CATEGORICITY

B. Zilber Gődel Lecture

AMS meeting at Chicago, 1 June 2003 1.Uniqueness, completeness and

categoricity

Description of an object in a language – informally

Naive assumption on a language: we can describe an object of our interest fully and completely, uniquely determining the object.

Newtonian physics: such a description, a theory, is possible.

Newtonian physics: The world is not a chaotic collection of objects and events but a smooth structure allowing a comprehensive description.

Newtonian physics: The world is not a chaotic collection of objects and events but a smooth structure allowing a comprehensive description.

What about modern physics?

In a formal logic we have Completeness and Categoricity.

Completeness = ’fully and completely’ in terms of the formal language itself. Categoricity = unique interpretation in reality. Categoricity – easier acceptable to practical mathematicians and physicists. Completeness – easier achievable on the theoretical level.

Do the objects described by a formal theory exists in reality?

Depends on the type of the description, complete or categorical.

2. Categoricity. What categoricity?

Must not be based on a list of all possible configuration in all possible locations of the structure.

We want a concise description of a large structure.

Thus the (absolute) categoricity for first order languages is uninteresting.

Definition A structure M in a language L is said to have λ-categorical theory if there is exactly one, up to isomorphism, structure of cardinality λ satisfying the L-theory Th(M) of M.

Theorem (M.Morley, 1964 confirming a conjecture by J.Los)

If a countable first-order T is λ-categorical for some λ > ℵ_{0} then it is categorical for all λ > ℵ_{0}.

Theorem (M.Morley, 1964 confirming a conjecture by J.Los)

If a countable first-order T is λ-categorical for some λ > ℵ_{0} then it is categorical for all λ > ℵ_{0}.

Theorem (S.Shelah, 1983)

If an L_{ω_{1},ω}-sentence is categorical in ℵ_{n} for all n then it is categorical (and has models) in all infinite λ.

3. Stability, homogeneity and smoothness.

Theorem (M.Morley, S.Shelah) Categoricity implies stability

Stability = dimension theory plus highly homogeneous models in all cardinalities).

Thesis This is a weak form of smoothness 4. Trichotmy Conjecture Classical first-order λ-categorical structures for uncountable λ :

(1) Trivial structures (= only)

(2) Linear structures (Abelian divisible torsion-free groups; Vector spaces over a countable division ring, …)

(3) Algebraically closed fields.

One can construct more complicated structures over the basic ones preserving the property of categoricity, e.g.

Algebraic groups

GL(n, ℂ), PGL(n, ℂ), ...

More generally, complex algebraic varieties V ⊆ ℂ^{n} equipped with polynomially defined relations

p(x̄_{1}, …, x̄_{m}) = 0

in n × m variables).

Observation – example: Compact complex spaces in the natural language are ω-stable of finite Morley rank.

If a compact complex space is Ka̋hler it is saturated.

Trichotomy Conjecture: An uncountably categorical structure must be classical.

Trichotomy Conjecture (Z. 1982) Given a f-o uncountably categorical structure M one and only one of the following holds:

(i) the geometry of M is trivial;

(ii) the geometry of M is isomorphic to an affine or projective geometry over a countable division ring;

(iii) M is a structure of algebraic geometry over an algebraically closed field K. (M is definably equivalent to the field K.)

The Trichotomy conjecture refuted in general (E.Hrushovski, 1989) proved under extra Zariski assumptions (E.Hrushovski and B.Zilber, 1994) Also, proved in the o-minimality – real algebraic geometry context (Y.Peterzil, S.Starchenko, 1996).

Both the Zariski and o-minimality proofs exploit heavily smoothness assumptions.

Both the Zariski and o-minimality proofs exploit heavily smoothness assumptions. In the Zariski context Pre-smoothness assumption on M (dimension theorem): For any closed irreducible

S_{1}, S_{2} ⊆ M^{n} dim S_{1}∩S_{2} ≥ dim S_{1}+dim S_{2}−dim M^{n} component-wise.

Hrushovski’s counter-examples

Given a class of structures M with a dimension notions ∂_{1}, and ∂_{2} we want to consider a new function f on M. On (M, f) introduce a predimension

δ(X) = ∂_{1}(X ∪ f(X)) − ∂_{2}(X).

Consider structures (M, f) which satisfy the Hrushovski inequality:

δ(X) ≥ 0 for any finite X ⊂ M.

Amalgamate all such structures to get a universal and homogeneous structure in the class. The resulting structure (M̃, f) will have a good dimension notion and a nice geometry.

Observation:

If M is a field and we want f = ex to be a group homomorphism

ex(x_{1} + x_{2}) = ex(x_{1}) · ex(x_{2})

then the corresponding predimension must be

δ(X) = tr.deg(X∪ex(X))−lin.dim_{ℚ}(X) ≥ 0.

The Hrushovski inequality, in the case of the complex numbers and ex = exp, is equivalent to

tr.deg(x_{1}, …, x_{n}, e^{x_{1}}, …, e^{x_{n}}) ≥ n

assuming that x_{1}, …, x_{n} are linearly independent. This is the Schanuel conjecture.

Pseudo-exponentiation

Consider the class of fields of characteristic 0 with a function ex: K_{ex} = (K, +, ·, ex) satisfying

EXP1: ex(x_{1} + x_{2}) = ex(x_{1}) · ex(x_{2}) EXP2: ker ex = πℤ

Consider the subclass satisfying the Schanuel condition

SCH : tr.deg(X∪ex(X))−lin.dim_{ℚ}(X) ≥ 0.

Amalgamation process produces K_{ex}, an Existentially Closed field with pseudo-exponentiation, Existential Closedness property EC: Any well-overdetermined system of equations in +, ·, ex has a solution in K_{ex}. And Countable Closure property CC: ’Analytic’ subsets of K^{n} of dimension 0 are countable.

Theorem Given λ > ℵ_{0}, there is a unique model of axioms ACF_{0} + EXP + SCH + EC + CC of cardinality λ.

This is a consequence of

Theorem A The L_{ω_{1},ω}(Q)-sentence ACF_{0} + EXP + SCH + EC + CC is axiomatising a quasi-minimal excellent class.

and

Theorem B (Essentially S.Shelah 1983) A quasi-minimal excellent class has models and is categorical in any uncountable cardinality.

There are a series of further pseudoanalytic and analytic examples.

Theorem (A.Wilkie, P.Koiran, B.Z.) There is an entire analytic function f which satisfies:

(i) Schanuel-type property (Hrushovski inequality) tr.deg(x_{1}, …, x_{n}, f(x_{1}), … f(x_{n})) ≥ n for distinct x_{1}, …, x_{n} ∈ ℂ (ii) existential closedness property, (iii) the first-order theory T_{f} of (ℂ, +, ·, f) is ω-stable, (iv) the L_{ω_{1},ω}(Q)-sentence T_{f}+ CC is categorical in all uncountable cardinals.

Conclusions.

1.Taking into account λ-categoricity and stability for stronger languages we can extend the list of basic geometries: Basic geometries of stability theory:

(1) Trivial geometry

(2) Linear geometry

(3) Algebraic geometry (*) “Classical analytic” geometries – fusions of (1)–(3).

2. We can predict for classical analytic geometries Schanuel-type property – a Hrushovski inequality.

3. We can predict for classical analytic geometries the existential closeness property –

– Any non-overdetermined and free (from obvious contradictions) system of equations has a solution.

Universal covers of semi-abelian varieties

0 ⟶ Λ ⟶^{i} ℂ^{g} ⟶^{exp} 𝔸(ℂ) ⟶ 1,

where exp is an analytic homomorphism from the additive group (ℂ^{g}, +) and Λ = ℤ^{N} is a discrete subgroup of ℂ^{g}, exp a group homomorphism.

Question Is the universal cover uniquely determined by algebraic data only?

0 ⟶ Λ ⟶^{i} V ⟶^{ex} 𝔸(F) ⟶ 1

(V, +) a group, F an algebraically closed field, 𝔸(F) the F -points of semiabelian variety with the structure induced by F, ex a group homomorphism.

Reformulation Is the obvious L_{ω_{1},ω}- sentence Σ_{A} describing the sequence uncountably categorical?

Theorem C Σ_{A} is uncountably categorical iff the following arithmetic conditions hold for good fields k:

(i) (Galois action on torsion points) for all but finitely many prime p the group Gal(k̃ : k) acts on the Tate module T_{p}(𝔸) as GL_{N}(ℤ_{p}), and for remaining finite number of p the group acts as a subgroup of GL_{N}(ℤ_{p}) of finite index; (i) (Galois action on torsion points) for all but finitely many prime p the group Gal(k̃ : k) acts on the Tate module T_{p}(𝔸) as GL_{N}(ℤ_{p}), and for remaining finite number of p the group acts as a subgroup of GL_{N}(ℤ_{p}) of finite index;

(ii) (Kummer theory and heights) given a mult-independent a_{1}, …, a_{n} there is an l ∈ ℕ such that for any m ∈ ℕ s.t. 𝔸(k) contains ml-torsion

Gal(k(a1^{\frac{1}{ml}}, …, an^{\frac{1}{ml}}) : k(a1^{\frac{1}{l}}, …, an^{\frac{1}{l}}))

≅ (ℤ/mℤ)^{Nn}.

Theorem C is a consequence of Keisler – Shelah theory of L_{ω_{1},ω}-categoricity (excellency).

Galois action on roots of unity, Kummer and height theories are known for some 𝔸.

This implies

Theorem A The L_{ω_{1},ω}(Q)-sentence ACF_{0} + EXP + SCH + EC + CC is axiomatising a quasi-minimal excellent class.

Conclusion 4 Categoricity can predict strong arithmetic properties.

Analytic context

Can the universal cover of the multiplicative group (torus) F^{∗} be compactified as an analytic Zariski structure

0 ⟶ Λ ⟶^{i} V ⟶^{ex} F^{∗} ⟶ 1?

This leads to the theory of toric geometry.

Toric geometry is the main model for string theory and mirror symmetry.

Hope

Uniqueness - categoricity criterion can help to find a true mathematical model of physics.