Title: Dimension theory in t-minimal and dp-minimal structures

Abstract:A theory T is "t-minimal" in the sense of Mathews if any model M is endowed with a definable topology with no isolated points, such that every definable set has finite boundary. For example, dense o-minimal and C-minimal theories are t-minimal. In this talk, I will present a dimension theory for definable sets in t-minimal structures, and highlight some analogies to Pierre Simon's work on dimension theory in dp-minimal structures. In both cases, a key role is played by a certain ideal of "narrow" sets. If time permits, I will also discuss the role played by t-minimality and narrow sets in the classification of fields of finite dp-rank.