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No. 504
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I would say it is that bad. They can't make you leave if you pass their qualifiers and maintain an acceptable GPA, but they can choose not to fund you as a TA/RA. There is a guy in my department who is going through this crises at the moment, I think because he was openly critical of some of the practices (and I would say rightly so) of the dept. On the other hand, if you are in good with some of the faculty you can be guaranteed some kind of success, e.g. extra qualifier attempts. I even know one guy who was a lab tech and was offered admission because he got close with a prof who liked his work. It can really help by just not being a social retard and shitting where you eat. Anyway, the result of being a pariah is that you've got to find work somewhere else and the best place is probably teaching at a C.C. to defray some of the expenses. Amusingly, the guy I know must be titled Professor because he does that.
It doesn't sound like you've actually got that problem though, so maybe the dept legitimately doesn't have money and over accepted students. I can't really see why they would care much about the direction of your study unless it is a really small dept and they were depending on someone joining a small program, but I'm physics so I can't really relate there. What'd happen in physics would be like "high energy is full, go find some other passion" or "I'm going to retire/die soon so atmospheric research is effectively ending". That means you'd potentially be spending more time waiting for positions to open up as others graduate which is bad if you're taking on debt for a math phd. If they laid it out like "you join this program and you get funding" you should just take it and milk it for all it is worth. Better to get paid and then switch uni than to pay and end up switching anyway.
I haven't really heard the bit about saying you want to teach. I can't see many faculty really care too much about it, as they're already set with a position and if they're actually making decisions they've been there for a decade and firmly entrenched. I don't think I have ever heard of a faculty member being proud of their doctoral students that went on to teach though, they usually talk about the ones that went to work with Intel or something like that. I can imagine that it is different in math though as most people would just assume math phds are going to teach.
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