Yeah, you will notice the following sentence in every single job in Canada: All qualified candidates are encouraged to apply; however, Canadian citizens and permanent residents will be given priority.
So how this materializes is that search committees have to create two piles of applicants: Canadians and non-Canadians. Only if there are no qualified Canadians do they move on to the non-Canadians. It is kind of well known that there are 2-3 schools that don't strictly follow this (I am not going to say their names in a public forum, but it would be easy to guess by looking at the composition of faculty). But essentially every other school in Canada follows this pretty closely.
So that will help get your foot in the door if you wanted to work in Canada, but you also have been really focused on brand recognition -- what about research fit? Do you see professors at both schools that fit what you are interested in? I think that is more important.
1) I want to live in Canada in long term (Which makes me close to Alberta)
2) While Alberta faculty members try their best to help my decision by explaining the whole PhD process, clarifying the questions in my mind, etc; A&M professors are seem like they do not even care whether I join their program or not. (Which again makes me close to Alberta)
This makes me feel that although the research fit is slightly better at A&M, I will not get enough attention from the professors. Besides, A&M has almost 2 times PhD students which also makes me think that I will need to compete with them to get the professors's attention in their limited time.
Long story short, I would be so happy If someone convince me on either Alberta has a strong recognition in Canada/EU/US or the ranking or recognition of the university do not matter much.
It is clearly a difficult choice for you.
It is my opinion that becoming Canadian will drastically increase your chances on the job market in Canada. But ultimately what will matter is how productive you can be in your PhD.
You need to also weigh specific professors you'd like to work with at each school. Who's in their network? Canadian/EU professors? How well known are they (this IMO is just as important as the reputation of the school itself)
Look, ultimately these are both good schools, and they will set up you for success, as long as you do your part and work hard in the PhD and set yourself up for success.
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