Short courses

Lecture courses:

Peter Aczel (Manchester): Constructive Set Theory
Larry Moss (Indiana): Coalgebra and Circularity
Jouko Vaananen (Amsterdam & Helsinki): Playing Games on Models
Alex Wilkie (Manchester): An introduction to O-minimal Structures

Timetable

Abstracts:

Aczel: Click for the abstract in pdf form; Course Materials
Moss: Coalgebra is primarily pursued as an active field of contemporary theoretical computer science. It might be described as a category theoretic generalization of parts of automata theory, modal logic, and other fields. This set of lectures will present a side of the subject that could be more relevant to students of mathematical logic. It will aim towards connections of coalgebra to recursion, and to the study of some 'circular' phenomena of general mathematical interest. It also will be a way to learn some of the basic notions of category theory.
I will try to write a set of lecture notes and to have slides available before the Summer School. Much of the conceptual content of the course may be found at http://plato.stanford.edu/entries/nonwellfounded-set-theory/but I'll say less about set theory and more about other topics. And as I mentioned, I'll be going further with the mathematics.; Course Materials; Preliminary set of slides available at:
http://cs.indiana.edu/cmcs/circintro.pdf
http://cs.indiana.edu/cmcs/categories.pdf
http://cs.indiana.edu/cmcs/finitary.pdf
http://cs.indiana.edu/cmcs/bisimulation.pdf
http://cs.indiana.edu/cmcs/measure.pdf
Vaananen: Games occur in model theory in many guises. Elementary equivalence in first order logic has a characterization in terms of the so-called Ehrenfeucht-Fraisse game; the truth of a sentence in a model can be characterized in terms of the so-called evaluation game; finally, the existence of a model for a sentence can be characterized in terms of the so-called model existence game. I will start with an overview of these games, and show that the first two games play an important role not only in model theory generally but also on finite structures. I will then show that these games provide a unified approach to the infinitary logic L_{omega_1,omega}. Finally I will move on to stronger infinitary logics and special features of games on uncountable models, involving stationary sets and properties of large trees without long branches. An important motivation for us is the result of Shelah that the uncountable models of a countable complete theory are either all identifiable by dimension-like invariants, or else there are in all uncountable cardinalities models that are so similar to each other that there is no hope to identify them in terms of dimension-like invariants of any kind. In the latter case transfinitely long Ehrenfeucht-Fraisse games are used to measure the similarity of models.; Course Materials:notes / paper
Wilkie: The notion of an o-minimal expansion of the ordered field of real numbers was invented by L van den Dries as a framework for investigating the model theory of the real exponential function and thereby settling an old problem of Tarski. Exactly how this problem was finally resolved, together with some applications, will be the subject of my fourth lecture, but for the first three lectures the notion is best motivated as being a candidate for Grothendieck's idea of "tame topology". The latter was not formulated precisely, but it seems clear that such a candidate should satisfy (at least) the following criteria.
(A) It should be a framework that is flexible enough to carry out many geometrical and topological constructions on real functions and on subsets of real euclidean spaces.
(B) But at the same time it should have built in restrictions so that we are a priori guaranteed that pathological phenomena can never arise. In particular, there should be a meaningful notion of dimension for all sets under consideration and for any that can be constructed from these by use of the operations allowed under (A).
(C) One must be able to prove finiteness theorems that are "uniform in parameters".
The simplest example illustrating what is meant by (C) is the fact that there is an upper bound on the number of roots of a polynomial of degree at most d that depends only on d and not on the coefficients. (The bound is, of course, just d itself.) Indeed, I shall begin with the algebraic case (where the sets under consideration are those definable in the language of ordered fields) and devote the first lecture to a new exposition (at least, new to me; I learnt it from Khovanskii at a recent conference) of Tarski's elimination theorem, before discussing o-minimality in general.