mathwonk said:
university of chicago has one of the world's best math departments. i am not crazy about the local environment there in that part of chicago. i.e. it is right in the city and not the nicest part of the city, but that is true of some other urban campuses. the mathematicians there are incredibly good. some I have known or known of for a long time are: Nori, Drinfeld, Ginzburg, May, Nygard, Fefferman, Sallky, Alperin... other younger people include Matthew Emerton, whom I have recently gotten to known through mathoverflow, and who is also very nice.
I believe the department at Chicago has long had a reputation as strong at undergraduate teaching. For a long time they were one of the few departments to continue to teach a very high powered introduction to calculus from Spivak's book, whereas other top places like Harvard discontinued it, under the (I think often false) assumption that a good grounding in beginning calculus is already known to all entering math types.
I don't think I will mind the location too much. One thing I appreciate with American towns is that all of them seem properly planned and everything is flat. At least, judging from what I see on TV shows and films, it looks so. I can imagine that from a bird's-eye-view, towns would seem as if they were chess boards. I am unusually fussy about such issues and it would make me happy to live some place where things are accessible and the roads are bicycle friendly. At any rate, I doubt I will have too many issues, location-wise.
http://math.uchicago.edu/~lind/161/
Yep, Spivak is indeed the prescribed text. It is interesting to note that it is merely intended to be used as a reference text. Students are expected to write a so-called "journal" in which they should each write their proofs. They call it "Inquiry Based Learning" (I think I got that right...) and it would
seem that the students are expected to do the bulk of the work. (i.e, absence of spoon-feeding) Sounds like a cracking course. I will definitely try to see if I can adapt their own method when I learn from Spivak's book in the near future.
Is it not just the "higher ranked colleges" who now have multi-variable calculus as their freshman honours calculus course? My understanding is that everywhere else, where an honours variant of freshman calculus is present, the first part deals with single variables? I think of the "top schools", MIT (they use Apostol) and UChicago are the only exceptions.
Another thing. As you have pointed out before, the students who went to high school around the same time as you had access to more advanced mathematics than those students of today. Save for those participating in Olympiads or those who spend some time reading about mathematics, I doubt many have heard of that result and countless others. According to Wikipedia, the "New Math" of the 60s was created largely as a response to the threat that Soviet engineers were posing.
I'm unsure as to whether the dumbed down high school mathematics curriculum is a good or a bad thing. Only a minority will ever use such mathematics, let alone be interested in it. I think it might be a good idea to have everyone take a rigorous course (say, geometry) in mathematics and then have the next courses at varying levels of complexity and content. I cannot recall who, but a Math PhD turned coder from Stanford, had a few notes on how to change the system. He proposed three streams. One for those aiming to pursue math at university or those just interested in math. One for those going into the natural/social sciences or engineering. One which focused on more day-to-day uses of mathematics.
Sankaku said:
Number theory is based around the study of the Natural numbers and, by extension, the integers. Higher-level number theory gets into other algebraic structures, but that is where it starts. With the Natural numbers, you can't always divide things the way you want. Much complexity comes out of this simple fact. They are also the quintessential countable set.
As you say, Analysis is based around the study of the Real numbers. Though the distinction seems small from the outside, it is actually huge. The real numbers are the prototypical complete ordered field and you get to grapple with the brain-bending properties of uncountable sets. Most people just accept it, but I think the Real numbers are actually the most frightening thing in all of mathematics.
Perhaps it is because I have limited exposure to them but as of right now, my view is simply that they are fascinating, and much less scary!
The book "Challenge and Thrill of Pre-College Mathematics", which may be of interest to other prospective math majors on here, does a good job at explaining numbers. First, the set of natural numbers and the operations that can be carried out with that type of number is presented. From there, the set of integers is introduced, and the authors also explain how this new set can overcome the limitations of the previous set but also explain that new set's own limitations. They do likewise up until complex numbers and have a nice chart which shows what was "gained and lost" by "expanding" (?) the respective sets each time. A preview is available on Google Books. In fact, most of the book can be viewed.
This text and the result/computation in the previous page have made me look forward to taking an analysis course.
dkotschessaa said:
Well you sound pretty conscientious for 20 Mepris, so I think you are doing alright. I'm 35 now so I'm way behind. It certainly isn't too late for you to make some good choices now.
I hope you find what's best for you, though of course I am heavily biased towards USF, and if you should come here, you would have some instant friends. (Just think, sunshine, girls in shorts all the time... oh and math.. lots of math). Here is the course flow chart:
http://i47.tinypic.com/2vltump.jpg Let me know if that's not readable and I'll re-size. Looks a bit fuzzy.
-Dave K
Sunshine and girls in shorts sounds awesome but then again, I might be liking the sound of it too much not inherently, but because of my new font. I'm currently running Xubuntu (a linux distrubution) and everything is in something which looks like "Consolas" or "Lucida" - not sure which.
Thank you for the flow chart. It's readable and helpful! The college I attend will depend more on the outcomes of my application, and much less on me, for getting aid (merit or need) is a massive crapshoot for international students. Nevertheless, I think I will apply to USF.