I'm doing my master's in mathematics, and I will be graduating next year. I'm looking to start a PhD soon, perhaps next year, most likely in two years. I'm really interested in category theory and topos theory as a framework for mathematics, mostly motivated by logic, but not limited to logic. Here are examples of the flavor of things that would be interesting to me:
- topoi in geometry and logic (yes, I'm working through Mac Lane & Moerdijk);
- higher categories and higher topoi;
- fancy category theory in general;
- synthetic differential geometry;
- abstract homotopy theory;
etc.
I've been looking into some of the "big names" in this area. However, a big problem I discovered in the middle of this is that most of these big names are very senior and likely to retire soon (or have already retired!). Meanwhile, the younger generation of mathematicians working in this area seems significantly smaller. I know of a few of them, but it seems that they are dispersed over the world (and practically speaking, many of them work in highly competitive departments which might mean that it will be difficult for me to get in).
But regardless, where are some good places to do a PhD in this area? By "good" I have two criteria:
- very preferably, have at least one faculty member working in the area of topos theory/category theory who is not retiring soon;
- have a concentration of (i.e., at least a few) faculty members interested in related topics (e.g., "Grothendieck-style" algebraic geometry, abstract algebraic topology, but also including mathematical physicists and theoretical computer scientists working in related areas, e.g. semantics of lambda calculi).