<p>I just finished my first year at the University of Michigan (Ann Arbor).
I kind of screwed myself over by not taking math first semester. I wanted to explore other interests, and I figured I could always go back to math if I wanted to.
Unfortunately, I completely missed the honors math sequence (the equivalent of algebra 1 & 2 and analysis 1 & 2, I believe). I talked with an advisor about this, and he said that I could still do it if I took an extra year (but financially I cannot do that, and the idea doesn't seem very appealing anyway).
So, during my second semester at UM, I took the math class I placed into, multivariable calculus. I did very well in it. The next logical step is linear algebra, which I'm taking in the fall (not at the graduate level); I'm also taking probability (not at the graduate level either).
I would like to take graduate level courses, but I find this basically impossible since I have no true experience with proofs or abstract math, and they often require/recommend that.
I made a tentative schedule for the rest of college (entailing my pure mathematics requirements) and at best I could take 2-3 graduate level courses by the time I graduate. This is partly because I want to double major with a foreign language, but also partly because I am not in the honors math program (which has a much, much better preparation for grad school than regular pure math at UM--for those who don't know). The grad courses I wish to take are in number theory and probability.
The non-honors pure mathematics program at UM is still pretty good - I'll have taken differential geometry, analysis, algebra, etc but none at graduate level. </p>
<p>So my main questions are:
1. What kind of graduate schools in math are realistic for someone who lacks graduate level courses?
2. What would be an appropriate amount of research to do? And what would be the best way going about it considering my lack of real math classes at this point? Does research actually compensate for a mediocre transcript?
3. I would just like some advice in general. </p>
<p>I am not in math myself, so I can’t answer your (very math specific) questions, but did you think about teaching math to yourself? </p>
<p>I was in a similar situation as you a few years ago, I am a bio major and could not take too many math classes due to scheduling conflicts.</p>
<p>I ended up buying myself a textbook for analysis and I worked partly through a linear algebra book ( [A</a> First Course in Linear Algebra (A Free Textbook)](<a href=“http://linear.ups.edu/index.html]A”>A First Course in Linear Algebra (A Free Textbook)) ), mainly because I wanted to understand some advanced math and hey, math problems are much more fun than your average sudoku
Just do a large number of exercises and maybe find somebody who is willing to look your stuff over (Or ask the internet - but try yourself first!). Especially building proofs is something you learn through experience, but the linear algebra book has good hints for building basic proofs.</p>
<ol>
<li><p>If you are talking about a PhD, I wouldn’t look at the top 10 since those are extremely competitive, but there is not reason why you wouldn’t be able to make it say a top 25 or so school, depending on GPA. </p></li>
<li><p>I don’t do research in math personally, but I know for other departments it is huge. For math they don’t really expect you to have any research but it is a plus. It probably can’t fix a mediocre transcript but it can certainly help it.</p></li>
<li><p>If you haven’t taken a proof based math course yet(Algebra/Analysis), there is a good chance taking these will change your mind about even considering grad school for pure math.</p></li>
</ol>
<p>I would just pick the following books and start reading them over the summer:</p>
<p>Rudin, Principles of Mathematical Analysis
Dummit & Foote, Abstract Algebra</p>
<p>If you’re talented enough to do well in a good Ph.D. program, you should be able to read those by yourself. I read the first 5 chapters of Rudin while still in high school, so you shouldn’t have any trouble doing the same. It’s hard work and you should probably not expect to digest more than a couple of pages per day maybe 5 if you’re lucky. The important thing is to understand everything you read before you go further. If you get stuck you can always ask for help on some math related discussion forum (there are a few like planetmath.org, physicsforums.org, mathlinks.ro etc.).</p>
<p>Thanks for all your replies. I want to study math over the summer but I’m not sure what I should do.</p>
<p>I have the following books to study from:
Spivak, Calculus
Rudin, Principles of Mathematical Analysis (as eof mentioned)
Various linear algebra textbooks (including the one jixani mentioned)</p>
<p>I’m kind of against studying linear algebra before I take the course because I don’t want to sit through a class in which I already know the material. I also have several books on constructing proofs with an emphasis on set theory; would it be better to study those over Analysis textbooks?</p>
<p>hey bro, so u gonna be a junior at fall 2010?</p>
<p>I am also a math major in Umich: i assume u have done with 215(281)-217(513), then u have just 8 math classes to go. If u still want to do honor, u should take 500 level courses (things like 490 dont satisfy electives). Otherwise its still wise to take some 500 classes.</p>
<p>Another idea is to go with the 295-396 sequense. I know a girl who was a transfer student and managed to finish the honor math taking 295 at her junior year, while same time taking other math courses. (you should talk to Stephen DeBacker about it, he has good advice, and also can give u necessary overrides to take the classes) </p>
<p>As grad school concerns, letter of rec and research experience is far more important. </p>
<p>also having some focused graduate level classes look actually better than lots of random grad level courses: so something like some 400 classes 2 500 geometry(590+537) + 1 600 level algebraic or differential geometry >>> than bunch of unrelated 500 classes.</p>
<p>Just go with analysis and abstract algebra during the summer. People often try to postpone reading “difficult” material, because they think they don’t have the necessary prerequisites. The problem is that difficult material rarely become easier without actually studying it. The exception is of course if some field is written in the language of some other field you know nothing about. E.g. you don’t start with algebraic geometry without a background in algebra.</p>
<p>Analysis and abstract algebra start from nothing and they are important and often hard for the beginner. The sooner you understand them the better. If you’re afraid of getting bored in linear algebra, I suggest you start with abstract. This way you will at least see lots of practical examples for stuff you saw in abstract when you do take linear and the summer is too short for you to reach the chapter on modules in Dummit and Foote, so you won’t see any linear algebra in advance… Knowing abstract algebra will help you see the big picture and give you a much better understanding of linear algebra and vice versa. Studying abstract probably also helps with getting a good grade. Analysis and linear algebra are often the first real proof based courses and how well you do compared to your peers often tells a lot about your natural talent. For this reason many graduate programs look at these grades closely, so if something makes it easier for you to get an A in these courses, then you should do it.</p>
<p>Dummit and Foote is pretty good. It has loads of examples and exercises of various levels of difficulty and will serve as a good reference for some time to come. I know some schools consider it a graduate level book because of the amount of content, but the presentation is undergraduate style (i.e. no category theory).</p>
