I'm a mathematics Master's student and I'm applying to Ph.D. programs this semester. I want to study in a field related to Geometry/Topology and I'm trying to come up with a list of possible universities to apply to from all over the world especially Europe, Australia, and the US.
I'm not sure what I want to do exactly yet, but I know that I like ideas with geometric intuition behind them, as there is in some parts of Algebraic topology and differential geometry, but I don't particularly like the mostly algebraic parts in AT or the long calculations in DG. (I can tolerate them as long as they're used to describe something geometric). I've been to some summer schools on low-dimensional topology and I've quite enjoyed the more geometric parts.
I'm looking for universities that have at least a few geometric-minded people working in low-dimensional topology and a relatively large research group in both topology and geometry in case I change my mind about low-dimensional topology. I would appreciate any suggestions!
Also a side question: What do you think my chances are of getting into Berkeley? It seems like the ideal school for me. I've been doing my Master's in UBC but in math-bio (working on theoretical stuff though) and I haven't published anything yet, nor have I presented my research anywhere. I'm in my third year. I feel like this makes my chances dim.