Luminy, Marseilles

France

Programme | Tutorials | Abstracts | List of open problems | List of participants |

I should like to thank all speakers and participants at
Simpleton 2002 - The Workshop. I hope you
have enjoyed the meeting as much as I have.

You can find the definitive programme below, as well as the updated list of abstracts. I shall add links to the slides of the talks in the abstract section when and if I get them from the speakers. Eventually, all talks except for Ziegler's (and the corresponding slides, if any) will be available here on the CIRM website. Finally, here is the conference photo.

Time | Monday | Tuesday | Wednesday | Thursday | Friday |
---|---|---|---|---|---|

7h00 - 9h00 | Breakfast | ||||

9h15 - 10h15 | Casanovas | Hart | Wagner | Ben-Yaacov | Chatzidakis |

10h15 - 10h45 | Coffee Break | ||||

10h45 - 11h45 | Onshuus | Kim | Tomasic | Pourmahdian | Shami |

11h55 - 12h25 | Iwanow | de Piro | Lippel | Evans | Krupinski |

12h30 | Lunch | ||||

14h00 - 14h30 | Ealy | ||||

14h35 - 15h30 | Newelski | ||||

15h40 - 15h50 | Prof. Shadoko | ||||

16h15 - 17h15 | Kikyo | Newelski | Berenstein | Coffee Break | |

17h15 - 17h45 | Coffee Break | Coffee Break | |||

17h45 - 18h45 | MacPherson | Ziegler | Pillay | ||

18h55 - 19h25 | Problem Session | Tsuboi | Ben-Yaacov | ||

19h30 | Dinner |

Enrique Casanovas : | Simplicity simplified |

We present the basics of simplicity theory, up to the independence theorem. The proofs will use several simplifications found recently by different people, which may also be of interest for experts. | |

Bradd Hart : | Hyperimaginaries and canonical bases |

Shelah's ^{eq}-construction gives a canonical
first-order way to deal with classes modulo definable equivalence
relations, which is all we need in the stable case. In the simple
context things may no longer be so simple : The
general theory requires us to consider classes modulo type-definable
equivalence relations, so-called hyperimaginaries - although
there is no first-order example where hyperimaginaries cannot be
eliminated in favour of ordinary imaginaries.I shall develop the model theory of hyperimaginaries, and present their most important occurrence, namely as canonical base for a Lascar strong type. | |

Frank Wagner : | Groups in simple theories Slides |

I shall present the basic properties of simple groups (normal subgroups, chain conditions, generic types, Schlichting's Theorem and variants). | |

Itaï Ben-Yaacov : | Simplicity in compact abstract theories Slides |

When working with simple theories, one may
get the impression that the
distinction between formulas and partial types is not so important (in
contrast with the stable case). In other words, the ability to take
negation is not so important, but compactness is. We turn this impression into a precise statement by working in the framework of compact abstract theories (cats): negation is taken out, but compactness is preserved for positive formulas. For the sake of simplicity we add the (very weak) assumption of thickness, namely that indiscernibility is type-definable (with positive formulas). I will describe the framework of thick cats, and how first order simplicity theory generalises to it fully, although some of the
proofs need to be modified.Examples of thick simple (or stable) cats include: simple (or stable) first order/Robinson theories, Hilbert spaces, etc. | |

Zoé Chatzidakis
: | Simplicity in fields |

I will recall some of the classical results on fields with a simple theory, and then present examples of such fields, maybe with additional structure. |

We observe that for every (complete) simple theory

I will sketch the proof of this result, and use it as an example for several notions that arise naturally when considering simple cats.

**Alexander Berenstein:** *Dividing in the
algebra of compact operators over a Hilbert space*

**Abstract:** Simplicity and
Stability inside a Hilbert space *H* can be studied from many
points of view. One approach is looking at *H* as a large
strongly homogeneous structure and apply the tools developed by
Buechler and Lessmann. We review some of the properties of
independence and dividing from this perspective inside a Hilbert space.
We also define the algebra of finite rank operators on *H* and of compact
operators on *H*. We study simplicity and stability inside these
structures and the relationship with independence in *H*.

**Tristam de Piro:** *The geometry of minimal
types in simple theories*

**Abstract:** I will present a geometric proof that
linearity implies one-basedness for rank 1 Lascar strong types,
and summarise closely related results.

**Clifton Ealy:** *þ-forking in simple
theories*

**Abstract:** I sketch the proof that þ-forking is the same
as forking in a simple theory with elimination of hyperimaginaries.

**David Evans:** *Ample
dividing*

**Abstract:**We construct a stable universal domain in which forking is
trivial, but which has a reduct which is *n*-ample for all
*n*, with respect to a reasonably well-behaved notion of independence. In
particular, the reduct is not *CM*-trivial. It is not clear whether
the structure (and therefore the reduct) is actually stable in the
full first-order sense.

**Alexander Iwanow:** *Strongly determined
types and simple theories*

**Abstract:** The notion of a strongly determined
type generalizes the notion of a definable type. In a stable theory
every type extends to a strongly determined one. The question when
this is true in a simple theory is central in the talk. In particular
I am going to discuss the case of simple theories obtained from stable
ones by adding a relation in some generic way.

**Kikyo, Hirotaka:** *Generic automorphisms
of unstable theories*

**Abstract:** I will talk on the conjecture stating that if *T*
is unstable and model complete, then the theory *T* + `ø is
an automorphism' does not have a model companion.

The conjecture is true if *T* has the strict order property, or if
*T* has the amalgamation property for automorphisms.

Allthough the first example for the conjecture was the
theory of the random graph, I unfortunately still don't have a general
argument valid for simple theories.

**Byunghan Kim:** *Non-finite axiomatizability of some
*w*-categorical rank-1 theories*

**Abstract:** We study the non-finite axiomatizability of
*w*-categorical rank-1 structures. We reach the desired conclusion
for the case of a trivial geometry, and for a generic
predicate. Extensions beyond the rank-1 level are expected.
We conjecture the result for any *w*-categorical 1-based theory.

(Joint work with de Piro and Young.)

**Krzysztof Krupinski:** *Profinite structures
interpretable in separably closed fields*

**Abstract:** Profinite structures are inverse limits of finite
permutational structures. We consider the question which profinite
structures are interpretable in separably closed fields. We introduce
the notion of profinite structures weakly interpretable in a
first-order theory. Using some model theory of separably closed fields
(and showing some new results) we show that profinite structures
interpretable in separably closed fields are the same as profinite
structures weakly interpretable in *ACF _{p}*. We then
find a strong connection between the last notion and the the main
problem of Galois theory. We also construct examples with particular
model-theortic properties (smallness,

**David Lippel:** *What *w*-categorical
theories are finitely axiomatizable ?*

**Abstract:** I shall survey results on finite axiomatizability of
*w*-categorical theories.

**Dugald Macpherson:** *Asymptotics of
definable sets in finite structures*

**Abstract:** A theorem of Chatzidakis, van den Dries and Macintyre
gives good asymptotic information on the sizes of definable sets in
finite fields, and yields a notion of measure (and dimension) for
definable sets in pseudofinite fields. I will describe joint work with
Charles Steinhorn, in which we consider arbitrary classes of finite
structures which satisfy the conclusion of the CDM theorem. This
leads to the notion of `measurable' supersimple theory, which will
be discussed.

**Ludomir Newelski:** *Local modularity in simple
theories***Abstract:** I will recall the definition of local
modularity in stable theories and will discuss this concept in the
simple case. I will show how to generalize the weakly minimal case to
some supersimple theories.

**Ludomir Newelski:** *The diameter of a
Lascar strong type*

**Abstract:** Let *R* be the reflexive, symmetric and
type-definable relation which holds of (*a _{0}, a_{1}*)
if

**Alf Onshuus:** *Properties and consequences of
þ-independence* Slides

**Abstract:** We develop a new notion of independence,
þ-independence, and show that in a large class of theories
(including all simple ones) it has many of the relevant geometric
properties. In stable, supersimple and *o*-minimal theories it
agrees with the usual notion of independence, thus giving a unified
approach to stability (simplicity) and *o*-minimality. Finally,
we discuss some connections between þ-independence and the stable
forking conjecture.

**Anand Pillay:** *Lovely pairs of models of a simple
theory*

**Abstract:** We study a simple analogue of Poizat's
*belles paires*, called *lovely pairs*. We give criteria for
a saturated model of the theory *T _{P}* of lovely pairs
(of models of a simple theory

(Joint work with Itaï Ben-Yaacov and Evgueni Vassiliev.)

**Massoud Pourmahdian:** *A simple homogeneous
model without the stable forking property*

**Abstract:** We use Fraïssé-Hrushovski amalgamation to
construct a simple homogeneous model without the stable forking property.

**Ziv Shami:** *Unidimensional simple theories*

**Abstract:** We will present some recent coordinatisation theorems
and new foreignness notions for simple theories, and some of their
applications to unidimensional simple theories.

**Ivan Tomasic:** *The group configuration in
simple theories and geometric simplicity theory*

**Abstract:** We shall give a survey of the recent work with
Ben-Yaacov and Wagner on the group configuration in simple theories
and its applications reminiscent of geometric stability theory.

**Tsuboi, Akito:** *Simple theories without
PAPA.*

**Abstract:** I will give some examples of simple theories without PAPA.
I also state some problems concerning such theories.

**Martin Ziegler:** *A new approach to Lascar strong
types***Abstract:** The title says it all. Or maybe it
doesn't. Oh well - this abstract was written by the organizer anyway.

- (Bradd Hart) Let
*T*be simple. Is there always a reduct*T*of_{0}*T*such that*T*is stable, and_{0}- for all models
*M*of_{0}, M_{1}, M_{2}*T*, if*M*and_{1}*M*depend over_{2}*M*, then this forking is witnessed by a (stable) formula in the language of_{0}*T*?_{0}

**False**for pseudofinite fields.

(Itaï Ben-Yaacov) Is there a map from the type-space of*T*to the type-space of a stable theory preserving independence both ways ?

This is true for*Psf, TP, TA*.

- (Anand Pillay) Is there an unstable theory (simple, supersimple)
with a unique theory of proper pairs ?

- (Ziv Shami) Let
*U*be an Ø-invariant (large) set. Call a type*p*__controlled__over*U*if there is a set*A*of parameters such that all realizations of*p*are in*DCL(U, A)*. Put*Û*= {*a*:*tp(a)*is controlled over*U*}, the__controlled hull__of*U*.- Determine the possibilities for
*Û*in a simple theory. - If
*I*(*U*) = {*a*:*tp(a)*is*U*-internal }, does*Û*contain*I*(*U*) in a simple theory ?

- Determine the possibilities for
- (David Evans) Let
*M*be*w*-categorical of*SU*-rank*1*with weak elimination of imaginaries. Suppose*A*,... is a chain of algebraically closed subsets of_{1}, A_{2}, A_{3}*M*with*dim*(*A*) =_{i}*i*. Let*l*be the minimal line size when localizing_{i}*M*at*A*. Note that_{i}*l*is non-decreasing._{i}

Can*l*go to infinity ?_{i}

Observations- If there is some
*q*such that |*acl*(*a*)| <_{1}, a_{2},..., a_{m}*q*, then^{m}*l*is eventually constant._{i} - If
*l*is eventually constant at least 3, we get a group configuration._{i}

- If there is some

Itaï Ben-Yaacov (Paris-7)

Alexander Berenstein (Notre Dame)

Thomas Blossier (Paris-7)

Elisabeth Bouscaren (CNRS, Paris-7)

Enrique Casanovas Ruiz-Fornells (Barcelona)

Zoé Chatzidakis (CNRS, Paris-7)

Tristam De Piro (MIT)

Françoise Delon (CNRS, Paris-7)

Clifton Ealy (UC Berkeley)

David Evans (East Anglia)

Bradd Hart (McMaster, Fields Institute)

Martin Hils (Paris-7)

Alexander Iwanow (Woclaw)

Kikyo, Hirotaka (Tokai)

Byunghan Kim (MIT)

Martin Körwien (Paris-7)

Piotr Kowalski (Wroclaw)

Krzysztof Krupinski (Wroclaw)

David Lippel (McMaster)

Angus Macintyre (Edinburgh)

Dugald Macpherson (Leeds)

Yerulan Mustafin (Lyon-1)

Ludomir Newelski (Wroclaw)

Herwig Nubling (East Anglia)

Alf Onshuus (UC Berkeley)

Assaf Peretz (UC Berkeley)

Anand Pillay (UI Urbana-Champaign)

Françoise Point (FNRS, Mons-Hainaut)

Juan Francisco Pons i Llopis (Barcelona)

Massoud Pourmahdian (East Anglia)

Serge Randriambobolona (ENS Lyon)

Cédric Rivière (Mons-Hainaut)

Olivier Roche (Lyon-1)

Ziv Shami (McMaster)

Katrin Tent (Würzburg)

Ivan Tomasic (Leeds)

Tsuboi, Akito (Tsukuba)

Alex Usvyatsov (Jerusalem)

Frank Wagner (Lyon-1)

Roman Wencel (Wroclaw)

Jessica Young (MIT)

Martin Ziegler (Freiburg)

Frank O. Wagner Institut Girard Desargues et Institut universitaire de France |
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