Best Cities for Mathematical Study This may sound silly, but... Suppose an aspiring amateur mathematician wanted to plan to move to another city...
What are some cities that are home to some of the largest number of the brightest mathematicians? I'm sure this may depend on university presence, or possibly industry presence, or possibly something surprising. Wondering where the best place to take a non-faculty job at a university and try to make friends with some sharp minds in the computer lab or at the nearby pub might be.
 A: New York City, in addition to a vibrant city life and diverse professional opportunities, has also a vibrant academic life, with an enormous number of colleges and universities, rivaling even Boston. The large number of math seminars and workshops at the CUNY Graduate Center, for example, are usually quite welcoming of outsiders, and the typical pattern there has high-level specialized seminars in whatever subject populated by faculty and advanced graduate students from diverse regional institutions. I am certain that you would find something suitable, challenging and rewarding.
A: An obvious answer here is Paris. Both historically and currently, Paris is a bit of a Mecca for mathematicians. With the 13 universities of Paris and ihes so close there are more seminars than could be attended per week. Additionally the level of the mathematicians is top tier if anything. 
Culturally, and generally, Paris is a real city. Lots going on, lots to see, and many people find the atmosphere very inviting. I have been told by many mathematicians, and experienced it myself, that doing math in Paris seems somehow more relaxing. Additionally, public transportation, food, and general facilities are very accommodating. 
Paris may be great, but it is no nirvana, it is known for it's insane housing issues, and busy-ness. Moreeover some actually dislike this French culture. 
A: Without sounding biased in any way, I would say Cambridge/Boston is a good choice for you. In the particular order of funded research/department size there is Harvard University, MIT, Boston University, Boston College, Northeastern University, Brandeis University, Tufts University, Bently University, University of Massachusetts at Boston, Curry College, Eastern Nazarene College, Pine Manor College, Hellenic College, Lesley University, Wheelock College, Lasell College, Simmons University, Cambridge College and Bunker Hill Community College (and many, many more) within the metropolis. See http://en.wikipedia.org/wiki/List_of_colleges_and_universities_in_metropolitan_Boston for a complete list.
A number of these institutions offer extension programs (with open enrollment and classes in the evening or weekends) suitable for life-long learners and aspiring amateur mathematicians. For example, the Masters for Mathematics Teaching Program at Harvard University offers courses in all major mathematics subject areas, taught by many instructors which hold separate positions in the university (like adjunct/junior faculty, preceptors, senior lecturers, post-doctoral or teaching fellows and even a senior graduate student).
A: I imagine that you could sit in on a lot of classes at Berkeley if you ask the professors. I doubt many would mind if you are very serious about mathematics. 
A: I would think it depends largely on your personality type. Do you thrive in slow, peaceful
  environments, or do you prefer a faster environment with more variety?. I am in NYC
  and love it, but my friend prefers U of Florida in Gainesville, which is not that slow
  of a place, but he has the option of going to the country--real country, on the weekends
  ( if you want real country in NYC , you need to travel a few hours). Also, there are
  school environments that are more casual than others. I took  a class at Columbia,
  and many of the people there were a bunch of a-holes, who pretended not to know you when they saw you outside of class, even after you had taken classes with them for a whole semester, even a whole year. I think there are books out there that rate graduate programs, using many different parameters. If you are interested in getting a hold of these books and can't find thme, let me know and I will get you the actual title(s). 
Good Luck!
A: I would recommend the University of Waterloo in Canada, one of the few universities in the world with a mathematics faculty rather than just a math department.
A: I would say Princeton, New Jersey. It has Princeton University. Its mathematics department is one of the best in the whole world. There are three Fields' medalists in the department: Andrew Wiles, Charles Fefferman, and Andrei Okounkov. IAS is closed by and it has another three Fields medalists: Jean Bourgain, Pierre Deligne, and Enrico Bombieri. Rutgers University is also closed by. 
