## Sunday, December 8, 2013

### Homotopy type theory on Images des mathématiques

This post will be a short advertisement to a longer general audience text about homotopy type theory that I published on the website Images des mathématiques.

In this text, I try to convey my excitement at the reading of the book published by the participants of last year's IAS program, under direction of Steve Awoodey, Thierry Coquand and Vladimir Voevodsky.  As I write there (this is the title of this article), this remarkable work is at the crossroads of foundations of mathematics, topology and computer science. Indeed, the new foundational setup for mathematics provided by type theory may not only replace set theory; it is also at the heart of the systems for computer proof checking, and gave birth to a new kind of synthetic homotopy theory'' which is totally freed of the general topology framework.

Also remarkable is the way this book was produced: written collaboratively, using technology well known in open source software's development, then published under a Creative commons's license, and printed on demand.

This is not the only general audience paper on this subject, probably not the last one neither. Here are links to those I know of:
Once more, here is the link towards my article on Images des mathématiques and that towards the HoTT Book!

## Friday, November 8, 2013

### Ari Hoenig concert

Among the three themes I planned to discuss, only math had some place here, and not a single word about jazz. Many concerts, though, and a few of them were good, but none really good to the point to grab a keyboard and write a notice.

Two of them were even a bit disappointing. One year ago, Wayne Shorter celebrated his 80th birthday with his quartet (Danilo Perez, John Pattitucci, Brian Blade) at the Salle Pleyel. But I found the music a bit cold. Only now do I begin to appreciate the CD Without a net that they published soon after.

In september, John Zorn celebrated his 60th birthday at La Villette with a kind of musical marathon: 3 concerts, 9 bands (even 10), some 5 hours of music. Alas. While a similar concert at Banlieues bleues in 2012 had been wonderful, that one was a great disappointment. Except for 3-4 bands (Holy visions, Acoustic Masada, The Dreamers, Bar Kokhba), the rest was boring (Alchemist), ridiculous (Song Project),  if not unbearable (Templars).

But I had the great pleasure to hear Ari Hoenig in Vincennes, with Gilad Hekselman on guitar and Noam Wiesenberg on bass. Ari is a nice young drummer from Philadelphia, with a very melodic touch; I had heared him twice at the Smalls (once with Pilc and Moutin, the other I don't remember!), and he is always very interesting. When I say that he his a melodist, this is not a metaphor. The last piece they played was Charlie Parker's theme Anthropology, and this is the first time that I heard the theme played on the drums. The introduction was rather variations at a slow tempo, but at some point, he played the theme at full speed, and that was really music! Incredible when you think that drums do not have many notes to offer; so for some strokes, Ari had to put his elbow on the drumhead, pressing strongly, so as to modify the pitch. There is a video on youtube where you can see him in action, playing Anthropology (at 7:44, you can guess what I'm talking about), you can also hear that on his CD Inversations. If you like that, his Punk Bop - Live at Smalls is also an excellent CD to listen to.

Enjoy!

## Friday, October 11, 2013

### Walls have ears—Random numbers, Diffie-Hellman, Tom Hales and the NSA

I've been silent for a while, sorry.

Today's message will be quite short, essentially a bunch of links to various other blog posts related to cryptography, the NSA, and how our electronic messages are potentially listened to by people we did not think of, at least before Snowden leaked all this information.

I learned of this story through Thomas Hales's web page and his blog post The NSA back door to NIST (to be published in the Notices of the AMS). There he explains how a standard protocol referenced by the NIST (National Institute for Standards and Technology) has a structural flaw that may well be intended.

All modern cryptographic protocols rely on some randomness. This is in particular the case of the Diffie-Hellman key exchange protocol which allows two people to share a secret without risking to let anybody else aware of this secret.

However, in our computers, randomness is not really random, becaused it is produced by algorithms, but is enough random-looking so that it can be used safely in applications.

As Hales explains, one of these standards for pseudo-random numbers has a back door. While it looks secure at first sight, it is possible that somebody possesses the back door'' information that allows him to understand the logic in the output of our series of pseudo-random numbers, all over the world, thus undermining the solidity of the algorithm.

Worse, this somebody might well be the NSA (National Security Agency).

There are two arguments in favor of this thesis:
1. The back door is very elementary, so elementary in fact, that specialists can't believe its existence is not on purpose. According to a New York Times paper (N.S.A. Able to Foil Basic Safeguards of Privacy on Web, September 5th, 2013) classified memos apparently confirm this.
2. By American law, the use of NIST-approved protocols is required to obtain various certifications. Moreover, the NSA is consulted for whatever cryptographic protocol is issued and has been a strong advocate of this protocol at the NIST and subsequently promoted this protocol at all major members of the International Standard Organization (ISO).
As Hales concludes: « An algorithm that has been designed by NSA with a clear mathematical structure giving them exclusive back door access is no accident, particularly in light of the Snowden documents. This is a work of experts. »

## Friday, March 8, 2013

### A presheaf that has no associated sheaf

In his paper Basically bounded functors and flat sheaves (Pacific Math. J, vol. 57, no. 2, 1975, p. 597-610), William C. Waterhouse gives a nice example of a presheaf that has no associated sheaf. This is Theorem 5.5 (page 605).  I thank François Loeser for having indicated this paper to me, and for his suggestion of explaining it here!

Of course, such a beast is reputed not to exist, since it is well known that any presheaf has an associated sheaf, see for example Godement's book Topologie algébrique et théorie des faisceaux, pages 110-111.
That is, for any presheaf $F$ on a topological space, there is a sheaf $G$ with a morphism of presheaves $\alpha\colon F\to G$ which satisfies a universal property: any morphism from $F$ to a sheaf factors uniquely through $\alpha$.

Waterhouse's presheaf is a more sophisticated example of a presheaf, since it is a presheaf on the category of affine schemes for the flat topology. Thus, a presheaf $F$ on the category of affine schemes is the datum,

• of a set $F(A)$ for every ring $A$,
• and of a map $\phi_*\colon F(A)\to F(B)$ for every morphism of rings $\phi\colon A\to B$,

subject to the following conditions:

• if $\phi\colon A\to B$ and $\psi\colon B\to C$ are morphism of rings, then $(\psi\circ\phi)_*=\psi_*\circ\phi_*$;
• one has ${\rm id}_A)_*={\rm id}_{F(A)}$ for every ring $A$.

Any morphism of rings $\phi\colon A\to B$ gives rise to two morphisms $\psi_1,\psi_2\colon B\to B\otimes B$ respectively defined by $\psi_1(b)=b\otimes 1$ and $\psi_2(b)=1\otimes b$, and the two compositions $A\to B\to B\otimes_A B$ are equal. Consequently, for any presheaf $F$, the two associated maps $F(A) \to F(B) \to F(B\otimes_A B)$ are equal.

By definition, a presheaf $F$ is a sheaf for the flat topology if for any faithfully flat morphism of rings, the map ${\phi_*} \colon F(A)\to F(B)$ is injective and its image is the set of elements $g\in F(B)$ at which the two natural maps $(\psi_1)_*$ and $(\psi_2)_*$ from $F(B)$ to $F(B\otimes_A B)$ coincide.

Here is Waterhouse's example.

For every ring $A$, let $F(A)$ be the set of all locally constant functions $f$ from $\mathop{\rm Spec}(A)$ to some von Neumann cardinal such that $f(\mathfrak p)<\mathop{\rm Card}(\kappa(\mathfrak p))$ for every $\mathfrak p\in\mathop{\rm Spec}(A)$.

This is a presheaf. Indeed, let $\phi\colon A\to B$ is a ring morphism, let $\phi^a\colon\mathop{\rm Spec}(B)\to \mathop{\rm Spec}(A)$ be the associated continuous map on spectra. For $f\in F(A)$, then $f\circ\phi^a$ is a locally constant map from ${\rm Spec}(B)$ to some von Neumann cardinal. Moreover, for every prime ideal $\mathfrak q$ in $B$, with inverse image $\mathfrak p=\phi^{-1}(\mathfrak q)=\phi^a(\mathfrak q)$, the morphism $\phi$ induces an injection from the residue field $\kappa(\mathfrak q)$ into $\kappa(\mathfrak p)$, so that $f\circ\phi^a$ satisfies the additional condition on $F$, hence $f\circ\phi^a\in F(B)$.

However, this presheaf has no associated sheaf for the flat topology. The proof is by contradiction. So assume that $G$ is a sheaf and $\alpha\colon F\to G$ satisfies the universal property.

First of all, we prove that the morphism $\alpha$ is injective: for any ring $A$, the map $\alpha_A\colon F(A)\to G(A)$ is injective. For any cardinal $c$ and any ring $A$, let $L_c(A)$ be the set of locally constant maps  from ${\rm Spec}(A)$ to $c$. Then $L_c$ is a presheaf, and in fact a sheaf. There is a natural morphism of presheaves $\beta_c\colon F\to L_c$, given by $\beta_c(f)(\mathfrak p)=f(\mathfrak p)$ if $f(\mathfrak p)\in c$, that is, $f(\mathfrak p)<c$, and $\beta_c(f)(\mathfrak p)=0$ otherwise. Consequently, there is a unique morphism of sheaves $\gamma_c\colon G\to L_c$ such that $\beta_c=\gamma_c\circ\alpha$. For any ring $A$, and any large enough cardinal $c$, the  map $\beta_c(A)\colon F(A)\to L_c(A)$ is injective. In particular, the map $\alpha(A)$ must be injective.

Let $B$ be a ring and $\phi\colon A\to B$ be a faithfully flat morphism. Let $\psi_1,\psi_2\colon B\to B\otimes_A B$ be the two natural morphisms of rings defined above. Then, the equalizer $E(A,B)$ of the two maps $(\psi_1)_*$ and $(\psi_2)_*$ from $F(B)$ to $F(B\otimes_A B)$ must inject into the equalizer of the two corresponding maps from $G(B)$ to $G(B\otimes_A B)$. Consequently, one has an injection from $E(A,B)$ to $G(A)$.

The contradiction will become apparent once one can find rings $B$ for which $E(A,B)$ has a cardinality as large as desired. If ${\rm Spec}(B)$ is a point $\mathfrak p$, then $F(B)$ is just the set of functions $f$ from the point $\mathfrak p$ to some von Neumann cardinal $c$ such that $f(\mathfrak p)<{\rm Card}(\kappa(\mathfrak p))$. That is, $F(B)$ is the cardinal ${\rm Card}(\kappa(\mathfrak p))$ itself. And since ${\rm Spec}(B)$ is a point, the coincidence condition is necessarily satisfied, so that $E(A,B)= {\rm Card}(\kappa(\mathfrak p))\leq G(A)$.

To conclude, it suffices to take a faithfully flat morphism $A\to B$  such that $B$ is field of cardinality strictly greater than $G(A)$. For example, one can take $A$ to be a field and $B$ the field of rational functions in many indeterminates (strictly more than the cardinality of $G(A)$).

What does this example show? Why isn't there a contradiction in mathematics (yet)?

Because the definition of sheaves and presheaves for the flat topology that I gave above was definitely defective: it neglects in a too dramatic way the set theoretical issues that one must tackle to define sheaves on categories. In the standard setting of set theory provided by ZFC, everything is a set. In particular, categories, presheaves, etc. are sets or maps between sets (themselves represented by sets).  But the presheaf $F$ that Waterhouse defines does not exist as a set, since there does not exist a set $\mathbf{Ring}$ of all rings, nor a set $\mathbf{card}$ of all von Neumann cardinals.

The usual way (as explained in SGA 4) to introduce sheaves for the flat topology consists in adding the axiom of universes — there exists a set $\mathscr U$ which is a model of set theory. Then, one does not consider the (inexistent) set of all rings, or cardinals, but only those belonging to the universe $\mathscr U$—one talks of $\mathscr U$-categories, $\mathscr U$-(pre)sheaves, etc.. In that framework, the $\mathscr U$-presheaf $F$ defined by Waterhouse (where one restricts oneself to algebras and von Neumann cardinals in $\mathscr U$) has an associated sheaf $G_{\mathscr U}$. But this sheaf depends on the chosen universe: if $\mathscr V$ is an universe containing $\mathscr U$, the restriction of $G_{\mathscr V}$ to algebras in $\mathscr U$ will no longer be a $\mathscr U$-presheaf.

### S. Bloch on Milne's Étale cohomology

A long time ago, Spencer Bloch wrote a review of James Milne's Étale cohomology for the Bulletin of AMS which can freely be obtained from the AMS web site. This review explains why one should study étale cohomology, how it grew up, and discusses some aspects of Milne's book. It features an unusual combination of humor and seriousness. Reading highly recommended once you have 15 minutes free!

## Saturday, February 16, 2013

### The Poisson summation formula, Minkowski's first theorem, and the density of sphere packings

I opened by accident a paper by Henry Cohn and Noam Elkies, New upper bounds on sphere packings I, Annals of Mathematics, 157 (2003), 689–714, and I had the good surprise to see a beautiful application of the Poisson summation formula to upper bounds for the density of sphere packings.

In fact, their argument is very close to a proof of the first theorem of Minkowski about lattice points in convex bodies which I had discovered in 2009. However, a final remark in their paper, appendix C, shows that this proof is not really new. Anyway, the whole story is nice enough to prompt me to discuss it in this blog.

1. The Poisson formula

I first recall the Poisson formula: let $f\colon\mathbf R^n\to \mathbf R$ be a continuous function whose Fourier transform $\hat f$ is integrable, let $L$ be a lattice in $\mathbf R^n$, and let $L'$ be the dual lattice of $L$. Then,
$\sum_{x\in L} f(x) = \frac1{\mu(\mathbf R^n/L)}\sum_{y\in L'} \hat f(y).$

In fact, one needs a bit more about $f$ that the above hypotheses (but we'll ignore this in the sequel). For example, it is sufficient that $f(x)$ and $\hat f(x)$ be bounded from above by a multiple of $1/(1+\| x\|)^{n+\epsilon}$, for some strictly positive real number $\epsilon$.

2. Minkowski's first theorem

Let $B$ a closed symmetric convex neighborhood of $0$. Minkowski's first theorem bounds from below the cardinality of $B\cap L$. A natural idea would be to apply the Poisson summation formula to the indicator function $f_B$ of $B$. However, $f_B$ is not continuous, so we need to replace $f_B$ by some function $f$ which satisfies the following properties:
• $f\leq f_B$, so that $\#(B\cap L)= \sum_{x\in L} f_B(x)\geq \sum_{x\in L} f(x)$;
• $\hat f\geq 0$, so that $\sum_{y\in L'} \hat f(y) \geq \hat f(0)$.
The second condition suggest to take for $f$ a function of the form $g*\check g$, so that $\hat f=|\hat g| ^2$. Then, the first condition will hold outside of $B$ if $g$ is supported in $\frac12 B$. Let's try $g=cf_{B/2}$ for some positive real number $c$. For all $x\in\mathbf R^n$,
$f(x) \leq f(0) =c^2 \int_{\mathbf R^n} f_{B/2}(x) f_{B/2}(-x) = c^2 \mu(B/2) = c^2 \mu(B)/2^n.$
It suffices to choose $c=(2^n/\mu(B))^{1/2}$.
On the other hand,
$\hat g(0) = \int_{\mathbf R^n} f_{B/2}(x) = \mu(B/2)=c\mu(B)/2^n=1/c.$
Finally, the Poisson formula implies
$\displaystyle \#(B\cap L) \geq \sum_{x\in L} f(x) = \frac1{\mu(\mathbf R^n/L)} \sum_{y\in L'}\hat f(y) \geq \frac1{\mu(\mathbf R^n/L)} \hat f(0)$
$\displaystyle = \frac1{\mu(\mathbf R^n/L)} | \hat g(0)|^2 = \frac{2^n}{\mu(B) \mu(\mathbf R^n/L)}.$

This is exactly Minkowski's first theorem!

Of course, one may consider apply this argument to other functions $f$. In the case where $B$ is an Euclidean ball, a natural choice consists in taking a Gaussian $f(x)=a\exp(-b\|x\|^2)$, since then $\hat f$ has the same form. This is what has essentially been done by Schoof and van der Geer in their paper Effectivity of Arakelov divisors and the analogue of the theta divisor of a number field, Selecta Math. New Ser. 6 (2000), 377–398, and Damian Rössler independently. Indeed, up to normalization factors, the left hand side of the Poisson summation formula is then interpreted as the exponential of the $h^0$ of a line bundle over an arithmetic curve, and the Poisson summation formula itself is the analogue of Serre's duality theorem in Arakelov geometry. As Jean-Benoît Bost explained to me, Gaussian functions provide an inequality for $\#(B\cap L)$ which is sharper than Minkowski's first theorem. The details of the computation can be found in my notes about Arakelov geometry (beware, these are mostly a work in slow progress). I have not tried to look for optimal functions beyond that case.

3. The theorem of Cohn-Elkies

We consider a sphere packing, that is a set of points in $\mathbf R^n$ with mutual distances at least 1,
and we want to bound from above its density, that is the ratio of volume occupied by balls of radius $1/2$ centered at these spheres. One may think of a lattice packing, the particular case where the centers of these spheres are exactly the points of a lattice $L$ or, more generally, of periodic packing when the centers of the spheres are finitely many translates $v_1+L,\dots,v_N+L$ of  a lattice $L$. In fact, Cohn and Elkies argue that it suffices to study such periodic packings (repeating periodically an arbitrarily large part of the sphere packing), so we shall do like them and assume that our sphere packing is periodic.

Now, let $f$ be a real valued function on $\mathbf R^n$ satisfying the following properties:
• $f$ is continuous, integrable on $\mathbf R^n$ as well as its Fourier transform.
• $f(0)>0$ and $f(x)\leq 0$ for $\| x\|\geq 1$;
• $\hat f(x)\geq 0$ for all $x$.
Then, the density $\Delta$ of the sphere packing satisfies
$\Delta \leq 2^{-n} \mu(B) \frac{f(0)}{\hat f(0)},$
where $\mu(B)$ is the volume of the unit ball.

We assume that no difference $v_i-v_j$ is a point of the lattice $L$ (otherwise, we can exclude one translate from the list). With the above notation, the fundamental parallelepiped of the lattice contains exactly $N$ balls of radius $1/2$, hence
$\Delta= 2^{-n}\mu(B) \frac N{\mu(\mathbf R^n/L)}.$
For any $v\in\mathbf R^n$, the Poisson summation formula for $x\mapsto f(x+v)$ writes
$\sum_{x\in L} f(x+v) = \frac1{\mu(L)} \sum_{y\in L'} e^{2\pi i \langle v,y\rangle} \hat f(y),$
hence
$\displaystyle \sum_{1\leq j,k\leq N} \sum_{x\in L} f(x+v_j-v_k) = \frac1{\mu(L)} \sum_{y\in L'}\hat f(y) \sum_{j,k=1}^N e^{2\pi i \langle v_j-v_k,y\rangle}$
$\displaystyle = \frac1{\mu(L)} \sum_{y\in L'}\hat f(y) \left| \sum_{j=1}^N e^{2\pi i \langle v_j,y\rangle} \right|^2.$
All terms of the right hand side are positive or null, so that we can bound it from below by
the term for $y=0$, hence
$\sum_{1\leq j,k\leq N} \sum_{x\in L} f(x+v_j-v_k) \geq N^2 \frac{\hat f(0)}{\mu(\mathbf R^n/L)}.$
Now, there is a sphere of our packing centered at $x+v_j$, and another at $v_k$,
so that $\| x+v_j-v_k\|\geq 1$ unless $x+v_j=v_k$, that is $v_k-v_j=x$ hence $x=0$ and $v_j=v_k$. In the first case, the value of $f$ at $x+v_j-v_k$ is negative or null; in the latter, it equal $f(0)$. Consequently, the left hand side of the previous inequality is at most $Nf(0)$.  Finally, $\displaystyle N f(0) \geq N^2 \hat f(0)/\mu(\mathbf R^n/L)$, hence the desired inequality.

## Monday, February 11, 2013

### The Poisson summation formula, arithmetic and geometry

François Loeser and I just uploaded a paper on arXiv about Motivic height zeta functions. That such a thing could be possible is quite funny, so I'll take this opportunity to break a long silence on this blog.

In Diophantine geometry, an established and important game consists in saying as much as possible of the solutions of diophantine equations. In algebraic terms, this means proving qualitative or quantitative properties of the set of integer solutions of polynomial equations with integral coefficients. In fact, one can only understand something by making the geometry more apparent; then, one is interested in integral points of schemes $X$ of finite type over the ring $\mathbf Z$ of integers. There are in fact two sub-games: one in which one tries to prove that such solutions are scarce, for example when $X$ is smooth and of general type (conjecture of Mordell=Faltings's theorem, conjecture of Lang) ; the other in which one tries to prove that there are many solutions —then, one can even try to count how many solutions there are of given height, a measure of their size. There is a conjecture of Manin predicting what would happen, and our work belongs to this field of thought.

Many methods exist to understand rational points or integral points of varieties. When the scheme carries an action of an algebraic group, it is tempting to try to use harmonic analysis. In fact, this has been done since the beginnings of Manin's conjecture when Franke, Manin, Tschinkel showed that when the variety is a generalized flag variety ($G/P$, where $G$ is semi-simple, $P$ a parabolic subgroup, for example projective spaces, grassmannians, quadrics,...), the solution of Manin's question was already given by Langlands's theory of Eisenstein series. Later, Batyrev and Tschinkel proved the case of toric varieties, and again with Tschinkel, I studied the analogue of toric varieties when the group is not a torus but a vector space. In these two cases, the main idea consists in introducing a generating series of our counting problem, the height zeta function, and establishing its analytic properties. In fact, this zeta function is a sum over rational points of a height function defined on the adelic space of the group, and the Poisson summation formula rewrites this sum as the integral of the Fourier transform of the height function over the group of topological characters. What makes the analysis possible is the fact that, essentially, the trivial character carries all the relevant information; it is nevertheless quite technical to establish what happens for other characters, and then to check that the behavior of the whole integral is indeed governed by the trivial character.

In mathematics, analogy often leads to interesting results. The analogy between number fields and function fields suggests that diophantine equations over the integers have a geometric analogue, which consists in studying morphisms from a curve to a given variety. If the ground field of the function field is finite, the dictionary goes quite far; for example, Manin's question has been studied a lot by Bourqui who established the case of toric varieties. But when the ground field is infinite, it is no more possible to count solutions of given height since they will generally be infinite.

However, as remarked by Peyre around 2000, all these solutions, which are morphisms from a curve to a scheme, form themselves a scheme of finite type. So the question is to understand the behavior of these schemes, when the height parameter grows to infinity. In fact, in an influential but unpublished paper, Kapranov had already established the case of flag varieties (without noticing)! The height zeta function is now a formal power series whose coefficients are algebraic varieties; one viewes them as elements of the Grothendieck ring of varieties, the universal ring generated by varieties with addition given by cutting-and-pasting, and multiplication given by the product of varieties. This ring is a standard tool of motivic integration (as invented by Kontsevich and developed by Denef and Loeser, and many people since). That's why this height zeta function is called motivic.

What we proved with François is a rationality theorem for such a motivic height function, when the variety is an equivariant compactification of a vector group. This means that all this spaces of morphisms, indexed by some integer, satisfy a linear dependence relation in the Grothendieck ring of varieties! To prove this result, we rely crucially on an analogue of the Poisson summation formula in motivic integration, due to Hrushovski and Kazhdan, which allows us to perform a similar analysis to the one I had done with Tschinkel in a paper that appeared last year in Duke Math. J.

Many things remain puzzling. The most disturbing is the following. If you read Tate's thesis, or Weil's Basic Number Theory, you'll see that the Riemann Roch formula and Serre's duality theorem for curves over finite fields are consequences of the Poisson summation formula in harmonic analysis. In motivic integration, things go the other way round: if one unwinds all the definitions (we explain this in our paper with François), the motivic Poisson summation formula boils down to the Riemann-Roch and Serre theorems. So, in principle, our proof could be understood just from these two theorems. But this is not clear at all how to do this directly: passing through the looking-glass to go computing our height zeta function in the Fourier world appears to be non-trivial and efficient...