Summer 2010 Course on Model Theory: Chapter 4

Summer 2010 Course on Model Theory : Chapter 5

Having seen that categoricity is a concept of somewhat limited usefulness, we now turn to a more versatile (and historically prior) technique for establishing model completeness, namely quantifier elimination. But more than just giving further criteria for showing that theories are model / completen, quantifier elimination is a fundamental concept in its own right (arguably more so than categor...

Summer 2010 Course on Model Theory and Its Applications

2010 Summer Course on Model Theory : Chapter 1

1) The Nullstellensatz and the R-Nullstellensatz. 2) Chevalley’s Theorem: the image of a constructible set is constructible. 3) (Grothendieck, Ax) An injective polynomial map from C to C is surjective. 4) Hilbert’s 17th problem: a positive semidefinite rational function f ∈ R(t1, . . . , tn) is a sum of squares. 5) Polynomially compact operators have invariant subspaces. (Let V be a complex Hil...

Coalgebras and Modal Logic

1 These lecture notes were written mainly during my stay as an ERCIM fellow at the Masaryk University, Brno, Czech Republic. I owe special thanks to Lubo s Brim, Ji r Rosick y, and the Faculty for Informatics. Preface These course notes present universal coalgebra as a general theory of systems. By`system' we understand some entity running in and communicating with an environment. We also assum...

A mini course on percolation theory

These are lecture notes based on a mini course on percolation which was given at the Jyväskylä summer school in mathematics in Jyväskylä, Finland, August 2009. The point of the course was to try to touch on a number of different topics in percolation in order to give people some feel for the field. These notes follow fairly closely the lectures given in the summer school. However, some topics c...