Nonstandard Analysis (NSA) was introduced around 1965 by Robinson as a formalization of the intuitive infinitesimal calculus which is in use to date in most of physics and historically in mathematics until the advent of Weierstrass’ epsilon-delta framework. Famous people like Connes and Bishop have derided NSA for its alleged utter lack of computational/effective/constructive content. In this talk I show that every theorem of ‘pure’ NSA can be (equivalently) converted to a theorem of computable mathematics. In many cases, the resulting theorem is even constructive in the sense of Bishop.

How much information about a group G is contained in the group ring K(G) for an arbitrary field K? Can one recover the algebraic or geometric structure of G from the ring? Are the algorithmic properties of K(G) similar to that of G? I will discuss all these questions in conjunction with the classical Kaplansky-type problems for some interesting classes of groups, in particular, for limit, hyperbolic, and solvable groups. At the end I will touch on the solution to the generalized 10th Hilbert problem in group rings and how equations in groups are related to equations in the group rings. The talk is based on joint results with O. Kharlampovich.

In this talk and its sequel I will construct models of arithmetic with exactly two expansions to a model of $ACA_0$. To do so, I will use a modified version of Keisler’s construction of a rather-classless model. This talk will focus on this construction, while in Part 2 I will show how to use this construction to get the result.

${\rm AD}_{\mathbb R}$ is a strengthening of the determinacy axiom that states that all games on the real numbers are determined. It is a Theorem of Solovay that under ${\rm ZF}+{\rm AD}_{\mathbb R}$ there is a fine, countably complete and normal filter on $P_{\omega_1}(\mathbb R)$, so $\omega_1$ is $\mathbb R$-supercompact. The exact consistency strength of the theory ${\rm ZF}+ {\rm AD}+$“$\omega_1$ is $\mathbb{R}$-supercompact” is, however, weaker than the one of ${\rm ZF}+{\rm AD}_{\mathbb R}$.
One central interest of Inner Model Theory is to construct/find canonical models for theories extending ${\rm ZF}$. A natural question is, then, whether there is a canonical model for the theory ${\rm ZF}+ {\rm AD}+$“$\omega_1$ is $\mathbb{R}$-supercompact”.
In this talk, we will discuss the consistency strength and minimal models of this theory. We will discuss the proof of the uniqueness of minimal models of this theory, under various appropriate hypotheses. And time permitting we will discuss the proof of the result that under ${\rm ZFC}$ there is at most one minimal model of this theory. This is joint work with Nam Trang.

I will give a standard introduction to the theory of Borel reducibility, including some details for non-logicians. The rest of the talk will be about some results in classification problems for countable nonstandard models of arithmetic due to Samuel Coskey and myself.

Connections between logic and operator algebras in the past century were few and sparse. Recently, some long-standing open problems on the structure of operator algebras were solved using methods from mathematical logic. I will survey some of these results, with a particular emphasis on applications of set theory.