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I’m biased as I live here, but I’d go with Hamilton Heights. It’s nicer than the other options, close to campus (both Columbia and City College), has a lot of grad students and young professionals, bars, restaurants, nice architecture (actually some of the best in the city), and you have 5 trains at 145th (A, B, C, D at Saint Nicholas and the 1 at Broadway).
Seconded. The only thing is I wouldn't try to walk to Columbia every day. It's not super-far but the hills in the area make it more strenuous. It''s a quick commute on the 1 train.