## Monday, May 11, 2015

### Model theory and algebraic geometry, 3 — Real closed fields and o-minimality

In this third post devoted to some interactions between model theory and algebraic geometry, we describe the concept of o-minimality and the o-minimal complex analysis of Peterzil and Starchenko.

1. Real closed fields and the theorem of Tarski-Seidenberg

To begin with, we work in the language $L_{\mathrm{or}}$ of ordered rings which is the language of rings $L_{\mathrm r}=\{+,-,\cdot,0,1\}$ enlarged with an order relation $\leq$.

Let us recall the definition of a real closed field: this is an field $K$ endowed with an ordering which is compatible with the field laws (the sum of positive elements is positive and the product of positive elements is positive) which satisfies the intermediate value theorem for polynomials: for every polynomial $P\in K[T]$, any pair $(a,b)$ of elements of $K$ such that $a<b$, $P(a)<0$ and $P(b)>0$, there exists $c\in K$ such that $P(c)=0$ and $a<c<b$. Observe that this property can be expressed by a sequence of first-order formulas, one for each degree.

The field $\mathbf R$ of real numbers is real closed, but there are many other. For example, the field of formal Puiseux series with real coefficients is also real closed.

A theorem of Artin-Schreier asserts that a field $K$ is real closed if and only if $\sqrt{-1}\not\in K$ and $K(\sqrt{-1})$ is an algebraic closure of $K$. This is also equialent to the fact that “the” algebraic closure of $K$ is a finite non-trivial extension of $K$. While the algebraic notion adapted to the language of rings is that of an algebraically closed field, the notion of a real closed field is the one which is adapted to the language of ordered rings. In model theoretic terms, the theory of real closed fields is the model companion of the theory of ordered fields.

The analogue of the theorem of Chevalley is the classical theorem of Tarski-Seidenberg:

Theorem (Tarski-Seidenberg). — The theory of real closed fields eliminates quantifiers in the language of ordered rings.

There is a very classical example of this theorem, namely, the resolution of polynomial equation of degree 2. Indeed, in a real closed field, every positive element has a square root (if $a>0$, then the polynomial $T^2-a$ is negative at $0$ and positive at $\max(a,1)$, so that it admits a positive root). The usual algebraic computation thus shows that the formula $\exists x, x^2+ax+b=0$ is equivalent to the formula $a^2-4b\geq 0$.

Corollary 1. — If $M$ is a real closed field and $A$ is a subset of $A$, then $\mathop{\rm Def}(M^n,A)$ is the set of all semi-algebraic subsets of $M$ defined by polynomials with coefficients in $A$.

Corollary 2. — If $M$ is a real closed field, the definable subsets of $M$ are the finite unions of intervals (open, closed or half-open, $\mathopen]a;b\mathclose[$, $\mathopen]a;b]$, $[\mathopen a;b\mathclose[$, $[a;b]$, possibly unbounded, possibly reduced to singletons).

2. O-minimality

The seemingly innocuous property stated in corollary 2 leads to a definition which is surprisingly important and powerful.

Definition. — Let $T$ be the theory of a real closed field $M$ in an expansion $L$ of the language of ordered rings. One says that $T$ is o-minimal if the definable subsets of $M$ are the finite unions of intervals.

It is a non-trivial result that the o-minimality is indeed a property of the theory $T$, and not a property of the model $M$: if it holds, then for every elementary extension $N$ of $M$, the definable subsets of $N$ still are finite unions of intervals.

By the theorem of Tarski-Seidenberg, the theory of real closed fields is o-minimal. The discovery of more complicated o-minimal theories is a remarkable fact from the 80s.

Example. — Let $L_{\mathrm{an},\mathrm{exp}}$ be the language obtained by adjoining to the language $L_{\mathrm{or}}$ of ordered rings symbols of functions $\exp$ and $f$, for every real analytic function $f\colon [0;1]^n\to\mathbf R$. The field of real numbers is viewed as a structure for this language by interpreting $\exp$ as the exponential function from $\mathbf R$ to $\mathbf R$, and every function symbol $f$ as the function from $\mathbf R^n$ to $\mathbf R$ that maps $x$ to $f(x)$ if $x\in [0;1]^n$, and to $0$ otherwise. The theory (denoted $\mathbf R_{\mathrm{an},\mathrm{exp}})$) of $\mathbf R$ in this language is o-minimal.

This is a thorem of van den Dries and Miller; the case of $L_{\mathrm{an}}$ (without the exponential function) had been established Denef and van den Dries, while the case of $L_{\mathrm{exp}}$ is due to Wilkie.

To give a non-example, let us consider the language obtained by adjoining a symbol $\sin$ and view $\mathbf R$ as a structure for this language, the symbol $\sin$ being interpreted as the sine function from $\mathbf R$ to $\mathbf R$. Then the theory of $\mathbf R$ in this language is not o-minimal. Indeed, the set $2\pi\mathbf Z$ is definable by the formula $\sin(x)=0$, but $2\pi\mathbf Z$ has infinitely many connected components, so is not a finite union of intervals.

One motivation for o-minimality is that it realizes (part of) Grothendieck quest towards tame topology as described in his Esquisse d'un programme. Indeed, sets which are definable in an o-minimal structure have many tameness properties:
• The interior, the closure, the boundary of a definable set is definable.
• Every definable set is homeomorphic to (the topological realization) of a simplicial complex
• Every definable set has a celllular decomposition. Precisely, let us call a cell of $\mathbf R^{n+1}$ any subset $C$ of the following form: one is given a definable subset $A$ of $\mathbf R^n$ and definable functions $f,g\colon A\to\mathbf R$ such that $f(x)<g(x)$ for every $x\in A$, and the set $C$ is defined by the condition $x\in A$, and by one of the conditions $t<f(x)$, or $t=f(x)$, or $f(x)<t<g(x)$, or $t>f(x)$.  Then for every finite family $(B_i)$ of definable subsets of $\mathbf R^{n+1}$, there is a finite partition of $\mathbf R^{n+1}$ into cells such that every $B_i$ is a union of cells.
• Every definable function is piecewise smooth.
• Definable continuous functions are definably piecewise trivial (theorem of Hardt): for every function $f\colon X\to Y$ between definable sets which is definable and continuous, there is a finite partition $(Y_i)$ of $Y$ into definable subsets such that the map $f_i\colon f^{-1}(Y_i)\to Y_i$ deduced from $f$ by restriction is isomorphic to a projection $Y_i\times S_i\to Y_i$.

Recently, o-minimality has had spectacular and fantastic applications via the approach of Pila-Zannier to the conjecture of Pink, leading to new proofs of the Manin-Mumford conjecture (Pila-Zannier), and to proofs of the André-Oort conjecture (Pila, Pila-Tsimerman, Klingler-Ullmo-Yafaev), and, more recently, to partial results towards the conjecture of Pink (Gao, Habegger-Pila,...). However, this is not the goal of that post, so let me refer the interested reader to Tom Scanlon's Bourbaki talk on that topic.

3. O-minimal complex analysis

The standard identification of the field $\mathbf C$ of complex numbers with $\mathbf R^2$ (associating with a complex number its real and imaginary parts) allows to talk of complex valued functions (on a subset of $\mathbf C^n$) which are definable in a given language. In a remarkable series of papers, Peterzil and Starchenko have shown that holomorphic functions which are definable in an o-minimal structure possess very rigid properties. Let us quote some of their theorems.

So we fix an expansion of the language $L_{\mathrm{or}}$ of which the field $\mathbf R$ is a structure whose theory is o-minimal. By “definable”, we mean definable in that language. The typical language considered in the applications here is the language $L_{\mathrm{an},\mathrm{exp}}$.

Theorem. — Let $A$ be a finite subset of $\mathbf C$ and let $f\colon \mathbf C\setminus A\to \mathbf C$ be a holomorphic function. If $f$ is definable, then it is a rational function.

Theorem. — Let $V\subset\mathbf C^n$ be a closed analytic subset. If $V$ is definable, then $V$ is algebraic.

Corollary (Theorem of Chow). — Let $V\subset\mathbf P^n(\mathbf C)$ be a closed analytic subset. Then $V$ is algebraic.

Indeed, working on the standard charts of $\mathbf P^n(\mathbf C)$, we see that $V$ is locally definable by analytic functions. By compactness of $\mathbf P^n(\mathbf C)$, it is thus definable in the language $L_{\mathrm{an}}$. Since the theory of $\mathbf R$ in this language is o-minimal, the corollary is a consequence of the previous theorem.

Let us finally give an important example. Let $X$ be an bounded symmetric domain. This means that $X$ is a bounded open subset of $\mathbf C^n$ such that for every point $p\in X$, there exists a biholomorphic involution $f\colon X\to X$ such that $p$ is an isolated fixed point of $f$. This implies that $X$ is a homogeneous space $G/K$ under a semisimple Lie group $G$ which acts by holomorphisms, and $K$ is a maximal compact subgroup of $G$. Moreover, $X$ has a canonical Kähler metric which is invariant under $G$.

The most classical example is given by the Poincaré upper half-plane on which $\mathrm{PGL}(2,\mathbf R)$ acts by homographies; of course, the upper half-plane is not bounded, but is biholomorphic to the open unit disk.

A more sophisticated example is given by the Siegel upper half-plane or, rather, its bounded version. That is, $X$ is the set of $n\times n$ symmetric complex matrices $Z$ such that $\mathrm I_n-Z^* Z$ is positive definite. It is a homogeneous space for the symplectic group $\mathrm{Sp}(2n,\mathbf R)$; the fixator of $Z=0$ is the unitary group $U(n)$.

Let now $\Gamma$ be an arithmetic subgroup of $\mathrm{Sp}(2n,\mathbf R)$; for example, let us take $\Gamma$ be a subgroup of finite index of $\mathrm{Sp}(2n,\mathbf Z)$. Then the quotient $S=X/\Gamma$ admits a structure of an analytic set and the projection $p\colon X\to S$ is an analytic map. If $\Gamma$ is “small enough” (torsion free, say), then $S$ is even complex manifold manifold, and $p$ is a covering. An important and difficult theorem of Baily-Borel asserts that $S$ is an algebraic variety.

In fact, it is classical in this context that there exist Siegel sets, which are explicit subsets $F$ of $X$ such that $\Gamma\cdot F=X$ and such that the set of $\gamma\in\Gamma$ such that $\gamma\cdot F\cap F\neq\emptyset$ is finite. So Siegel sets are almost fundamental domains. An important remark is that they are semi-algebraic, that is, definable in the language of ordered rings. For example in the upper half-plane, one may take $F$ to be the set of all $z\in\mathbf C$ such that $-\frac12\leq \Re(z)\leq \frac12$ and $\Im(z)\geq \sqrt 3/2$. One may even take “fundamental sets” (which are fundamental domains up to something of empty interior) such as the one defined by the inequalities $-\frac12\leq \Re(z)\leq\frac12$ and $\lvert z\rvert \geq1$.

Peterzil and Starchenko have proved that there restriction to $F$ of the projection $p$ is definable in the language $L_{\mathrm{an},\mathrm{exp}}$. An immediate consequence is that $S$ is definable in this language, hence is algebraic.

These results have been generalized by Klinger, Ullmo and Yafaev to any bounded symmetric domain. This is an important technical part of their proof of the hyperbolic Ax-Lindemann conjecture.

## Saturday, May 2, 2015

### Model theory and algebraic geometry, 2 — Definable sets, types; quantifier elimination

This is the second post in a series of 4 devoted to the exposition of interactions between model theory and algebraic geometry. In the first one, I explained the notions of language, structures and theories, with examples taken from algebra. Here, I shall discuss the notion of definable set, of types, as well as basic results from dimension theory ($\omega$-stability).

So we fix a theory $T$ in a language $L$. A definable set is defined, in a given model $M$ of $T$, by a formula. More precisely, we consider definable sets in cartesian powers $M^n$ of the model $M$, which can be defined by a formula in $n$ free variables with parameters in some subset $A$ of $M$. By definition, such a formula is a formula of the form $\phi(x;a)$, where $\phi(x;y)$ is a formula in $n+m$ free variables, split into two groups $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_m)$ and $a=(a_1,\dots,a_m)\in A^m$ is an $m$-tuple of parameters; the formula $\phi(x;y)$ can have quantifiers and bounded variables too. Given such a formula, we define a subset $[\phi(x;a)]$ of $M^n$ by $\{ x\in M^n\mid \phi(x;a)\}$. We write $\mathrm{Def}(M^n;A)$ for the set of all subsets of $M^n$ which are definable with parameters in $A$.

Let us give examples, where $L$ is the language of rings and $T$ is the theory $\mathrm{ACF}$ of algebraically closed fields:
• $V_1=\{x\mid x\neq 0 \}\subset M$, given by the formula “$x\neq 0$” with 1 variable and $0$ parameter;
• $V_2=\{x\mid \exists y, 2xy=1\} \subset M$, given by the formula “$\exists y, 2xy=1$” with 1 free variable $x$, and one bounded variable $y$;
• $V_3=\{(x,y)\mid x^2+\sqrt 2 y^2=\pi \}\subset \mathbf C^2$, where the model $\mathbf C$ is the field of complex numbers, $\phi((x,y),(a,b))$ is the formula $x^2+ay^2=b$ in 4 free variables, and the parameters are given by $(a,b)=(\sqrt 2,\pi)$.
Theorem (Chevalley). — Let $L$ be the language of rings, $T=\mathrm{ACF}$ and $M$ be an algebraically closed field; let $A$ be a subset of $M$. The set $\mathrm{Def}(M^n;A)$ is the smallest boolean algebra of subsets of $M^n$ which contains all subsets of $M^n$ of the form $[P(x;a)]$ where $P$ is a polynomial in $n+m$ variables with coefficients in $\mathbf Z$ and $a=(a_1,\dots,a_m)$ is an $m$-tuple of elements of $A$. In other words, a subsets of $M^n$ is definable with parameters in $A$ if and only if it is constructible with parameters in $A$.

The reason behind this theorem is the following set-theoretic interpretation of quantifiers and logical connectors. Precisely, if $\phi$ is a formula in $n+m+p$ variables, and $a\in A^p$, the definable subset $[\exists y \phi(x,y,a)]$ of $M^n$ coincides with the image of the definable subset $[\phi(x,y;a)]$ of $M^{n+m}$ under the projection $p_x \colon M^{n+m}\to M^n$. Similarly, if $\phi(x)$ and $\psi(x)$ are two formulas in $n$ free variables, then the definable subset $[\phi(x)\wedge\psi(x)]$ is the union of the definable subsets $[\phi(x)]$ and $[\psi(x)]$. And if $\phi(x)$ is a formula in $n$ variables, then the definable subset $[\neg\phi(x)]$ is the complement in $M^n$ of the definable subset $[\phi(x)]$.

For example, the subset $V_2=[\exists y, 2xy=1]$ defined above can also be defined by $M\setminus [2x=0]$.

One says that the theory ACF admits elimination of quantifiers: modulo the axioms of algebraically closed fields, every formula of the language $L$ is equivalent to a formula without quantifiers.

An important consequence of this property is that for every extension $M\hookrightarrow M'$ of models of ACF, the theory of $M'$ is equal to the theory of $M$—one says that every extension of models is elementary.

Let $p$ be either $0$ or a prime number. Observe that every algebraically closed field of characteristic $p$ is an extension of $\overline{\mathbf Q}$ if $p=0$, or of $\overline{\mathbf F_p}$ if $p$ is a prime number. As a consequence, for every characteristic $p\geq0$, the theory $\mathrm{ACF}_p$ of algebraically closed fields of characteristic $p$ (defined by the axioms of $\mathrm{ACF}$, and  the axiom $1+1+\dots+1=0$ that the characteristic is $p$ if $p$ is a prime number, or the infinite list of axioms that assert that the characteristic is $\neq \ell$, if $p=0$) is complete: this list of axioms determines everything that can be said about algebraically closed fields of characteristic $p$.

Definition. — Let $a\in M^n$ and let $A$ be a subset of $M$. The type of $a$ (with parameters in $A$) is the set $\mathrm{tp}(a/A)$ of all formulas $\phi(x;b)$ in $n$ free variables with parameters in $A$ such that $\phi(a;b)$ holds in the model $M$.

Definition. — Let $A$ be a subset of $M$. For every integer $n\geq 0$, the set $S_n(A)$ of types (with parameters in $A$) is the set of all types $\mathrm{tp}(a/A)$, where $N$ is an extension of $M$ which is a model of $T$ and $a\in N^n$. One then says that this type is realized in $N$.

Gödel's completeness theorem allows us to give an alternative description of $S_n(A)$. Namely, let $p$ be a set of formulas in $n$ free variables and parameters in $A$ which contains the diagram of $A$ (that is, all formulas which involve only elements of $A$ and are true in $M$). Assume that $p$ is consistent (there exists a model $N$ which is an extension of $M$ and and element $a\in M^n$ such that $\phi(a)$ holds in $N$ for every $\phi\in p$) and maximal (for every formula $\phi\not\in p$, then for every model $N$ and every $a\in N^n$ such that $p\subset \mathrm{tp}(a/A)$, then $\phi(a)$ does not hold). Then $p\in S_n(A)$.

For every formula $\phi\in L(A)$ in $n$ free variables and parameters in $A$, let $V_\phi$ be the set of types $p\in S_n(A)$ such that $\phi\in p$. Then the subsets $V_\phi$ of $S_n(A)$ consistute a basis of open sets for a natural topology on $S_n(A)$.

Theorem. — The topological space $S_n(A)$ is compact and totally discontinuous.

Let us detail the case of the theory ACF in the langage of rings. I claim that if $K$ is a field, then $S_n(K)$ is homeomorphic to the spectrum $\mathop{\rm Spec}(K[T_1,\dots,T_n])$ endowed with its constructible topology. Concretely, for every algebraically closed extension $M$ of $K$ and every $a\in M^n$, the homeomorphism $j$ maps $\mathrm{tp}(a/K)$ to the prime ideal $\mathfrak p_a$ consisting of all polynomials $P\in K[T_1,\dots,T_n]$ such that $P(a)=0$.

A type $p=\mathrm{tp}(a/K)$ is isolated if and only if the prime ideal $\mathfrak p_a$ is maximal. Consequently, if $n=1$, there is exactly one non-isolated type in $S_1(K)$, corresponding to the generic point of the spectrum $\mathop{\rm Spec}(K[T])$.

As for any compact topological space, a space of types can be studied via its Cantor-Bendixson analysis, which is a decreasing sequence of subspaces, indexed by ordinals, defined by transfinite induction. First of all, for every topological space $X$, one denotes by $D(X)$ the set of all non-isolated points of $X$. One then defines $X_0=X$, $X_{\alpha}=D(X_\beta)$ if $\alpha=\beta+1$ is a successor-ordinal, and $X_\alpha=\bigcap_{\beta<\alpha} X_\beta$ if $\alpha$ is a limit-ordinal. For $x\in X$, the Cantor-Bendixson rank of $x$ is defined by $r_{CB}(x)=\alpha$ if $x\in X_\alpha$ and $x\not\in X_\beta$ for $\beta>\alpha$, and $r_{CB}(x)=\infty$ if $x\in X_\alpha$ for every ordinal $\alpha$. The set of points of infinite rank is the largest perfect subset of $X$.

Let us return to the example of the theory ACF. If a type $p\in S_n(K)$ corresponds to a prime ideal $\mathfrak p=j(p)$ of $\mathop{\rm Spec}(K[T_1,\dots,T_n])$, its Cantor-Bendixson rank is the Zariski dimension of $V(I)$. More generally, if $F$ is a constructible subset of $\mathop{\rm Spec}(K[T_1,\dots,T_n])$, then $r_{CB}(F)$ is the Zariski-dimension of the Zariski-closure of $F$. Moreover, the points of maximal Cantor-Bendixson rank correspond to the generic points of the irreducible components of maximal dimension; in particular, there are only finitely many of them.

Definition. — One says that a theory $T$ is $\omega$-stable if for every finite or countable set of parameters $A$, the space of 1-types $S_1(A)$ is finite or countable.

The theory ACF is $\omega$-stable. Indeed, if $K$ is the field generated by $A$, then $K[T]$ being
a countable noetherian ring, it has only countably many prime ideals.

Since any non-empty perfect set is uncountable, one has the following lemma.

Lemma. — Let $T$ be an $\omega$-stable theory and let $M$ be a model of $T$. Then the Cantor-Bendixson rank of every type $x\in S_n(M)$ is finite.

Let us assume that $T$ is $\omega$-stable and let $F$ be a closed subset of $S_n(M)$. Then $r_{CB}(F)=\sup \{ r_{CB}(x)\,;\, x\in F\}$ is finite, and the set of points $x\in F$ such that $r_{CB}(x)=r_{CB}(F)$ is finite and non-empty.

This example gives a strong indication that the model theory approach may be extremly fruitful for the study of algebraic theories whose geometry is not as well developed than algebraic geometry.

## Thursday, April 23, 2015

### Model theory and algebraic geometry, 1 — Structures, languages, theories, models

Last november, I had been invited to lecture at the GAGC conference on the use of model theoretic methods in algebraic geometry. In the last two decades, important results of “general mathematics” have been proved using sophisticated techniques, see for example Hrushovski's proofs of the Manin-Mumford and of the Mordell-Lang conjecture over function fields, or Chatzidakis-Hrushovski's proof of a descent result in algebraic dynamics (generalizing a theorem of Néron for abelian varieties), or Hrushovski-Loeser's approach to the topology of Berkovich spaces, or Medvedev-Scanlon's results on invariant varieties in polynomial dynamics, or Hrushovski's generalization of the Lang-Weil estimates, or the applications to the André-Oort conjecture (by Pila and others) of a theorem of Pila-Wilkie in o-minimal geometry... All these wonderful results were however too complicated to be discussed from scratch in this series of lectures and I decided to discuss a beautiful paper of Scanlon that “explains” why coverings from analytic geometry lead to algebraic differential equations.
There will be 4 posts:
1. Structures, languages, theories, models (this one)
2. Definable sets, types, quantifier elimination
3. Real closed fields and o-minimality
4. Elimination of imaginaries
Model theory — a branch of mathematical logic — has two aspects:
• The first one, that one could name “pure”, studies mathematical theories as mathematical objects. It introduced important concepts, such as quantifier elimination, elimination of imaginaries, types and their dimensions, stability theory, Zariski geometries, and provides a rough classification of mathematical theories.
• The second one is “applied”: it studies classical mathematical theories using these tools. It may be for algebraic theories, such as fields, differential fields, valued fields, ordered groups or fields, difference fields, etc., that it works the best, and for theories which are primitive enough so that they escape indecidability à la Gödel.
Let us begin with an empirical observation; classical mathematical theories feature:
• sets (which may be receptacles for groups, rings, fields, modules, etc.);
• functions and relations between those sets (composition laws, order relations, equality);
• certain axioms which are well-formed formulas using these functions, these relations, basic logical symbols ($\forall$, $\exists$, $\vee$, $\wedge$, $\neg$) or their variants ($\Rightarrow$, $\Leftrightarrow$, $\exists!$, etc.).
Model theory (to be precise, first-order model theory) introduces the concepts of a language (the letters and symbols that allow to express a mathematical theory), of a theory (sets of formulas in a given language, using a fixed infinite supply of variables), of a structure (sets, functions and relations that allow to interpret all formulas in the language) and finally of a model of a theory (a structure where the formulas of the given theory are interpreted as true). The theory of a structure is the set of all formulas which are interpreted as true. A morphism of structures is a map which is compatible with all the given relations.

Let us give three examples from algebra: groups, fields, differential fields

a) Groups

The language of groups has one symbol $\cdot$ which represents a binary law. Consequently, a structure for this language is just a set $S$ together with a binary law $S\times S\to S$. In this language, one can axiomatize groups using two axioms:
• Associativity: $\forall x \forall y \forall z \quad x\cdot (y\cdot z)= (x\cdot y)\cdot z$
• Existence of a neutral element and of inverses: $\exists e\forall x \exists y \quad (x\cdot e=e\cdot x \wedge x\cdot y=y\cdot x=e)$.
Observe that in writing these formulas, we allow ourselves the usual shortcuts to which we are used as mathematicians. In fact, the foundations of model theory require to spend a few pages to discuss how formulas should be written, with or without parentheses, that they can be unambiguously read, etc.

However, it may be more useful to study groups in a language with 3 symbols $\cdot,e,i$, where $\cdot$ represents the binary law, $e$ the neutral element and $i$ the inversion. Then a structure is a set together with a binary law, a distinguished element and a self-map; in particular, what is a structure depends on the language. In this new language, groups are axiomatized with three axioms:
• Associativity as above;
• Neutral element: $\forall x \quad x\cdot e=e\cdot x=x$;
• Inverse: $\forall x\quad x\cdot i(x)=i(x)\cdot x=e$.
The two theories of groups are essentially equivalent: one can translates any formula of the first language into the second, and conversely. Indeed, if a formula of the second language involves the symbols $e$, it suffices to copy $\exists e x\cdot e=e\cdot x$ in front of it; and if a formula involves $i(x)$, it suffices to add $\exists y$ in front of it, as well as the requirement $x\cdot y=y\cdot x=e$, and to replace $i(x)$ by $y$. Since the neutral element and the inverse law of a group are unambiguously defined by the composition law, this shows that the new formula is equivalent, albeit longer and less practical, to the initial one.

The possibility of interpreting a theory in a language in a second language is a very important tool in mathematical logic.

b) Rings

The language used to study rings has 5 symbols: $+,-,0,1,\cdot$. In this language, structures are just sets with three binary laws and two distinguished elements. One can of course axiomatize rings, using the well-known formulas that express that the law $+$ is associative and commutative, that $0$ is a neutral element and that $-$ gives subtraction, that the law $\cdot$ is associative and commutative with $1$ as a neutral element, and that the multiplication $\cdot$ distributes over addition.

Adding the axioms $\forall x (x\neq 0 \Rightarrow \exists y \quad xy=1)$ and $1\neq 0$ gives rise to fields.

That a field has characteristic 2, say, is axiomatized by the formula $1+1=0$, that it has characteristic 3 is axiomatized by the formula $1+1+1=0$, etc. That a field has characteristic 0 is axiomatized by an infinite list of axiom, one for each prime number $p$, saying that $1+1+\cdots+1\neq 0$ (with $p$ symbols $1$ on the left). We will see below why fields of characteristic 0 must be axiomatized by infinitely  axioms.

That a field is algebraically closed means that every monic polynomial has a root. To express this property, one needs to write down all possible polynomials. However, the language of rings does not give us access to integers, nor to sets of polynomials. Consequently, we must write down an infinite list of axioms, one for each positive integer $n$: $\forall x_1\forall x_2\cdots \forall x_n \exists y \quad y^n+x_1 y^{n-1}+\cdots+x_{n-1}y+x_n=0$. Here $y^m$ is an abbreviation for the product $y\cdot y \cdots y$ of $m$ factors equal to $y$.

As we will see, the language of rings and the theory ACF of algebraically closed fields is well suited to study algebraic geometry.

c) Differential fields

A differential ring/field is a ring/field $A$ endowed with a derivation $\partial\colon A\to A$, that is, with an additive map satisfying the Leibniz relation $\partial(ab)=a\partial(b)+b\partial(a)$. They can be naturally axiomatized in the language of rings augmented with a symbol $\partial$.

There is a notion of a differentially closed field, analogous to the notion of an algebraically closed field, but encompassing differential equations. A differential field is differentially closed if any differential equation which has a solution in some differential extension already has a solution. This property is analogous to the consequence of Hilbert's Nullstellensatz according to which a field is algebraically closed if any system of polynomial equations which has a solution in an extension already has a solution. At least in characteristic zero, Robinson showed that their theory DCF$_0$ can be axiomatized by various families of axioms. For example, the one devised by Blum asserts the existence of an element $x$ such that $P(x)=0$ and $Q(x)\neq0$, for every pair $(P,Q)$ of non-zero differential polynomials in one indeterminate such that the order of $Q$ is strictly smaller than the order of $P$. This study requires the development of important and difficult results in differential algebra due to Ritt and Seidenberg.

At this level, there are two important basic theorems to mention: Gödel completeness theorem, and the theorems of Löwenheim-Skolem.

Completeness theorem (Gödel). — Let $T$ be a theory in a language $L$. Assume that every finite subset $S$ of $T$ admits a model. Then $T$ admits a model.

There are two classical proof of this theorem.

The first one uses ultraproducts and consists in choosing a model $M_S$ for every finite subset $S$ of $T$. Let then $\mathcal U$ be a non-principal ultrafilter on the set of finite subsets of $T$ and let $M$ be the ultraproduct of the family of models $(M_S)$. It inherits functions and relations from those of the models $M_S$, so that it is a structure in the language $L$. Moreover, one deduces from the definition of an ultrafilter that for every axiom $\alpha$ of $T$, the structure $M$ satisfies the axiom $\alpha$. Consequently, $M$ is a model of $T$.

A second proof, due to Henkin, is more syntactical. It considers the set of all terms in the language $L$ (formulas without logical connectors), together with an equivalence relation that equates two terms for which some axiom says that they are equal, and with symbols representing objets of which an axiom affirms the existence. The quotient set modulo the equivalence relation is a model. In essence, this proof is very close to the construction of a free group as words.

It is important to obseve that the proof of this theorem uses the existence of non-principal ultraproducts, which is a weak form of the axiom of choice. In fact, as in all classical mathematics, the axiom of choice — and set theory in general — is used in model theory to establish theorems. That does not prevent logicians to study the model theory of set theory without choice as a particular mathematical theory, but even to do that, one uses choice.

Theorem of Löwenheim-Skolem.Let $T$ be a theory in a language $L$. If it admits an infinite model $M$, then it admits a model in every cardinality $\geq \sup(\mathop{\rm Card}(L),\aleph_0)$.

To show the existence of a model of cardinality $\geq\kappa$, one enlarges the language $L$ and the theory $T$ by adding symbols $c_i$, indexed by a set of cardinality $\kappa$, and the axioms $c_i\neq c_j$ if $i\neq j$, giving rise to a theory $T'$ in a language $L'$. A structure for $L'$ is a structure for $L$ together with distinguished elements $c_i$; such a structure is a model of $T'$ if and only if it is a model of $T$ and if the elements $c_i$ are pairwise disintct. If the initial theory $T$ has an infinite model, then this model is a model of every finite fragment of the theory $T'$, because there are only finitely many axioms of the form $c_i\neq c_j$ to satisfy, and the model is assumed to be infinite. By Gödel's completeness theorem, the theory $T'$ has a model $M'$; forgetting the choice of distinguished elements, $M'$  is a model of the theory $T$, but the mere existence of the elements $c_i$ forces its cardinality to be at least $\kappa$.

To show that there exists a model of cardinality exactly $\kappa$ (assumed to be larger than $\sup(\mathop{\rm Card}(L),\aleph_0)$), one starts from a model $M$ of cardinality $\geq\kappa$ and defines a substructure by induction, starting from the constant symbols and adding step by step only the elements which are required by the function symbols, the axioms and the elements already constructed. This construction furnishes a model of $T$ whose cardinality is equal to $\kappa$.

## Monday, March 23, 2015

### When Lagrange meets Galois

Jean-Benoît Bost told me a beautiful proof of the main ingredient in the proof of Galois correspondence, which had been published by Lagrange in his 1772 “Réflexions sur la résolution des résolutions algébriques”, almost 60 years before Galois. (See Section 4 of that paper, I think; it is often difficult to recognize our modern mathematics in the language of these old masters.)

In modernized notations, Lagrange considers the following situation. He is given a polynomial equation $T^n + a_{n-1} T^{n-1}+\cdots + a_0 = 0$, with roots $x_1,\dots,x_n$, and two “rational functions” of its roots  $f(x_1,\dots,x_n)$ and $\phi(x_1,\dots,x_n)$. (This means that $f$ and $\phi$ are the evaluation at the $n$-tuple $(x_1,\dots,x_n)$ of two rational functions in $n$ variables.) Lagrange says that $f$ and $\phi$ are similar (“semblables”) if every permutation of the roots which leaves $f(x_1,\dots,x_n)$ unchanged leaves $\phi(x_1,\dots,x_n)$ unchanged as well (and conversely). He then proves that $\phi(x_1,\dots,x_n)$ is a rational function of $a_0,\dots,a_{n-1}$ and $f(x_1,\dots,x_n)$.

Let us restate this in a more modern language. Let $K\to L$ be a finite Galois extension of fields, in the sense that $K= L^{G}$, where $G=\mathop{\rm Aut}_K(L)$. Let $a, b\in L$ and let us assume that every element $g\in G$ which fixes $a$ fixes $b$ as well; then Lagrange proves that $b\in K(a)$.

Translated in our language, his proof could be as follows. In formula, the assumption is that $g\cdot a=a$ implies $g\cdot b=b$; consequently, there exists a unique *function* $\phi\colon G\cdot a\to G\cdot b$ which is $G$-equivariant and maps $a$ to $b$. Let $d=\mathop{\rm Card}(G\cdot a)$ and let us consider Lagrange's interpolation polynomial —the unique polynomial $P\in L[T]$ of degree $d$ such that $P(x)=\phi(x)$ for every $x\in G\cdot a$. If $h\in G$, the polynomial $P^h$ obtained by applying $h$ to the coefficients of $P$ has degree $d$ and coincides with $\phi$; consequently, $P^h=P$. By the initial assumption, $P$ belongs to $K[T]$ and $b=P(a)$, hence $b\in K(a)$, as claimed.

Combined with the primitive element theorem, this allows to give another short, and fairly elementary, presentation of the Galois correspondence.

## Saturday, February 28, 2015

### Galois Theory, Geck's style

This note aims at popularizing a short note of Meinolf Geck, On the characterization of Galois extensions, Amer. Math. Monthly 121 (2014), no. 7, 637–639 (Article, Math Reviews, arXiv), that proposes a radical shortcut to the treatment of Galois theory at an elementary level. The proof of the pudding is in the eating, so let's see how it works. The novelty lies in theorem 2, but I give the full story so as to be sure that I do not hide something under the rug.

Proposition 1. Let $K\to L$ be a field extension. Then $L$ is not the union of finitely many subfields $M$ such that $K\to M\subsetneq L$.
Proof. It splits into two parts, according whether $K$ is finite or infinite.

Assume that $K$ is finite and let $q=\mathop{\rm Card}( K)$. Then $L$ is finite as well, and let $n=[L:K]$ so that $\mathop{\rm Card}(L)=q^n$. If $M$ is a subextension of $L$, then $\mathop{\rm Card}( L)=q^m$, for some integer $m$ dividing $n$; moreover, $x^{q^m}=x$ for every $x\in L$. Then the union of all strict sub-extensions of $L$ has cardinality at most $\sum_{m=1}^{n-1} q^m =\frac{q^n-q}{q-1}<q^n$.

It remains to treat the case where $K$ is infinite; then the proposition follows from the fact that a finite union of strict subspace of a $K$-vector space $E$ is not equal to $E$. Let indeed $(E_i)_{1\leq i\leq n}$ be a family of strict subspaces of $E$ and let us prove by induction on $n$ that $E\neq \bigcup_{i=1}^n E_i$. The cases $n\leq1$ are obvious. By induction we know that for every $j\in\{1,\dots,n\}$, the union $\bigcup_{i\neq j}E_i$ is distinct from $E$, hence select an element $x\in E$ such that $x\not\in E_2\cup \dots\cup E_n$. The desired result follows if, by chance, $x\not\in E_1$. Otherwise, choose $y\in E\setminus E_1$. For $s\neq t\in K$, and $i\in\{2,\dots,n\}$, observe that $y+sx$ and $y+tx$ cannot both belong to $E_i$, for this would imply that $(s-t)x\in E_i$, hence $x\in E_i$ since $s\neq t$. Consequently, there are at most $n-1$ elements $s\in K$ such that $y+sx\in \bigcup_{i=2}^nE_i$. Since $K$ is infinite, there exists $s\in K$ such that $y+sx\not\in\bigcup_{i=2}^n E_i$. Then $y+sx\not\in E_1$, neither, since $x\in E_1$ and $y\not\in E_1$. This proves that $E\neq \bigcup_{i=1}^nE_i$.

Let $K\to L$ be a field extension and let $P\in K[T]$. We say that $P$ is split in $L$ if it is a product of linear factors in $L[T]$. We say that $P$ is separable if all of its roots (in some extension where it is split) have multiplicity $1$. We say that $K\to L$ is a splitting extension of $P$ if $P$ is split in $L$ and if $L$ is the subextension of $K$ generated by the roots of $P$ in $L$. Finally, we let $\mathop{\rm Aut}_K(L)$ be the set of $K$-linear automorphisms of $L$; it is a group under composition.

Theorem 2. Let $K\to L$ be a finite extension of fields and let $G=\mathop{\rm Aut}_K(L)$. Then $\mathop{\rm Card}( G)\leq [L:K]$. Moreover, the following conditions are equivalent:

1. One has $\mathop{\rm Card}( G)=[L:K]$;
2. There exists an irreducible separable polynomial $P\in K[T]$ such that $\deg(P)=[L:K]$ and which is split in $L$;
3. The extension $K\to L$ is a splitting extension of a separable polynomial in $K[T]$;
4. One has $K=L^G$.

Remark 3. In the conditions of (2), let us fix a root $z\in L$ of $P$. One has $L=K(z)$. Moreover, the map $f\mapsto f(z)$ is a bijection from $\mathop{\rm Aut}_K(L)$ to the set of roots of $P$ in $L$.

Proof of Theorem 2.
(a) Let us prove that $\mathop{\rm Card} (G)\leq [L:K]$. Let $m\in\mathbf N$ be such that $m\leq \mathop{\rm Card}( G)$ and let $\sigma_1,\dots,\sigma_m$ be distinct elements of $G$. For $1\leq i<j\leq m$, let $M_{i,j}$ be the subfield of $L$ consisting of all $x\in L$ such that $\sigma_i(x)=\sigma_j(x)$. It is a strict subextension of $L$ because $\sigma_i\neq\sigma_j$. Consequently, $L$ is not the union of the subfields $M_{i,j}$ and there exists an element $z\in L$ such that $\sigma_i(z)\neq \sigma_j(z)$ for all $i\neq j$. Let $P$ be the minimal polynomial of $z$. Then the set $\{\sigma_1(z),\dots,\sigma_m(z)\}$ consists of distinct roots of $P$, hence $\deg(P)\geq m$. In particular, $m\leq [L:K]$. Since this holds for every $m\leq \mathop{\rm Card}( G)$, this shows that $\mathop{\rm Card}( G)\leq [L:K]$.

(b) If one has $\mathop{\rm Card}( G)=[L:K]$, then taking $m=\mathop{\rm Card}( G)$, we get an irreducible polynomial $P\in K[T]$ of degree $m$, with $m$ distinct roots in $L$. Necessarily, $P$ is separable and split in $L$. This gives (1)$\Rightarrow$(2).

The implication (2)$\Rightarrow$(3) is obvious.

(1)$\Rightarrow$(4). Let $M=L^G$. One has $\mathop{\rm Aut}_K(L)=\mathop{\rm Aut}_M(L)=G$. Consequently, $\mathop{\rm Card}(G)\leq [L:M]$. Since $\mathop{\rm Card}( G)=[L:K]=[L:M][M:K]$, this forces $M=K$.

(4)$\Rightarrow$(3). There exists a $G$-invariant subset $A$ of $L$ such that $L=K(A)$. Then $P=\prod_{a\in A}(T-a)$ is split in $L$, and is $G$-invariant. Consequently, $P\in K[T]$. By construction, $P$ is separable and $L$ is a splitting extension of $P$.

(3)$\Rightarrow$(1). Let $M$ be a subextension of $L$ and let $f\colon M\to L$ be a $K$-morphism. Let $a\in A$ and let $Q_a$ be the minimal polynomial of $a$ over $M$. The association $g\mapsto g(a)$ defines a bijection between the set of extensions of $f$ to $M(a)$ and the set of roots of $Q_a$ in $L$. Since $P(a)=0$, the polynomial $Q_a$ divides $P$, hence it is separable and split in $L$. Consequently, $f$ has exactly $\deg(Q_a)=[M(a):M]$ extensions to $M(a)$.

By a straightforward induction on $\mathop{\rm Card}(B)$, for every subset $B$ of $A$, the set of $K$-morphisms from $K(B)$ to $L$ has cardinality $[K(B):K]$. When $B=A$, every such morphism is surjective, hence $\mathop{\rm Card}(\mathop{\rm Aut}_K(L))=[L:K]$.

If these equivalent conditions hold, we say that the finite extension $K\to L$ is Galois.

Corollary 4. Let $K\to L$ be a finite Galois extension. The maps $H\to L^H$ and $M\to \mathop{\rm Aut}_M(L)$ are bijections, inverse one of the other, between subgroups of $\mathop{\rm Aut}_K(L)$ and subextensions $K\to M\subset L$.
Proof. a) For every subextension $K\to M\subset L$, the extension $M\subset L$ is Galois. In particular, $M=L^{\mathop{\rm Aut}_M(L)}$ and $\mathop{\rm Aut}_M(L)=[L:M]$.

b) Let $H\subset\mathop{\rm Aut}_K(L)$ and let $M=L^H$. Then $M\to L$ is a Galois extension and $[L:M]=\mathop{\rm Aut}_M(L)$; moreover, one has $H\subset\mathop{\rm Aut}_M(L)$ by construction. Let us prove that $H=\mathop{\rm Aut}_M(L)$. Let $z\in L$ be any element whose minimal polynomial $P_z$ over $M$ is split and separable in $L$. One has $\mathop{\rm Card}(\mathop{\rm Aut}_M(L))=\deg(P_z)$. On the other hand, the polynomial $Q_z=\prod_{\sigma\in H}(T-\sigma(z))\in L[T]$ divides $P_z$ and is $H$-invariant, hence it belongs to $L^H[T]=M[T]$. This implies that $P_z=Q_z$, hence $\mathop{\rm Card}(H)=\deg(P_z)=\mathop{\rm Card}(\mathop{\rm Aut}_M(L))$. Consequently, $H=\mathop{\rm Aut}_M(L)$.

Corollary 5. Let $K\to L$ be a Galois extension and let $K\to M\to L$ be an intermediate extension. The extension $M\to L$ is Galois too. Moreover, the following assertions are equivalent:

1. The extension $K\to M$ is Galois;
2. $\mathop{\rm Aut}_M(L)$ is a normal subgroup of $\mathop{\rm Aut}_K(L)$;
3. For every $\sigma\in\mathop{\rm Aut}_K(L)$, one has $\sigma(M)\subset M$.

Proof. (a) Let $P\in K[T]$ be a separable polynomial of which $K\to L$ is a splitting field. Then $M\to L$ is a splitting extension of $P$, hence $M\to L$ is Galois.

(b) (1)$\Rightarrow$(2): Let $\sigma\in \mathop{\rm Aut}_K(L)$. Let $z$ be any element of $M$ and let $P\in K[T]$ be its minimal polynomial. One has $P(\sigma(z))=\sigma(P(z))=0$, hence $\sigma(z)$ is a root of $P$; in particular, $\sigma(z)\in M$. Consequently, the restriction of $\sigma$ to $M$ is a $K$-morphism from $M$ to itself; it is necessarily a $K$-automorphism. We thus have defined a map from $\mathop{\rm Aut}_K(L)$ to $\mathop{\rm Aut}_K(M)$; this map is a morphism of groups. Its kernel is $\mathop{\rm Aut}_M(L)$, so that this group is normal in $\mathop{\rm Aut}_K(L)$.

(2)$\Rightarrow$(3): Let $\sigma\in\mathop{\rm Aut}_K(L)$ and let $H=\sigma\mathop{\rm Aut}_M(L)\sigma^{-1}$. By construction, one has $\sigma(M)\subset L^G$. On the other hand, the hypothesis that $\mathop{\rm Aut}_M(L)$ is normal in $\mathop{\rm Aut}_K(L)$ implies that $G=\mathop{\rm Aut}_M(L)$, so that $L^G=M$. We thus have proved that $\sigma(M)\subset M$.

(3)$\Rightarrow$(1): Let $A$ be a finite subset of $M$ such that $M=K(A)$ and let $B$ be its orbit under $\mathop{\rm Aut}_K(L)$. The polynomial $\prod_{b\in B}(T-b)$ is separable and invariant under $\mathop{\rm Aut}_K(L)$, hence belongs to $K[T]$. By assumption, one has $B\subset M$. This implies that $K\to M$ is Galois.

Remark 6. Let $L$ be a field, let $G$ be a finite group of automorphisms of $L$ and let $K=L^G$. Every element $a$ of $L$ is algebraic and separable over $K$; inded, $a$ is a root of the separable polynomial $\prod_{b\in G\cdot a}(T-b)=0$, which is $G$-invariant hence belongs to $K[T]$. There exists a finite extension $M$ of $K$, contained in $L$, such that $G\cdot M=M$ and such that the map $\mathop{\rm Aut}_K(L)\to \mathop{\rm Aut}_K(M)$ is injective. Then $K\to M$ is Galois, and $G=\mathop{\rm Aut}_K(M)$. Indeed, one has $G\subset\mathop{\rm Aut}_K(M)$, hence $K\subset M^{\mathop{\rm Aut}_K(M)}\subset M^G\subset L^G=K$. This implies that $K\to M$ is Galois and the Galois correspondence then implies $G=\mathop{\rm Aut}_K(M)$. The argument applies to every finite extension of $K$ which contains $M$. Consequently, they all have degree $\mathop{\rm Card}(G)$; necessarily, $L=M$.

Remark 7 (editions). Matt Baker points out that the actual novelty of the treatment lies in theorem 2, the rest is standard. Also, remark 6 has been edited following an observation of Christian Naumovic that it is not a priori obvious that the extension $K\to L$ is finite.

## Monday, January 26, 2015

I just had the chance to attend two sets with Vijay Iyer and Wadada Leo Smith tonight! That happened at The Stone, a small music room in NYC owned by John Zorn that features avant-garde jazz music (but not only).

The first set was a plain duet of these two artists. The Stone was packed and we had to sit on the floor. After 10 quite boring minutes during which Vijay played electronics only, he took on the piano and music emerged. Although Vijay had sheets of music prepared, this set sounded very free, especially concerning Wadada Leo Smith's playing—it seems he used all what a trumpet allows to create sound. However, the atmosphere was peaceful. For those who know some of Wadada Leo Smith's music, this was closer to Kulture Jazz than to Ten Freedom Summers which I had discussed on this blog last year

For the second set, came along Reggie Workman at the bass, Nitin Mitta on the tablas, and Patricia Franceschy on vibes. This made the music sound quite differently. The musicians had decided of a few melodic lines and ostinatos, and grooved on that. The tablas gave a wonderful color to the music, similar as the one on Tirtha (with Prasanna on the guitar, and Nitin Mitta on the tablas). The vibes also gave a good touch. It seems that there are nice vibes players in free jazz nowadays; I'm thinking for example of Jason Adasiewicz who plays in Nicole Mitchelle's Ice Crystals group.

It was my first night in New York City since 2 years. I am happy to have had the opportunity to hear these great artists. Tomorrow night, if the announced snow storm permits, I'll go listen to Ari Hoenig at the Smalls!

## Wednesday, October 1, 2014

### Book review: Contemplative Practices in Higher Education

Contemplative Practices in Higher Education, by Daniel P. Barbezat and Mirabai Bush.
Jossey-Bass, 2014.
[Center for Contemplative Mind in Society] [Library of Congress] [Amazon] [Fnac] [Barnes&Noble]

“Contemplative practices in higher education”? what the f...? Does this means that we should have to have our students meditate instead of practicing mathematics by doing more and more exercises? Again, what the f... ? And is it really appropriate, in our universities (which, in France, are mostly laïques et républicaines) to experiment such practices?

The subtitle of the book under review should perhaps reassure us: Powerful methods to transform teaching and learning. Indeed, as its authors explain to us in the very first lines of its preface, contemplative practices always has a well established place in the intellectual inquiry, a place which goes well beyond their vital role in all the major religions and spiritual traditions. The authors acknowledge many objectives to these practices, pointing out 4 of them whose importance can difficultly be denied:
• Development of attention and focus;
• Deeper understanding of the content of the course;
• Compassion, relation with self; deepening of the moral and spiritual component of education;
• Development of personality, and of creativity.
The largest part of this book develops twenty years of experiments of various contemplative practices in higher education, that were put forward to strengthen that quality of teaching, especially in the first grades of American college, and in almost all fields (law, economics, physics, chemistry, environmental sciences, music, literature, psychology).

Daniel Barbezat, a professor in economics at Amherst, explains for example how these methods allowed him to solve the following contradiction: how is it possible that his field (economics) pretends studying the mechanisms of decision that are supposed to lead people to well-being, without every considering the nature of well-being? He proposed to his class various alternatives, of the following kind:
1. The class is divided in ten groups of three people; the member of one group receive $1000 each, the other nothing 2. Everybody receives$200
He then asked everyone to choose between these two possibilities, and to guess with which proportion each possibility would be chosen. He returned to that exercise later following a meditation exercise about gratitude (think to things you are grateful for, then think to someone who is at the source of this gratitude). The results were not at all the same, therefore opening a way for thinking on the place of individual in society.

David Haskell, who teaches environmental sciences and biology, adapted the reading method of monks (as he says, lectio without too much divina) to have his class study problems of hunger and development. He asked his students to alternate between periods of quiet rest (say one minute) and the reading of one or two sentences of the text (each one reads by turns) and to brief commentaries by the students, etc. Other teachers propose the students to behold some text, or some graphic representation, and then to comment it. Examples are given of the probability distribution of the hydrogen bromide atom, according to its energy levels, or to two charts of industrial production (in absolute vs relative value). The authors claim that such exercises deepen the relation with self, with the studied document, and with other materials of the course.

Mathematics are absent from this book. In a blog post hosted by the American mathematical society, Luke Wolcott evokes this possibility, but acknowledges that he did not go further than personal meditation. In fact, I could not find other explicit examples in various sources, even none in the archives of the Center for contemplative mind in society that the authors of this book lead. However, it seems to me that some practical exercises organised by a teacher such as Adrien Guinemer in his middle/high school classes go in that direction (notably, the study of sections of cubes, cones, cylinder made from plasticine).

There are at least two methods that I find interesting and that could easily be implemented in our classes:
• Meditation exercises at the beginning of the class — first have everybody focus his attention on its breath during five minutes, and then report it on the subject of the class.
• Introspection techniques to fight failure anxiety — the student is asked to solve an exercise while writing on his sheet everything that comes to his mind, whatever relation it has with the exercise.
Moreover, isn't it our role to develop a profound sense of compassion to our students, especially those who prepare themselves to become teachers?

The first part of the book proposes a theoretical and practical background that is necessary to appreciate the variety of these methods, as well as some issues that need to be avoided. Three of them seem particularly crucial to me, all of them requiring from the teacher a quite deep personal involvement in these contemplative practices:
1. Assign to the contemplative exercises a clear pedagogical goal, whose impact can be evaluated;
2. Disjoint the practice of these exercises from the cultural and religious backgrounds in which they were first devised;
3. Be able of managing students who would not be at ease, or even would reject, such practices.
Anyway, the variety of possibilities that is described in this book is an invitation from its two authors that we embrace these millenary-old techniques to deeply transform our teaching. So, to the question that begins this book review, the author do much better than answering “Why not?” since they tell us “Follow us, try, and see!”.

So let us try, and see.