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UPPSALA UNIVERSITET Matematiska institutionen Vera Koponen |
Model theory studies general properties that may be shared
by different kinds of structures.
For example, graphs and groups are different kinds of structures
since the former involves one binary symmetric relation and the latter
involves one binary function (or 'operation').
But nevertheless there are properties which some graph can have in common
with some group, but not with some other graph.
In model theory such properties often measure the complexity
of the structure relative to what can be expressed about it within
some formal language, usually first-order logic
or variations of it.
In this spirit a rich set of methods and results for classifying
infinite structures, called "classification theory" or "stability theory", has evolved.
There is also extensive work in areas bordering between model theory
and other subjects in mathematics such as algebra, analysis, geometry and discrete mathematics;
much of this work is inspired by ideas from classification theory.
The above concerns mostly infinite structures.
The model theory of finite structures has developed mainly in connection with
issues in computer science, in particular descriptive complexity theory,
but there are also interactions with probabilistic methods in discrete mathematics,
such as random graph theory.
In the logic group, connections between finite and infinite model theory have been studied.
On the one hand, by applying ideas and methods from infinite model theory
to the context of finite structures. On the other hand, by studying "finiteness properties"
that infinite structures may have, such as being approximated by sequences
of finite structures, by a blend of stability theoretic and probabilistic methods.
There is interest in further exploration of structures (finite and infinite)
with a combination of model theoretic, probabilistic and computational methods.
More about Model theory in Wikipedia.