Transcendence properties of the Painlevé and Schwarzian differential equations

The paper of Joel Nagloo, Model Theory and Differential Equations, just published in the Notices of the AMS, is a beautiful portrait of an area of research that encompasses the end of the 19th century, the full 20th century, up to today. It conflates two totally distinct line of thought. The first line is the theory of differential equations, such as the ones that appear in trigonometry (the sine and cosine functions are solutions of the differential equation $y''+y=0$) or in elaborations of trigonometry (Schwarzian or Painlevé equations). 

The Painlevé equations, named after the mathematician and statesman Paul Painlevé (1863–1933) — he has been minister of war and prime minister during the French 3rd Republic — are a series of 6 differential equations of the second order, of the form $y''=R(t,y,y')$, where $R$ is a rational function. They are characterized by the Painlevé property that all “movable singularities” of their solutions— those which depend on the initial conditions and are not imposed by the form of the equation— are poles. Painlevé classified these equations: up to computation errors later corrected by Gambier and Fuchs, this gives the following irreducible list of 6 equations, in which the variable is $t$, the unknown is $y$, and the Greek letters $\alpha, \beta,\gamma,\delta$ representing parameters:

1. $y''=6y^2+t$
2. $y''=2y^3+ty+\alpha$
3. $tyy''=t(y')^2-yy'+\delta t+\beta y+\alpha y^2+\gamma ty^4$
4. $y y''=\frac12 (y')^2+\beta+2(t^2-\alpha)y^2+4ty^3+\frac32 y^3$
5. $y''=(\dfrac1{2y}+\dfrac1{y-1})(y')^2-\dfrac1t y'+\dfrac{(y-1)^2}{t^2} (\alpha y+\beta \dfrac1y)+\gamma\dfrac yt+\delta \dfrac{y(y+1)}{y-1}$
6. $y''=\frac12(\dfrac1y+\dfrac1{y-1}+\dfrac1{y-t})(y')^2-(\dfrac1t+\dfrac1{t-1}+\dfrac1{y-t})y'+\dfrac{y(y-1)(y-t)}{t^2(t-1)^2}(\alpha+\beta \frac t{y^2}+\gamma \dfrac{t-1}{(y-1)^2}+\delta\dfrac{t(t-1)}{(y-t)^2}$

By irreducible list, it is meant that all equations satisfying the Painlevé property are reducible to either previously known equations (involving elliptic functions or the Ricatti equations), and that those equations are not. Therefore, the solutions of these equations were classically called “transcendental” because they were generally (meaning except for some particular choices of parameters) not rational functions, nor could be derived by algebraic equations. The basic questions that people want to understand here is how much these functions are transcendental.

In this direction, a 2020 theorem of Joel Nagloo is that if you take $n$ distinct solutions $y_1,\dots, y_n$ of a “generic” Painlevé equation of type III or VI, then there is no (nontrivial) algebraic relation between the $2n$ functions $y_1,\dots,y_n$ and their derivatives $y_1',\dots,y_n'$. Here, generic means that the parameters involved in the differential equation they satisfy are algebraically independent over the field of rational numbers.

With Guy Casale and James Freitag, Joel Nagloo has proved a similar theorem for the third order Schwarzian equations associated with a Fuchsian subgroup $\Gamma$ of $\mathrm{PSL}(2,\mathbf R)$. The situation is more subtle: there are possible algebraic relations, but they are completely classified by Hecke correspondences.

The second line of thought is model theory, an area of mathematics classically attached to mathematical logic that tries to understand the intrinsic properties of mathematical theories, whatever they are. To that aim, they do geometry: they consider “definable subsets”, that is, loci defined (in some object of the considered theory) by the type of equations that the theory affords.

For example, if the theory involves fields only, the object could be a large algebraically closed field (such as the field complex numbers) and the equations are essentially polynomial equations, with the exception that they allow quantifiers, and negations, to define their loci. In fact, in this case, it is a theorem, classically attributed to Chevalley, that quantifiers are not needed — one only gets so-called constructible sets.

Another kind of interesting structure is that of differential fields, in which one has the classical field operations, but also an abstract derivation operator (satisfying the standard rules for derivations). Then, definable sets are essentially differential equations.

What model theory does in general is trying to define ways to categorize the definable subsets. One such important invariant would be an analogue of the Zariski dimension of constructible sets in algebraic geometry: the Morley rank. Roughly, a definable set has Morley rank $0$ if it is finite. It has Morley rank at least $n$ if it admits an infinite family of disjoint definable subsets of Morley rank $\geq n-1$. A particular case of sets of Morley rank $1$ are the strongly minimal definable subsets which are infinite and such that any definable subset of it is finite or cofinite (the complementary set is finite).

More importantly, model theory identifies structural properties of theories that make them “neater”, and this gives rise to an “universe of theories”. As you can see on this “map of the universe” (made by Gabriel Conant), the theory ACF of algebraically closed fields is one of the simplest ones that exists, and recent work in model theory tries to clarify what happens in much more complicated ones, characterized by incredible names or acronyms, strongly minimal, stable, distal, o-minimal, NIP, supersimple, NTP₂…

One of the fundamental results in model theory that emerged in the years 1980-2000 is the theory of Zariski geometries and Boris Zilber's “trichotomy principle” that indicates that 1-dimensional (strongly minimal) objects classify in 3 disjoint classes:

• geometrically trivial: there are essentially no other relation between distinct points of that object, beyond the fact that they belong to it. The generic Painlevé equations furnish important examples of this case, but proving strong minimality is the hard step.
• group-like: some algebraic group is hidden in the picture, and constrains all possible relations.
• field-like: this is essentially algebraic geometry over some hidden field.

I must be terse here, stability theory in model theory is an immense area with which I am not enough familiar, and I just insert pictures of three big names in the field, namely Morley, Shelah and Hrushovski:

For more details, and definitely more insight, I refer you to Joel Nagloo's paper quoted above. I also refer you to the Freedom Math Dance blog posts on Model theory and algebraic geometry.

Model theory and algebraic geometry, 5 — Algebraic differential equations from coverings

In this final post of this series, I return to elimination of imaginaries in DCF and explain the main theorem from Tom Scanlon's paper Algebraic differential equations from covering maps.

The last ingredient to be discussed is jet spaces.

Differential algebra is seldom used explicitly in algebraic geometry. However, differential techniques have furnished a crucial tool for the study of the Mordell conjecture over function fields (beginning with the proof of this conjecture by Grauert and Manin), and its generalizations in higher dimension (theorem of Bogomolov on surfaces satisfying $c_1^2>3c_2$), or for holomorphic curve (conjecture of Green-Griffiths). They are often reformulated within the language of jet bundles.

Let us assume that $X$ is a smooth variety over a field $k$. Its tangent bundle $T(X)$ is a vector bundle over $X$ whose fiber at a (geometric) point $x$ is the tangent space $T_x(X)$ of $X$ at $x$. By construction, every morphism $f\colon Y\to X$ of algebraic varieties induces a tangent morphism $Tf\colon T(Y)\to T(X)$: it maps a tangent vector $v\in T_y(Y)$ at a (geometric) point $y\in Y$ to the tangent vector $T_yf(v)\int T_{f(y)}(X)$ at $f(y)$. This can be rephrased in the language of differential algebra as follows: for every differential field $(K,\partial)$ whose field of constants contains $k$, one has a derivative map $\nabla_1\colon X(K)\to T(X)(K)$. Here is the relation, where we assume that $K$ is the field of functions of a variety $Y$. A derivation $\partial$ on $K$ can be viewed as a vector field $V$ on $Y$, possibly not defined everywhere; replacing $Y$ by a dense open subset if needed, we assume that it is defined everywhere. Now, a point $x\in X(K)$ can be identified with a rational map $f\colon Y\dashrightarrow X$, defined on an open subset $U$ of $Y$. Then, we simply consider the morphism from $U$ to $T(X)$ given by $p\mapsto T_pf (V_p)$. At the level of function fields, this is our point $\nabla_1(x)\in T(X)(K)$.

If one wants to look at higher derivatives, the construction of the tangent bundle can be iterated and gives rise to jet bundles which are varieties $J_m(X)$, defined for all integers $m\geq 0$, such that $J_0(X)=X$,  $J_1(X)=T(X)$, and for $m\geq 1$, $J_m(X)$ is a vector bundle over $J_{m-1}X$ modelled on the $m$th symmetric product of $\Omega^1_X$.  For every differential field $(K,\partial)$ whose field of constants contains $k$, there is a canonical $m$th derivative map $\nabla_m\colon X(K) \to J_m(X) (K)$.

The construction of the jet bundles can be given so that the following three requirements are satisfied:

• If $X=\mathbf A^1$ is the affine line, then $J_m(X)$ is an affine space of dimension $m+1$, and $\nabla_m$ is just given by $ \nabla_m (x) = (x,\partial(x),\dots,\partial^m(x)) $ for $x\in X(K)=K$;
• Products: $J_m(X\times Y)=J_m(X)\times_k J_m(Y)$;
• Open immersions: if $U$ is an open subset of $X$, then $J_m(U)$ is an open subset of $X$ given by the preimage of $U$ under the projection $J_m(X)\to J_{m-1}(X)\to \dots\to J_0(X)=X$.
• When $X$ is an algebraic group, with origin $e$, then $J_m(X) $ is canonically isomorphic to the product of $X$ by the affine space $J_m(X)_e$ of $m$-jets at $e$.

We now describe Scanlon's application.

Let $G$ be a complex algebraic group acting on a complex algebraic variety $X$; let $S\colon X\to Z$ be the corresponding generalized Schwarzian map. Here, $Z$ is a complex algebraic variety, but $S$ is a differential map of some order $m$. In other words, there exists a constructible algebraic map $\tilde S\colon J_m(X)\to Z$ such that $S(x)=\tilde S(\nabla_m(x))$ for every differential field $(K,\partial)$ and every point $x\in X(K)$.

Let $U$ be an open subset of $X(\mathbf C)$, for the complex topology, and let $\Gamma$ be a Zariski dense subgroup of $G(\mathbf C)$ which stabilizes $U$. We assume that there exists a complex algebraic variety $Y$ and a biholomorphic map $p\colon \Gamma\backslash U \to Y(\mathbf C)$.

Locally, every open holomorphic map $\phi\colon\Omega\to Y(\mathbf C)$ can be lifted to a holomorphic map $\tilde\phi\colon \Omega\to U$. Two liftings differ locally by the action of an element of $\Gamma$, so that the composition $S\circ\tilde\phi$ does not depend on the choice of the lifting, by definition of the generalized Schwarzian map $S$. This gives a well-defined differential-analytic map $T\colon Y\to Z$. Let $m$ be the maximal order of derivatives appearing in a formula defining $T$. Then one may write $T\circ\phi =\tilde T\circ \nabla_m\tilde\phi$, where $\tilde T$ is a constructible analytic map from $J_m(Y)$ to $Z$.

Theorem (Scanlon). — Assume that there exists a fundamental domain $\mathfrak F\subset U$ such that the map $p|_{\mathfrak F}\colon \mathfrak F\to Y(\mathbf C)$ is definable in an o-minimal structure. Then $T$ is differential-algebraic: there exists a constructible map $\tilde T\colon J_m(Y)\to Z$ such that $T\circ \phi=\tilde T \circ J_m(\phi)$ for every $\phi$ as above.

For the proof, observe that the map $\tilde T$ is definable in an o-minimal structure, because it comes, by quotient of a definable map from the preimage in $J_m(U)$ of $\mathfrak F$, and o-minimal structures allow elimination of imaginaries. By the theorem of Peterzil and Starchenko, it is constructible algebraic.

Model theory and algebraic geometry, 4 — Elimination of imaginaries

The fourth post of this series is devoted to an important concept of model theory, that of elimination of imaginaries. The statement of Scanlon's theorem will appear in a subsequent one.

Definition. — Let $T$ be a theory in a language $L$. One says that $T$ eliminates imaginaries (resp. weakly eliminates imaginaries) if for every model $M$ and every formula $f(x;a)$ with parameters $a\in M^p$, there exists a formula $g(x;y)$ such that $\{ b\in M^q\;;\; \forall x, f(x;a)\Leftrightarrow g(x;b)\}$ is a singleton (resp. is a non-empty finite set).

What does this mean? View the formula $f(x;y)$ as defining a family of definable subsets, where $f(x;a)$ is the slice given by the choice of parameters $a$. It may happen that many fibers are equal. The property of elimination of imaginaries asserts that one can define the same family of definable subsets via another formula $g(x;y)$, with different parameters, so that every definable set in the original family appears once and only once. For the case of weak elimination, every definable set of the initial family appears only finitely times.

There is an alternative, Galois theoretic style, description: a theory $T$ (weakly) eliminates imaginaries if and only if, for every formula $f(x;a)$ with parameters in a model $M$, there exists a finite subset $B\subset M$ such that for every elementary extension $N$ of $M$ and every automorphism $\sigma$ of $N$, then $\sigma$ preserves the formula (meaning $f(x;a)\leftrightarrow f(\sigma(x);a)$, or, equivalently, $\sigma$ leaves globally invariant the definable subset of $N^n$ defined by the formula $f(x;a)$) if and only if  $\sigma$ leaves $B$ pointwise (resp. globally) invariant. One direction is obvious: take for $B$ the coordinates of the elements of the singleton (resp. the finite set) given by applying the definition. For the converse, elementary extensions must enter the picture because some models are too small to possess the necessary automorphisms that should exist; under “saturation hypotheses”, the model $M$ will witness them already.

This property is related to the possibility of representing equivalence classes modulo a definable equivalence relation. Namely, let $M$ be a model and let $E$ be an equivalence relation on $M^n$ whose graph is a definable subset of $M^n\times M^n$. Assume that the theory $T$ eliminates imaginaries and allows to define two distinct elements. Then there exists a definable map $f_E\colon M^n\to M^m$ such that for every $y,z\in M^n$, $y \mathrel{E} z$ if and only if $f_E(y)=f_E(z)$. In particular, the quotient set $M^n/E$ is represented by the image of the definable map $f_E$.

Conversely, let $f(x;a)$ be a formula with parameters $a\in M^p$ and consider the equivalence relation $E$ on $M^p$ given by $yEz$ if and only if $\forall x,\ f(x;y)\Leftrightarrow f(x;z)$. Its graph is obviously definable. Assume that there exists a definable map $f_E\colon M^p\to M^q$ such that $yEz$ if and only if $f_E(y)=f_E(z)$. Then an automorphism of (an elementary extension of) $M$ will fix the definable set defined by $f(x;a)$ if and only if it fixes $f_E(a)$, so that one has elimination of imaginaries.

Theorem (Poizat). — The theory of algebraically closed fields eliminates imaginaries.

This is more or less equivalent to Weil's theorem on the field of definition of a variety. It is my feeling, however, that this property is under-estimated in algebraic geometry. Indeed, it is closely related to a theorem of Rosenlicht that asserts that given a variety $X$ and an algebraic group $G$ acting on $X$, there exists a dense $G$-invariant open subset $U$ of $X$ such that a geometric quotient $U/G$ exists in the sense of Mumford's Geometric Invariant Theory.

Examples. — Let $K$ be an algebraically closed field.

a) Let $X$ be a Zariski closed subset of $K^n$ and let $G$ be a finite group of (regular) automorphisms of $X$. Let us consider the formula $f(x;y)=\bigwedge_{g\in G} (x=g\cdot y)$ which asserts that $x$ belongs to the orbit of $G$ under the given action, so that $f(x;y)$ parameterizes $G$-orbits. Since $G$ is finite, weak elimination of imaginaries is a trivial matter, but elimination of imaginaries is possible. Let indeed $A$ be the affine algebra of $X$; this is a $K$-algebra of finite type with an action of $G$ and the algebra $A^G$ is finitely generated. Consequently, there exists a Zariski closed subset $Y$ of some $K^m$ and a polynomial morphism $\phi\colon K^n\to K^m$ such that, for every $y,z\in X$, $\phi(y)=\phi(z)$ if and only if there exists $g\in G$ such that $z=g\cdot y$. Consequently, for $a\in X$, $b=\phi(a)$ is the only element such that the formula $f(x;a)$ be equivalent to the formula $g(x;b)=(b\in Y) \wedge (\exists y\in X)(\phi(y)=b) \wedge f(x;y))$.

The simplest instance would be the symmetric group $G=\mathfrak S_n$ acting on $K^n$ by permutation of coordinates. Then $G$-orbits are unordered $n$-tuples of elements of $K$, and it is a both trivial and fundamental fact that the orbit of $(x_1,\dots,x_n)$ is faithfully represented by the first $n$ elementary symmetric functions of $(x_1,\dots,x_n)$, equivalently, by the coefficients of the polynomial $\prod_{j=1}^n (T-x_j)$.

b) Let $X=K^{n^2}$ be the set of all $n\times n$ matrices under which the group $G=\mathop{\rm GL}(n,K)$ acts by conjugation. The Jordan decomposition gives a partition of $X$ into constructible sets, stable under the action of $G$, and on each of them, there exists a regular representation of the equivalence classes. For example, the set $U$ of all matrices with pairwise distinct eigenvalues is Zariski open — it is defined by the non-vanishing of the discriminant of the characteristic polynomial — and on this set $U$, the conjugacy class of a matrix is represented by its characteristic polynomial.

Theorem. — An o-minimal theory eliminates imaginaries. More precisely any surjective definable map $f\colon X\to Y$ between definable sets admits a definable section.

This follows from the fact that one can define a canonical point in every non-empty definable set. By induction on dimension, it suffices to prove this for a subset $A$ of the line. Then, let $J_A$ be the leftmost interval of $A$ (if the formula $f$ defines $f$, then $J_A$ is defined by the formula $y\leq \rightarrow f(y)$); let $u$ and $v$ be the “endpoints” of $J_A$; if $u=-\infty$ and $v=+\infty$, set $x_A=0$; if $u=-\infty$ and $v<\infty$, set $x_A=v-1$; if $-\infty<u\leq v<+\infty$, set $x_A=(u+v)/2$. It is easy to write down a formula that expresses $x_A$ in terms of a formula for $A$. Consequently, in a family $A_t\subset M$ of non-empty definable sets, the function $t\mapsto x_{A_t}$ is definable.

Theorem (Poizat). — The theory of differentially closed fields eliminates imaginaries in the language $\{+,-,\cdot,0,1,\partial\}$.

Examples. — Let $K$ be an algebraically closed differential field. Let $X$ be an algebraic variety with the action of an algebraic group $G$, all defined over the field of constants $C=K^\partial$. We can then endow $X(K)$ with the equivalence relation given by $x\sim y$ if and only if there exists $g\in G(C)$ such that $y=g\cdot x$. The following three special instances of elimination of imaginaries in DCF are classical results of function theory:

a) If $X=\mathbf A^1$ is the affine line and $G=\mathbf G_a$ is the additive group acting by translation, then the map $\partial\colon x\mapsto \partial (x)$ gives a bijection from $X(K)/G(C)$ to $K$. Indeed, two elements $x,y$ of $K$ differ by the addition of a constant element if and if $\partial(x)=\partial(y)$. (Moreover, every element of $K$ has a primitive.)

b) Let $X=\mathbf A^1\setminus\{0\}$ be the affine line minus the origin and let $G=\mathbf G_m$ be the multiplicative group acting by multiplication. Then the logarithmic derivative $\partial\log\colon x\mapsto \partial(x)/x$ gives a bijection from $X(K)/G(C)=K^\times/C^\times$ to $K$ — two elements $x,y$ of $K^\times$ differ by multiplication by a constant if and only if $\partial(x)/x=\partial(y)/y$, and every element of $K$ is a logarithmic derivative.

c) Let $X=\mathbf P^1$ be the projective line endowed with the action of the group $G=\operatorname{\rm PGL}(2)$. Then two points $x,y\in X(K)$ differ by an action of $G(C)$ if and only if their Schwarzian derivatives are equal, where the Schwarzian derivative of $x\in K$ is defined by
\[ S(x) = \partial\big(\partial^2 (x)/\partial (x)\big) -\frac12 \big(\partial^2(x)/\partial(x)\big). \]

Link to Part 5 — Algebraic differential equations from coverings

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.

Link to Part 4 — Elimination of imaginaries

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.

Link to Part 3 — Real closed fields and o-minimality

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 $\kappa$.

Link to Part 2 — Definable sets, types; quantifier elimination

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.