toposes.

I haven’t made any important contributions in that theory, for me it merely remained a tool.

(47:49)

——————————————————-

Laurent Lafforgue says he spend hundredths and hundredths of hours talking to Olivia Caramello about topos theory.

She must have been quite convincing. The last couple of years Lafforgue is a fierce advocate of Caramello’s work.

Her basic idea is that the same topos can arise from two very different mathematical settings (that is, two different categories with Grothendieck topologies can have equivalent categories of sheaves).

The hope then is to translate results from one theory to the other, or as she expresses it, toposes can be used as “bridges” between different mathematical topics.

At the moment though, is seems a bit far fetched for this idea to be relevant to the Langlands programme.

Caramello and Lafforgue have just a paper out: Sur la dualit´e des topos et de leurs pr´esentations et ses applications : une introduction.

The paper is based on a lecture Lafforgue gave in April in Nantes. Here’s the video:

In the introduction they write:

“It is our conviction that the theory of toposes and their representations, with its essential and structural ambiguity, is destined to have an impact on mathematics comparable to the impact group theory has had from the moment, some decades after its discovery by Galois, the mathematical community began to understand it.”

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The group algebra of all algebraic numbers

Some weeks ago, Robert Kucharczyk and Peter Scholze found a topological realisation of the ‘hopeless’ part of the absolute Galois group $\mathbf{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. That is, they constructed a compact connected space $M_{cyc}$ such that etale covers of it correspond to Galois extensions of the cyclotomic field $\mathbb{Q}_{cyc}$. This gives, at least in theory, a handle on the hopeless part of the Galois group $\mathbf{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}_{cyc})$, see the previous post in this series.

Here, we will get halfway into constructing $M_{cyc}$. We will try to understand the topology of the prime ideal spectrum $\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}])$ of the complex group algebra of the multiplicative group $\overline{\mathbb{Q}}^{\times}$ of all non-zero algebraic numbers.

[section_title text=”Pontryagin duals”]

Take an Abelian locally compact group $A$ (for example, an Abelian group equipped with the discrete topology), then its Pontryagin dual $A^{\vee}$ is the space of all continuous group morphisms $A \rightarrow \mathbb{S}^1$ to the unit circle $\mathbb{S}^1$ endowed with the compact open topology.

There are these topological properties of the locally compact group $A^{\vee}$:

– $A^{\vee}$ is compact if and only if $A$ has the discrete topology,

– $A^{\vee}$ is connected if and only if $A$ is a torsion free group,

– $A^{\vee}$ is totally disconnected if and only if $A$ is a torsion group.

If we take the additive group of rational numbers with the discrete topology, the dual space $\mathbb{Q}^{\vee}$ is the one-dimensional solenoid

It is a compact and connected group, but is not path connected. In fact, it path connected components can be identified with the finite adele classes $\mathbb{A}_f/\mathbb{Q} = \widehat{\mathbb{Z}}/\mathbb{Z}$ where $\widehat{\mathbb{Z}}$ is the ring of profinite integers.

Keith Conrad has an excellent readable paper on this fascinating object: The character group of $\mathbb{Q}$. Or you might have a look at this post.

[section_title text=”The multiplicative group of algebraic numbers”]

A torsion element $x$ in the multiplicative group $\overline{\mathbb{Q}}^{\times}$ of all algebraic numbers must satisfy $x^N=1$ for some $N$ so is a root of unity, so we have the exact sequence of Abelian groups

$0 \rightarrow \pmb{\mu}_{\infty} \rightarrow \overline{\mathbb{Q}}^{\times} \rightarrow \overline{\mathbb{Q}}^{\times}_{tf} \rightarrow 0$

where the last term is the maximal torsion-free quotient of $\overline{\mathbb{Q}}^{\times}$. By Pontryagin duality this gives us an exact sequence of compact topological groups

$0 \rightarrow (\overline{\mathbb{Q}}^{\times}_{tf})^{\vee} \rightarrow (\overline{\mathbb{Q}}^{\times})^{\vee} \rightarrow \pmb{\mu}^{\vee}_{\infty} \rightarrow 0$

Here, the left-most space is connected and $\pmb{\mu}^{\vee}_{\infty}$ is totally disconnected. That is, the connected components of $(\overline{\mathbb{Q}}^{\times})^{\vee}$ are precisely the translates of the connected subgroup $(\overline{\mathbb{Q}}^{\times}_{tf})^{\vee}$.

[section_title text=”Prime ideal spectra”]

The short exact sequence of Abelian groups gives a short exact sequence of the corresponding group schemes

$0 \rightarrow \mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}_{tf}]) \rightarrow \mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}] \rightarrow \mathbf{Spec}(\mathbb{C}[\pmb{\mu}_{\infty}]) \rightarrow 0$

The torsion free abelian group $\overline{\mathbb{Q}}^{\times}_{tf}$ is the direct limit $\underset{\rightarrow}{lim}~M_i$ of finitely generated abelian groups $M_i$ and as the corresponding group algebra $\mathbb{C}[M_i] = \mathbb{C}[x_1,x_1^{-1},\cdots, x_k,x_k^{-1}]$, we have that $\mathbf{Spec}(\mathbb{C}[M_i])$ is connected. But then this also holds for

$\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}_{tf}]) = \underset{\leftarrow}{lim}~\mathbf{Spec}(\mathbb{C}[M_i])$

The underlying group of $\mathbb{C}$-points of $\mathbf{Spec}(\mathbb{C}[\pmb{\mu}_{\infty}])$ is $\pmb{\mu}_{\infty}^{\vee}$ and is therefore totally disconnected. But then we have

$\pi_0(\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}]) \simeq \pi_0(\mathbf{Spec}(\mathbb{C}[\pmb{\mu}_{\infty}]) \simeq \pmb{\mu}_{\infty}^{\vee}$

and, more importantly, for the etale fundamental group

$\pi_1^{et}(\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}],x) \simeq \pi_1^{et}(\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}_{tf}],y)$

So, we have to compute the latter one. Again, write the torsion-free quotient as a direct limit of finitely generated torsion-free Abelian groups and recall that connected etale covers of $\mathbf{Spec}(\mathbb{C}[M_i])=\mathbf{Spec}(\mathbb{C}[x_1,x_1^{-1},\cdots,x_k,x_k^{-1}])$ are all of the form $\mathbf{Spec}(\mathbb{C}[N])$, where $N$ is a subgroup of $M_i \otimes \mathbb{Q}$ that contains $M_i$ with finite index (that is, adjoining roots of the $x_i$).

Again, this goes through the limit and so a connected etale cover of $\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}_{tf}])$ would be determined by a subgroup of the $\mathbb{Q}$-vectorspace $\overline{\mathbb{Q}}^{\times}_{tf} \otimes \mathbb{Q}$ containing $\overline{\mathbb{Q}}^{\times}_{tf}$ with finite index.

But, $\overline{\mathbb{Q}}^{\times}_{tf}$ is already a $\mathbb{Q}$-vectorspace as we can take arbitrary roots in it (remember we’re using the multiplicative structure). That is, $\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}])$ is simply connected!

[section_title text=”Bringing in the Galois group”]

Now, we’re closing in on the mysterious space $M_{cyc}$. Clearly, it cannot be the complex points of $\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}])$ as this has no proper etale covers, but we still have to bring the Galois group $\mathbf{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}_{cyc})$ into the game.

The group algebra $\mathbb{C}[\overline{\mathbb{Q}}^{\times}]$ is a commutative and cocommutative Hopf algebra, and all the elements of the Galois group act on it as Hopf-automorphisms, so it is natural to consider the fixed Hopf algebra

$H_{cyc}=\mathbb{C}[\overline{\mathbb{Q}}^{\times}]^{\mathbf{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}_{cyc})}$

This Hopf algebra has an interesting alternative description as a subalgebra of the Witt ring $W(\mathbb{Q}_{cyc})$, bringing it into the realm of $\mathbb{F}_1$-geometry.

This ring of Witt vectors has as its underlying set of elements $1 + \mathbb{Q}_{cyc}[[t]]$ of formal power series in $\mathbb{Q}_{cyc}[[t]]$. Addition on this set is defined by multiplication of power series. The surprising fact is that we can then put a ring structure on it by demanding that the product $\odot$ should obey the rule that for all $a,b \in \mathbb{Q}_{cyc}$ we have

$(1-at) \odot (1-bt) = 1 – ab t$

In this mind-boggling ring the Hopf algebra $H_{cyc}$ is the subring consisting of all power series having a rational expression of the form

$\frac{1+a_1t+a_2t^2+ \cdots + a_n t^n}{1+b_1 t + b_2 t^2 + \cdots + b_m t^m}$

with all $a_i,b_j \in \mathbb{Q}_{cyc}$.

We can embed $\pmb{\mu}_{\infty}$ by sending a root of unity $\zeta$ to $1 – \zeta t$, and then the desired space $M_{cyc}$ will be close to

$\mathbf{Spec}(H_{cyc} \otimes_{\mathbb{Z}[\pmb{\mu}_{\infty}]} \mathbb{C})$

but I’ll spare the details for another time.

In case you want to know more about the title-picture, quoting from John Baez’ post The Beauty of Roots:

“Sam Derbyshire decided to to make a high resolution plot of some roots of polynomials. After some experimentation, he decided that his favorite were polynomials whose coefficients were all 1 or -1 (not 0). He made a high-resolution plot by computing all the roots of all polynomials of this sort having degree ≤ 24. That’s $2^{24}$ polynomials, and about $24 \times 2^{24}$ roots — or about 400 million roots! It took Mathematica 4 days to generate the coordinates of the roots, producing about 5 gigabytes of data.”

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