
Academic Guidance Academic Guidance  Academic guidance for those pursuing a college degree... what college? Grad school? PhD help? 
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December 24th, 2016, 03:33 PM  #1 
Newbie Joined: Jan 2016 From: United Kingdom Posts: 16 Thanks: 0  How do I get the best mathematician out of myself?
Hello all, I'm an undergrad on the first year of his masters maths program. Right now, I'm really interested in doing a PhD and being a researcher. Not sure what area, but as a first year I gather that's tomorrows problem. My issue is with how I spend my time learning and developing as a mathematician. If I'm going to do this I want to be the best that I can, but I sometimes feel that I'm not going about it the right way. My current philosophy ( in practice right now) is this : Assuming all of the compulsory "bookkeeping" has been done ( by which I mean weekly assignments, lecture notes all written up, lecture proofs gone through) focus your time on asking yourself hard proofs. I provide two examples: 1. In linear algebra, we were taught that " the rank of any two row echelon forms of the same matrix are equal". The proof was omitted so I decided to work on trying to prove it. 2. In calculus, we were taught about integrability of a function, but the definition of this property was not in terms of a limit. I decided to try to unify the definition we were given with one which concerned a limit  specifically that the limit of the Riemann sum of the function existed. For number 1, I spent a few hours toiling before producing what I thought was a proof. I showed it to my lecturer, and low and behold, he spotted a technical issue I wouldn't have. He said to wait until I'd learned more and return to it. For number 2, similar outcome but self  realized. I noticed that I hadn't been told the definition of the limit of a sum, rather the limit of a function. I felt out of depth immediately, even more so because I wasn't even sure my hypothesis was correct to begin with. In both cases, I'm sort of setting myself mini projects to best recreate the conditions of being a senior researcher. My concern is with how I'm developing a knack for biting off more than I can chew. Is it more important to drill the questions set on the degree and get these foundations down like a machine? Should I leave the deep questions until later on? (seeing as I'm usually told the same thing each time) I mean, what are the odds that I'd ask myself a deeper proof which can be conveniently solved all with first year knowledge, that wasn't covered in lectures? In summary, I have two main questions. Should I even be doing this during term time ( when exams on compulsory content are around the corner) and in the summer when I can do my own thing, should I carry on throwing myself off these cliffs without a harness, or just buy some more advanced books and try the proofs in those? Do I continue down the guided path, or is there merit to going large? It would seem going maverick has it's drawbacks. Best regards to all, and a merry Christmas. James. 
December 24th, 2016, 04:21 PM  #2 
Senior Member Joined: Sep 2015 From: CA Posts: 780 Thanks: 407 
I assume you are given sets of homework problems that are subsets of the problems listed in your text. After completing these finish the rest of the problems listed, especially the more difficult problems towards the end or so marked. This way you will explore each subject in depth w/o running into areas you haven't learned about yet. Another option is to get a textbook from your library that is essentially the same course but at a higher/deeper level. You're an undergrad now, so grab yourself a graduate text on linear algebra or whatever and go through that. The difference shouldn't be that significant (except maybe for probability) but here and there you will be exposed to deeper material. I assume you've got access to a library with graduate textbooks, if not it's remarkable what can be found online these days. 
December 25th, 2016, 04:54 AM  #3 
Banned Camp Joined: Nov 2016 From: St. Louis, Missouri Posts: 28 Thanks: 4 Math Focus: arithmetic, fractions 
Although you're right to care a lot about getting better in math, you must successfully balance all of your studies and keep in mind that there is life outside of academics.

December 26th, 2016, 05:46 PM  #4 
Senior Member Joined: Feb 2016 From: Australia Posts: 782 Thanks: 287 Math Focus: Yet to find out. 
I like to think of math as one big accumulating body of knowledge, and to conquer it (or at least, feel like you can) you need to tackle it from all angles. This is at least while you are still searching for some specific area that you are really interested in and really want to pursue further. As romsek has said, i would finish all the prescribed homework and assigned tasks for the unit before venturing out into the unguided, and unknown world. That being said, i believe it's very important to do your so called 'mini projects'. Perhaps you don't need to go all the way with them though. Dive into an attempted proof by all means, but if you begin to become stuck, don't let it consume you. As you begin to learn more you will probably end up going back to the things you were once stuck on, and solve them with ease. 

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