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

Vijay Iyer and Wadada Leo Smith at The Stone

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.

Saturday, May 31, 2014

The evolution of higher education

After a few months of silence, a short blog post to indicate a few web links that I found interesting, rising concern about the evolution of higher education.

In February 2014, Counterpunch published a series of remarks by Noam Chomsky under the title On Academic Labor. (I found it first on Alternet, under the alternate title How America's Great University System Is Getting Destroyed.)

More recently (May 2014), the New York Times published an editorial, Fat-Cat Administrators at the Top 25, where they quote a report from the Institute for Policy Studies indicating that "student debt and low-wage faculty labor are rising faster at state universities with the highest-paid presidents."

In fact, I had been made aware of the problem by a few posts from the blog The Homeless Adjunct, notably this post from 2012 that clearly explains how the American university system was killed in five easy steps:
  1. Defund the university system;
  2. Deprofessionalize and impoverish the  professors;
  3. Install a managerial/administrative class who take over governance of the university;
  4. Move in corporate culture and corporate money;
  5. Destroy the students.
This probably looks a radical point of view, and had looked a bit radical to me at that time. Except that it is really how it now happens in France where we are clearly somewhere in between steps 3 and 4. Of course, the fact that our university system is mainly public delays the process a little bit, but look:
  1. Decisive progress towards defunding was made in 2009 by the Sarkozy-Pécresse LRU-law. While the acronym stands for Liberty and Responsability of Universities, this law has been infamously referred to Autonomy of Universities. The French public universities are now allocated a global budget by the State, which they are now supposed to manage as they wish, except that the allocated budget is insufficient, and that they have almost no control of whatever. Many universities are on the edge of defaulting. So what we have under the eyes is nothing but a defunding of the system disguised as a change of allocation model.
  2. The number of permanent positions is sharply decreasing. Of course, the age-pyramid of the present professors is also a cause for this evolution, since almost all baby-boomers have now retire. But the decrease is not at all the same in all fields—for example, this year, there were many more open positions in applied mathematics than in pure mathematics. Probably, when it comes about cutting positions, the "applied"-color makes it nicer for university boards. Probably too, applied mathematicians have been better at explaining their rôle in society.
  3. Meanwhile, the administration is getting fatter. To manage the global budget, it has been necessary to hire full-time "managers". And to be able to attract them, it seems that their pay has nothing to do with the usual range among French public servants. At the same time, a new law reorganizes the higher-education system by forcing universities (as well as our innumerous engineering schools) to regroup themselves. This will create enormous beasts that will look like the Lernean Hydra. For example, the Paris-Saclay University regroups 22 higher education schools, among which 2 universities and 10 "grandes écoles"; it will host around 50.000 students and more than 10.000 professors and researchers! No doubt that it will require a heavy bureaucracy to manage this high number of people. And since we're split in many institutions, it will be hard to have the voice of academic freedom be listened to.
  All of this is very depressing...

Friday, February 14, 2014

A map of the universe

If you'd be asked to tell what a map of the universe looks like, I'm pretty sure you'd imagine something on a dark background, with many dots representing planets, and shaded areas corresponding to galaxies. That map of the universe, drawn by Gabriel Conant,  a graduate student at Berkeley,  is of more or less like that. Except that dots are mathematical theories, and galaxies correspond to some stability properties defined in model theory.

Here theories have esoteric nicknames, such as ACF, ACVF, SCF$_p^n$, or ``universal graph omitting a bowtie'' (an homage to Tom S. ? :-)), and properties have even more esoteric nicknames — NIP, o-minimal, NSOP$_{n+1}$, or superstable.  To make it something more than an enjoyable invitation au voyage, Gabriel indicated important specific examples, with their definitions and references.

By the way, this is also a beautiful illustration of the power of HTML5.