[FOM] FOM: Zilber's Program

[Dave Marker]

At the ASL meeting Boris Zilber gave a thought provoking Godel
Lecture on his program to find "natural" manifestations of
certain interesting model theoretic phenomena. Part of his work
includes a program that might lead to a new understanding of the
complex exponential function.

Below is a summary of what I consider to be some of the
most interesting points of the program. Some references can
be found at the bottom. Any unexplained terms can be found in the
references.

Dave Marker


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I. Some history

Recall that a definable set is strongly minimal if it is infinite.
every definable subset is either finite or co-finite and this
remains true in elementary extensions. If D is a strongly minimal set
x is in D and A is a subset of D, we say that x is algebraic over
A if x is in a finite set defined over A.

The Baldwin-Lachlan analysis of aleph_1-categorical theories and Morley's
theorem showed the crucial role played by strongly minimal sets.
A key part of their proof was the observation that in a strongly minimal
set algebraic closure satisfies exchange (if x in acl(A,b) and x not in
acl(A),  then b is in acl(A,x)). This means that strongly minimal sets
with algebraic closure determine a pregeometry and there is a well defined
dimension theory (special cases are the usual dimension in linear algebra or
transcendence  degree in algebraically closed fields). While early
applications only  exploited the existence of a good notion of dependence,
Zilber noticed that interesting information could be obtained by studying
the fine  structure of the geometry.

We say that the geometry is "trivial" if
	x in acl(A) iff x in acl(a) for some a in A

The geometry is "locally modular" if
	x in acl(A,a) iff there is b in acl(A) such that x in acl(a,b)


Strongly minimal sets can be trivial, non-trivial locally modular, or
non-locally modular.

* The integers with successor is trivial as is an infinite set with
no structure
* A vector space over a division ring is non-trivial locally modular
* An algebraically closed field is non-locally modular

Zilber's original conjecture was that a nontrivial pregeometry is an
avatar of  ambient algebraic structure. In particular he conjectured
that non-trivial strongly minimal sets are essentially either
vector spaces or algebraically closed fields. These conjectures
motivated a great deal of research. The key results were:


(+) Hrushovski, building on earlier work of Zilber and others,
showed that non-trivial locally  modular strongly minimal sets are
essentially vector spaces.

(-) Hrushovski showed the conjectures were false by constructing
non-locally modular strongly minimal sets in which there are no
interpretable groups (and strange strongly minimal sets that support
two field structures of conflicting characteristics). Constructions
of this type were used by Baldwin, Baudish, Tent and others to construct
a wealth of interesting, somewhat pathological, examples.

(+) Hrushovski and Zilber showed that in the presence of a nice topology
(similar to the Zariski topology in algebraically closed fields) the
conjecture is true. Hrushovski exploited these results in interesting
applications in Diophantine geometry.

(+) Starchenko and Peterzil proved the analog of the conjecture for
o-minimal theories.

[It's probably worth mentioning that we don't really have any good
understanding of the trivial strongly minimal sets even in important
special cases like differentially closed fields.]

After these results one could (and I did) take the position that while
there are many pathological strongly minimal sets they don't arise in
nature.  Over the last decade Zilber has promoted the opposing view that
we should be searching for natural manifestation of the Hrushovski
construction.  Of course the key question is what is "natural"?
This is something of a moving target and I won't give a definitive answer,
but I  will describe two concrete examples that I find particularly
interesting.

II. Liouville Functions

Let p_n be the polynomial X^n+a_{n-1}X^{n-1}+...+a_0. Where
a_0,...,a_{n-1} are algebraically independent over the rationals.
Add a function symbol p to the language of fields. Zilber and Pascal
Koiran each showed that any sentence is either true or false for
all but finitely many n. The limit theory can be thought of as
the theory of a generic polynomial.

This proof uses a Hrushovski style construction. I need to say a
little about these construction. The key idea is to build structures
by Fraisse-style amalgamation constructions. We carefully
fix a collection of structures, a notion of good extension, and
build structures by  amalgamating. Usually this will build an infinite
rank structure, but in some cases by carefully controlling the
amalgamation
we can build a structure of finite rank. The key choice is the notion
of "good extension". This is done by defining a new notion of dimension
and insisting it by preserved.
 The idea is that often we have two notions
of dimension. For example in a field  for finite X we have
	td(X)=transcendence degree of X over the rationals
	ld(X)=linear dimension of X over the rationals
	|X|
We build a new notion by combining these.

In the generic polynomial example we define
	d(X)=td(X union f(X))-|X|
We only consider structures where d(X) in nonnegative for all finite X.
This is sometimes called the Hrushovski inequality. [more on
this in part III]

The theory we build is omega-stable of infinite rank. Do models
arise naturally? While the construction seems artificial, Zilber
conjectured
that there are entire functions f on the complex numbers C such that
(C,+,x,f) is a model of this theory.
Wilkie and Koiran showed that this is right by constructing entire
analytic functions satisfying the axioms. Zilber has showed that
adding axioms asserting that the solutions to systems of equations is
countable (stated using the quantifier "exists uncountably many")
is categorical in all infinite powers. Thus these models are in some
sense "canonical".

 "complex analytic" is one possible interpretation of
"natural", unfortunately it is too restrictive to try to build "natural"
finite rank structures. Using the Big Picard theorem, any nonpolynomial
entire
function f is infinite-to-one and this implies infinite rank.


III. Exponentiation

What can be said about the model theory of the complex field with
exponentiation?
>From one point of view this is going to be a bad theory.
{x:exp(x)=1}=\{2\pi i n: n an integer} by taking quotients
we define the rationals and, via Julia Robinson, we can define
the integers. Thus we have an undecidable theory and no good
definability theory. For some time model theorists stopped at this
point, but there are still some very basic interesting questions:

i) Is this as bad as it gets? In the real field if we add a predicate
for the integers we interpret all of second order arithmetic and
the definable sets are the projective sets of descriptive set theory.
Does that happen here or is first order arithmetic as bad as it gets?
Here is one conjecture that says that things aren't too bad:

i) (Quasiminimality conjecture) Any subset of C definable in (C,+,x,exp)
is either countable or co-countable?

ii) (Mycielski) Other than complex conjugation is there a nontrivial
automorphism of (C,+,x,exp)?

iii) Is there quantifier elimination or model completeness in some
reasonable language? (there isn't in the original language)

Zilber came up with a very novel approach to complex exponentiation.
He tried constructing algebraically closed fields with exponentiation
via a Hrushovski construction. The dimension he used was
	dim(X)=td(X union exp(X))-ld(X)
With this dimension function the Hrushovski inequality has a familiar
ring. If x_1,...,x_n are linearly independent over the rationals
then we want the transcendence degree of
Q(x_1,...,x_n,exp(x_1),...,exp(x_n))
to be at least n. This is Shannuel's conjecture!

Zilber gave a set of infinitary axioms using the quantifier exists
uncountably
many. In addition to Shannuel's conjecture (SC) there are axioms asserting
algebraic closedness (ACF)
the kernel of exponentiation is free on one generator (ker),
subsets of the line defined as solutions to systems of equations are
countable (CC), and an axiom asserting that certain non-overdetermined
systems of equations have solutions (EC). Using Hrushovski constructions
he showed that the theory is satisfiable. He then proved that it is
categorical
in all uncountable powers. This later proof is  a special
case of Shelah's work on categoricity in excellent classes.

These structures are quasi-minimal (definable subsets of the line are
countable or co-countable) and they have many automorphisms.

This leaves an intriguing question. This theory has a unique model of size
continuum. Is it (C,+,x,exp) the unique model of size continuum?

The axioms ACF+ker+CC are true in the complex numbers. But Shannuel's
conjecture is a well known open problem and the EC axioms, if true,
seem at least as daunting.

While it might seem like heresy to think that these combinatorial model
theoretic constructions will lead us to new understanding of one of the
most concrete mathematical objects, Zilber's work has open some
fascinating new possibilities.

IV. And beyond

This is only the starting point of Zilber's work. Here are a few other
ideas raised in his Godel lecture:

* Using dense translations of real analytic curves we can build
a bicolored field (an algebraically closed field with a unary predicate)
of infinite nonmonomial rank.

* At present we really only have "natural" examples of new infinite rank
structures. To build finite rank structures Zilber suggests we need find
a way of adding subsets of fractional Hausdorff dimension.

* At the more speculative end, Zilber suggests that some of the Hrushovski
style constructions may prove interesting in studying mirror symmetry.


Some references:

For background on the geometry of strongly minimal sets see Chapter 8
of my book "Model Theory: An Introduction". There is also a
survey paper "Strongly minimal sets and geometry" on my webpage
	http://www.math.uic.edu/~marker

Zilber's webpage has many interesting preprints on various aspects of the
program.
	http://www.maths.ox.ac.uk/~zilber/

"Analytic and pseduoanalytic structures" is a survey

"Pseudo-exponentiation on algebraically closed fields of characteristic
zero"
is his main work on exponential fields.

Koiran's webpage also has many interesting papers including his work on
Liouville functions.
	http://www.ens-lyon.fr/~koiran/