Abstracts

Here are some abstracts submitted for talks in the Seminar.

Homogeneous ordered graphs

Gregory Cherlin, May 17, 2013

We outline the classification of homogeneous ordered graphs. This answers a very natural question of Lionel Nguyen Van Thé, in a context that mixes model theory and topological dynamics. Lachlan and Woodrow classified the homogeneous (unordered) graphs. The modern theory finds particularly striking connections to topological dynamics when an ordering is present. I will sketch Lionel's reasons for wanting to see the classification, indicate why I thought the question was unreasonable, and explain what made me change my mind.

References:

Descartes as model theorist

David Pierce, May 16, 2013

In his Geometry of 1637, Rene Descartes gave a geometric justification of algebraic manipulations of symbols. He did this by interpreting a field in a vector-space with a notion of parallelism. At least this is how we might describe it today. I alluded to this in the abstract for my February 28 seminar, but did not actually talk about it. Now I want to talk about it.

By fixing a unit, Descartes defines the product of two line segments as another segment. He relies on a theory of proportion for this. Presumably this is the theory developed in Book V of Euclid's Elements—the theory that inspired Dedekind's definition of real numbers as “cuts” of rational numbers.

But this theory has an “Archimedean” assumption: for any two given segments, some multiple of the smaller exceeds the larger.

In fact this assumption is not needed, as Hilbert observed in Foundations of Geometry. Hilbert uses instead Pappus's Theorem. This work may be known now as “interpreting a field in a projective plane”.

I tracked down Pappus's original argument (from the 4th century), and yesterday I wrote an account of it on Wikipedia.

As for model theory, another result that comes out of these considerations is that there are model-complete theories of Lie-rings equipped with an endomorphism of the abelian-group-structure.

Covering definable open sets by open cells

Oleg Belegradek, May 9,2013

Wilkie (2004) proved that in any o-minimal expansion of a real closed field any bounded open definable set is a finite union of open cells. Andrews (2009) and Edmundo, Eleftheriou, Prelli (2012) generalized this result to certain o-minimal expansions of ordered abelian groups. In my talk I will tell about these results.

C-minimal modules

Gönenç Onay, May 2, 2013

C-minimality (in the context of fields) isolates the algebraically closed valued fields among the all valued fields by expressing that these are exactly the ones where the one-dimensional definable sets are simply those quantifier-free definable in a weak reduct just sufficient to describe the ultrametric (this is a result by Macpherson and Haskell); analogously to strong-minimalty (resp. o-minimality) for algebraically closed fields (resp. real closed fields).

We will consider ultrametric modules case. The main examples comes from valued difference fields: the module structure consists of two ingridients: the vector space structure over some subfield and the action of a distinguished field endomorphism (e.g. for any valued field (K,val) of char p>0, the structure given by the additive group <K,+> as F_p-vector space together with Frob:x→x^p and the ultrametric induced by the function val, is a valued module structure on K). I will explain the complete characterization of such C-minimal modules and relate it to the main ingredient in almost all well understood results of “relative” quantifier elimination (yielding Ax–Kochen and Ershov type theorems) for positive characteristic valued fields or difference valued fields.

Generically sharply transitive actions in the finite Morley rank context

Ayşe Berkman, April 25, 2013

Motivated by a question raised by Borovik and Cherlin, I am trying to understand generically sharply transitive actions in the finite Morley rank context. In my talk, first I plan to give all the definitions, examples and some motivation, then I shall discuss a possible approach on how to classify such actions. This is joint with Borovik.

Two remarks on shattering types and an application to groups which are hyperdefinable in a theory without the independence property

Cédric Milliet, March 14, 2013

I plan to propose a generalisation of the independence property to partial types, originally motivated by the fact that some objects which are definable in a stable group G are type-definable when G is merely without the independence property (i.e. NIP). I will give an application to groups which are quotient of a type-definable set by a type-definable equivalence relation in a NIP theory.

Chains of theories and group (scheme) actions

Piotr Kowalski, March 7, 2013

I plan to expand a comment I made at the end of David's talk last Thursday. I will interpret automorphisms and HS-derivations as group scheme actions and discuss connections between properties of these actions and chains of the corresponding theories.

Chains of Theories

David Pierce, February 28, 2013

I shall speak about properties of theories that are preserved under taking unions of increasing chains of theories. I talked about these things last fall, on November 29, but I shall try both:

not to assume that you were at that talk or remember it,
to say things that were not in that talk.

The main examples are the following.

For each natural number m, the theory, called m-DF, of fields with m commuting derivations has a model companion (which is the theory of the existentially closed models of m-DF, this class being elementary). Call this model-companion m-DCF. The theories m-DCF are mutually inconsistent, and therefore the union of the m-DF, which is consistent, has no model companion.

However, the theory of fields of characteristic 0 with omega-many commuting derivations does have a model-companion, which preserves the quantifier-elimination, completeness, and stability of the theories m-DCF₀, but not their omega-stability.

The theory of vector-spaces with the scalar-field as an additional sort is highly dependent on the chosen signature. If there are 2-ary, 3-ary, …, m-ary predicates for linear dependence of vectors, let the theory be m-VS. Then the existentially closed models of m-VS have dimension m, and the scalar field is algebraically closed. Thus the union of the model-companions of the m-VS is inconsistent. But the union of the m-VS has a model-companion (as I said last time), whose completions, obtained by specifying a characteristic for the field, are omega-stable. If m is at least 2, the m-ary predicate for linear dependence can be used to interpret the scalar field in the vector space (if this has dimension at least m) in such a way that, in the foregoing observations, m-VS can be replaced with the the theory of (one-sorted) abelian groups with the appropriate m-ary predicate.

Canonical Base Property (Zariski geometries vs jet spaces)

Piotr Kowalski, February 21, 2013

A theory satisfies Zilber's dichotomy if each non-modular strongly minimal type is internal to a definable field. Zilber's dichotomy holds in Zariski geometries (Hrushovski–Zilber). Pillay have noticed that Zilber's dichotomy for the theory of compact complex spaces follows directly from a theorem of Campana (due independently to Fujiki). After this observation, Pillay and Ziegler developed the jet space method which gives Zilber's dichotomy for several other theories. Actually, the jet space method yields a stronger model-theoretic property called the canonical base property. I will describe the canonical base property (giving all the necessary definitions). I will also present a possible connection between this property and Hasse-Schmidt derivations.

Representations of finite Morley rank

Adrien Deloro, November 1, 2012

The Cherlin–Zilber conjecture has received most of the attention devoted to groups of finite Morley rank. But groups of fMR are not interesting only as abstract groups, also as permutation groups. The talk is about modules of finite Morley rank, i.e. groups acting on abelian groups. I shall review the various results obtained in this vein.

A Jordan decomposition for minimal simple groups of finite Morley rank

Tuna Altınel, November 1, 2012

The Jordan decomposition of a matrix as the sum of two multiplicatively commuting matrices, one diagonalizable and the other nilpotent (or unipotent if the matrix is invertible), is fundamental in the analysis of semisimple Lie algebras and simple algebraic groups. It connects algebraic and geometric aspects of these structures.

Infinite simple groups of finite Morley rank, of which the only known examples are simple linear algebraic groups over algebraically closed fields, motivate the question whether one can propose an abstract Jordan decomposition using only group-theoretic properties, said more model-theoretically a decomposition definable in the group language.

In a joint work with Jeffrey Burdges and Olivier Frecon, we have proposed an approach for a Jordan decomposition of finite Morley rank coherent with the one in algebraic groups. The definition, quite natural for the expert, involves Carter subgroups. We have established that the decomposition enjoys the expected basic properties over a subclass of minimal simple groups of finite Morley rank. This subclass is rather restricted. The amount of effort is considerable. In my talk, I will try to illustrate why one has to work so much for so little, as well as why one can still expect much more with so little.

Homogeneity in relatively free groups

Oleg Belegradek, October 4, 2012

We show that any torsion-free, residually finite relatively free group of infinite rank is not aleph-one-homogeneous. This generalizes Sklinos' result that a free group of infinite rank is not aleph-one-homogeneous, and, in particular, gives a new simple proof of that result.

On model-theoretic connected components

Jakub Gismatullin, April 19, 2012

In my talk I will survey some recent results on model-theoretic connected components of groups and their quotients. I will concentrate mostly on the following classes of examples:

finitely generated nilpotent groups,
split linear groups (that is Chevalley groups) over infinite fields and some local rings
some central extensions of SL_2 over ordered fields.

It is a join work with C. Chen, N. Bowler and K. Krupinski.