Past Logic Seminar

Countable Borel equivalence relations arise naturally as orbit equivalence

relations for countable groups. For each countable Borel equivalence relation E

there is an infinitary sentence such that E is equivalent to the isomorphism

relation on countable models of that sentence. For first order theories the

question is open.

I'll include a rather short proof of this connectedness in a group of finite

Morley rank, but I'll maybe spend most of the time talking about related matter

without giving proofs.

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Let M be an ordered vector space over an ordered division ring, and G a definably compact, definably connected group definable in M. We show that G is definably isomorphic to a definable quotient U/L, where U is a convex subgroup of M^n and L is a Z-lattice of rank n. This is a joint work with Panelis Eleftheriou.

Much is now known about exp-log series, and their connections with o-

minimality and Hardy fields. However applied mathematicians who work with

differential equations, almost invariably want series involving

trigonometric functions which those theories exclude. The seminar looks at

one idea for incorporating oscillating functions into the framework of

Hardy fields.

The first seminar will be given with the new students in

mind. It will begin with a brief overview of quantifier elimination and its

relation to the back-and-forth property.I shall then discuss complexity issues

with particular reference to algebraically closed fields.In particular,how much

does the height and degree of polynomials in a formula increase when a

quantifier is eliminated? The precise answer here gave rise to the definition

of a `generic' transcendental entire function,which will also be

discussed.