(Ayers is important to Obama, and it's possible they met in Manhattan; but there's not much to really link Ayers to Columbia itself.)
WIKI:
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After the September 11 attacks in 2001, Brzezinski was criticized for his role in the formation of the Afghan mujaheddin network. He countered that blame ought to be laid at the feet of the Soviet Union's invasion, which radicalized the relatively stable Muslim society. ...
In August 2007, Brzezinski endorsed Democratic presidential candidate Barack Obama. He stated that Obama "recognizes that the challenge is a new face, a new sense of direction, a new definition of America's role in the world."[23] also saying, "What makes Obama attractive to me is that he understands that we live in a very different world where we have to relate to a variety of cultures and people."[24] In September 2007 during a speech on the Iraq war, Obama introduced Brzezinski as "one of our most outstanding thinkers,"
Tiny cough of poster's remorse from me. Still workin' on it!
That Brzezinski Link is very important ... as you point out, Obama's stay at Columbia was rather suspiciously brief for him to study his way to the degree ... and of course, we are not allowed to see what courses he actually took.