Ilya Shapirovsky
"Locally tabular modal logics"
Levon Haykazyan, Department of Pure Mathematics, University of Waterloo
We will prove that groups that carry locally modular homogeneous pregeometries are commutative.
MC 5403
Rahim Moosa, Department of Pure Mathematics, University of Waterloo
"Bounding solutions to first-order differential equations"
In this talk I will try to show how a recent theorem of myself and James Freitag on the model theory of differentially closed fields answers a question of Eremenko's from 1998 on algebraic solutions to a differential equation of the form P(x,x')=0 where P is a polynomial over C(t).
MC 5403
Ian Payne, Department of Pure Mathematics, University of Waterloo
Michael Deveau, Department of Pure Mathematics, University of Waterloo
We will investigate the notion of a Turing degree of a structure. However, since it is not as useful as we would hope, we instead use enumeration degrees, which prove more fruitful. We then establish some results concerning existential atomicity of a structure.
MC 5403
Michael Deveau, Department of Pure Mathematics, University of Waterloo
In this talk, I will explain some basic results about jump inversion as applied to a non-standard reducibility, the bounded Turing reduction. We will see that some standard results fail, and attempt to remedy this by modifying the jump operator to make it more compatible with the reduction. Along the way, we will encounter the Ershov hierarchy, problems with relativization, and a frightening definition.
MC 5403
Mohammad Mahmoud, University of Waterloo
Pantelis Eleftheriou, University of Konstanz
Let (M, P) be an expansion of an o-minimal structure M by a dense subset P of M, such that three tameness conditions hold. We prove that the induced structure on P by M eliminates imaginaries. As a corollary, we obtain that every small set X definable in M can be definably embedded into some P^n, uniformly in parameters. We then verify the tameness conditions in three examples: dense pairs, expansions of M by a dense independent set, and expansions by a dense divisible multiplicative group with the Mann property.
Matthew Harrison-Trainor, Department of Pure Mathematics, University of Waterloo
Every countable structure has a sentence of infinitary logic, called a Scott sentence, which describes it up to isomorphism among countable structures. We can characterize the complexity of a structure by the complexity of the simplest description of that structure. A finitely generated structure always has a $\Sigma^0_3$ description. We show that there is a finitely generated group which has no simpler description.
Laurent Bienvenu, University of Montpellier
This talk will elaborate on the Oct 30 colloquium. I will give details on several proofs, in particular how to show a $\Pi^0_1$ class is deep, applications of depth to other computability-theoretic notions, and how to use fireworks arguments with other forcing notions than Cohen-type forcing. Though attending the colloquium would give motivation and context for the topics we will discuss, the talk will be self-contained.
MC 5413