Hello! Welcome to the website for the Logic Seminar in the Ohio State Department of Mathematics. We normally meet on

Olivier's construction of the universal commutative (von Neumann) regular ring over a commutative ring is generalized to obtain the universal *-regular ring over a noncommutative ring (R, *) with involution. The construction of a universal *-regular ring proceeds similarly with the Moore–Penrose inverse replacing the role of the group inverse in the construction of universal abelian regular rings.

The involution of (R, *) induces an involution on the modular lattice L(R, 1) of positive primitive formulae in the language of left R-modules. It is shown that *-regular ring coordinatizes the quotient lattice of L(R, 1) modulo the least congruence for which the involution designates an orthogonal complement.

Some explicit examples will be given in the context of some algebras, as the Jacobson algebra. This is joint work with Ivo Herzog.

Classical theorems of Novikov, Boone, and Higman establish important connections between recursive and recursively enumerable sets on the one hand and the theory of finitely presented groups on the other hand. Despite the groundbreaking nature of these results, not much is known about possible extensions of those results to the realm of higher Turing degrees. My talk will be a contribution in this direction.

In my talk, I will discuss how one can view Turing degrees through invariants coming from geometric group theory. In particular, I will introduce a new quasi-isometric invariant, called (λ, μ)-taut filling spectra, and will show how this invariant can grasp Turing degrees of arbitrary degrees. Through these means, we will also introduce an interesting special class of functions, called Turing saturated maps, which will be useful for our applications.

As a main application, we will discuss a purely geometric group theoretical result that establishes quasi-isometric diversity of f.g. left-orderable simple groups, which is an important class of groups discovered by Hyde and Lodha. Previous attempts to obtain this result through more usual geometric/algebraic tools have not succeeded, which emphasizes the importance of the idea of Turing degrees in this context.

The talk is aimed to have an expository component on concepts from geometric group theory.

The countable abelian p-groups that have no divisible summands are determined, up to isomorphism, by their Ulm invariants. This classification can be used to determine the homogeneous countable abelian p-groups. One such abelian p-group turns out to be a universal countable abelian p-group for purity, i.e., every countable abelian p-group admits a pure embedding into it. It is the last step needed to complete the solution to Fuchs' Problem 5.1 below ℵ

We will start off with some background, including how Ulm's Theorem is used to obtain a Scott sentence as well as some motivating examples. This is joint work with Marcos Mazari Armida.

This page is currently maintained by Nigel Pynn-Coates.