[Turkmath:1236] IMBM Model Theory Day II

[Ozlem Beyarslan]

Dear All,
We will have the second of our IMBM Model Theory Meetings on 18 April Monday,
starting at 13:30. I attach below the program.
We thank IMBM for their hospitality,

Özlem Beyarslan
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IMBM Model Theory Meetings, April 18

David Pierce (Mimar Sinan) 13:30 -- 15:00

Title: Spaces and fields

Abstract: Using areas, Euclid proved results that today we consider
as algebraic.  We consider them so, because Descartes justified
algebra by showing how it could be considered as geometry.

Such observations can be understood as resulting from the equivalence
of certain categories.

Models of a given first-order theory T are the objects of two
different categories:

*  Mod(T), in which the morphisms are embeddings, and

*  Mod*(T), in which the morphisms are elementary embeddings.

The latter category is closed under direct limits; if the former is
likewise closed, then T has universal-existential axioms (and
conversely).  T is called model-complete if the two categories are the
same.  T is called companionable if it includes a model-complete
theory, called the model companion of T, in a model of which each
model of T embeds.

A vector space here is a pair (K,V), where K is a field, V is an
abelian group, and K acts on V.  The theory of vector spaces in this
sense has a model companion, which is theory of one-dimensional vector
spaces.

If T is the theory of vector spaces of dimension at least two, and U
is the theory of abelian groups with an appropriate notion of
parallelism, then Mod(T) and Mod(U) are equivalent.  If S is field
theory, and T_n is the theory of n-dimensional vector spaces (where
n>0) with a given basis, then Mod(S) and Mod(T_n) are equivalent.

In a vector space (K,V), V may also act on K as a Lie ring of
derivations; then (K,V) becomes a Lie--Rinehart pair.  Such pairs can
be given universal-existential axioms, using only the signature of
abelian groups for each of K and V, along with a symbol for the action
of each on the other.  In his 2010 dissertation, Özcan Kasal showed
that the resulting theory is not companionable, although if predicates
for certain definable relations are introduced, the theory becomes
companionable, and the model companion is not stable.  It turns out
that like the theory of the integers as a group, the model companion
even has the so-called tree property.


Gönenç Onay (Mimar Sinan) 15:30 -- 17:00

Title: On the elementary theory of F_p((t))

Abstract:

The famous Ax-Kochen and Ershov theorem gives in the complete
theory of the fields of p-adics numbers Q_p for every prime p and hence
proves that this theory is decidable. While seems to be so similar, the
elementary theory of F_p((t)) is widely unknown and it is one of the
challenging questions of contemporary model theory and arithmetics. In this
talk I will sketch historical approaches to this problem and give my humble
contribution.

-- 
Özlem Beyarslan

Bogazici Universitesi Matematik Bolumu
Bebek, Istanbul 34342
Tel: 90 212 359 6535
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