Zoom link: https://huji.zoom.us/j/82228144561?pwd=7DMWOF6kYawtg61EUzsZYPiNlGOJZU.1
Meeting ID: 822 2814 4561
Passcode: 653161
Title: Approximate rings
Abstract: I will start with some history concerning the notion of approximate subgroup which is central in additive combinatorics. In particular, I will discuss locally compact models and mention their role for structural results. Then I will concentrate on approximate subrings. I will discuss my result on the existence of locally compact models for arbitrary approximate subrings. The rest of the talk will be devoted to applications of this theorem to structural results on approximate subrings, including my very recent (not circulated yet) theorem with Simon Machado describing the structure of finite $K$-approximate subrings. This can also be viewed as a unified general form of the so-called sum-product phenomenon, which I will also briefly discuss.
Basic model theory plays an essential role in this research. The construction of locally compact models is obtained via model-theoretic connected components of definable groups and rings. Structural results on approximate subrings are obtained either using the aforementioned components or a pseudofinite context with some non-standard analysis.