Preparing for math REUs next summer

[zpconn]

Hi all!

I've posted here a few times before and remember these forums as being immensely helpful. Here's my situation. I've finished my first year of college as a pure math major. Technically I also am a computer science major, but I've recently come to the final decision to put all my focus on math, which means I'll probably be dropping the CS major at some point (whenever it keeps me from taking as many math classes as I please ). I definitely plan on going to graduate school in math.

I love math, and I really want to go to a math REU next summer. The one at Williams College is my top choice, but there are many that would be great.

I've done a lot of reading and studying in my spare time, so my knowledge isn't well represented by the classes I've taken. Nevertheless, the courses I've taken so far are calculus I-III, differential equations, and linear algebra. Next semester I'm signed up for analysis and abstract algebra. The semester after that I plan on taking complex analysis, topology, and hopefully another class, either number theory or abstract linear algebra.

Since I've got a year before applying, though, this schedule is malleable, but I'll have to finalize it soon (especially next semester's schedule). I thought I'd therefore ask here for advice on my schedule for next year.

I've spoken with my advisor about this, and he didn't seem totally knowledgeable about the REUs. However, he will also be the teacher for abstract algebra next semester, so I have my target set on him for a possible letter of recommendation.

So I'm basically looking for advice on what would be good to do with my schedule next year with the goal of getting into the best summer REUs.

I also have another specific question. My calculus III instructor (from last semester) was for some reason unusually impressed with me and even had the director of the Honors College here write me a personal letter congratulating me. Would a letter of recommendation from her be good for the REUs even though she taught calculus III (and not a higher course)? I'm asking now because she won't be here next year. In fact, she's not here right now, but she will be back for a bit in July, so that'll be my only chance; I'd have to have her write the letter this summer even though I won't be applying for a year.

But that also would pose another problem. I don't know how many or which REUs specifically I'll be applying to, so I'm not sure how I'd have her write the letter now.

I also have some general questions. Like I said, I love math. It's why I'm here. I know for a fact that I want to go to graduate school in math--there's no doubt about it. Honestly, I just want to learn as much about math as I can. If it was up to me I'd take nothing but math classes all four years. That's not possible, though, since the math department requires a large number of non-math courses. I'd be in heaven if I could just take nothing but math classes, though. Perhaps this is asking too much, but would it be reasonable for me to request this possibility from the math department or at least request that I perhaps take extra math courses in place of the present requirement for non-math courses? I haven't spoken with the math department about this yet because I've been afraid that it could be considered "rude" or simply out of line or ridiculous. What do you guys think?

Finally, sorry for the length of this post and thanks for reading!

 

[snipez90]

Huh, why would asking to take extra math courses be considered rude? There is probably a high chance that they will reject your request though. Remember, no one is stopping you from sitting in on extra math courses (and of course, you are learning math on your own right), provided that you are flexible with your schedule. Also, this is my own opinion, but I don't see any particularly good reason to confine yourself to math courses only. Even the smartest math students I know at my school have a variety of intellectual pursuits.

 

[zpconn]

Well, maybe I took the "math classes only" thing a bit too far. I certainly have other interests, especially in computer science, and I do really enjoy the computer science classes. Basically I just want to take slightly more math classes than is presently possible by substituting extra math classes for some of the non-math requirements. For example, next semester I'm enrolled in a course entitled modern Israeli film. I'm sure it's an interesting course, but I'm not paying so much money to be here so that I can study modern Israeli film. I'd like to substitute the requirement for this extra humanities course for a math class.

As for the request being rude, perhaps "rude" wasn't the best word, but my statement was based on my experiences in high school. My high school was extremely resistant to any changes to their pre-set curriculum. They absolutely would not budge in the least in any way, and the best way I can describe my counselor's reaction is that she was offended in some way, as if by requesting to do things slightly differently I'd insulted the school.

An additional reason that I wish to take more math classes is that the math curriculum at my school is what I would consider slow-paced. Certainly it's a good school, and I learn much in the classes, but I have very high goals (including going to a top graduate school), so I consequently have very high goals for how much material I'm able to cover as an undergraduate.

I can be more specific if it helps. I want to take a courses on algebraic topology and differential geometry as an undergraduate (presumably my senior year and at the graduate level since these aren't even offered at the undergraduate level). Accomplishing this requires me to take more math classes than is presently possible at my institution as an undergraduate.

This is the pace I'd like to be able to go at. I'd like to be able to take number theory, analysis, abstract algebra, topology, abstract linear algebra, and complex analysis over the next year. That sounds like a lot, but it's only three classes a semester. It's simply a higher density of math classes than is customary. The following summer I'd like to do an REU focusing on geometry and topology. My junior year I'd like to take graduate courses on analysis, algebra, and topology. I'd like to do a second REU the following summer. And I'd like to conclude my senior year with a bang: algebraic topology and differential geometry.

The subject I'd be most interested in is complex manifolds, complex dynamics, and Teichmuller theory (I've been inspired by the work of Curtis McMullen at Harvard), but no such courses are offered here unfortunately. There is an undergraduate course on dynamics, so I could possibly substitute that for number theory; however, number theory is pervasive, and I'd like to have a decent foundation in it. My dream, then, would be to study under him at Harvard. Getting there will require me to take more math classes than is customary.

So that's why.

I'm quite confident I'm capable of this pace simply because I so thoroughly enjoy the subject. However, this pace really isn't possible simply because I'd be unable to fulfill the other degree requirements (I wouldn't have enough non-math classes at the end unless I took an unreasonable number of credit hours per semester).

What do you think about me basically telling my advisor what I just said about my goals? It's hard for me to find a way to make a request like this without sounding like, "I'm displeased with the way things are done here; I want to do it my way." This isn't true and has the wrong tone. I really like my school and enjoy the classes. I just I guess am unusually ambitious and want to challenge myself more and really focus on achieving a particular goal within math as an undergraduate.

I haven't really told him much besides that I want to go to graduate school and that I want to go to an REU next summer.

Thanks for your response!

 

[owlpride]

The only geometry and topology REUs I know are Williams and U of Tennessee. There is a reason why there are so many algebra/discrete math and math modeling REUs but hardly any geometry ones: most students do not learn enough topology and geometry before their junior year to have a productive REU experience. Williams is among the most selective REUs in the country and their geometry group is not going to accept you with just a basic course in point-set topology. To even stand a chance you would need one of your references to say that you know a *lot* more about those subjects.

Talking about references, your letters of recommendation are the single most important element of the REU application. If your Calc III professor is the one professor who has been more impressed by your performance than anyone else, go ahead and ask him for a letter. If professors in your higher-level courses will be as excited about you, they might be a better option because higher-level classes tend to reveal more about the math skills that are relevant to REUs. I don't think you have to decide right now who you are going to ask for a letter. You can ask the professor now if he would be willing to write you a letter for your REU applications next spring, and then e-mail him a list of the programs you want to apply to later. If you change your mind about the references, you could give him only a partial list of the programs you are applying to.

I just finished my sophomore year and I was advised to apply to ~10 programs because many REUs prefer juniors. I was lucky and was admitted to 7 programs. Another sophomore at my college got into 3/8 programs and she had taken a decent amount of math as well. (Something like 2 courses each in linear algebra, abstract algebra and real analysis; number theory, cryptology, graph theory, diff eq and multivariable calc. She is interested in algebra and discrete math.)

I think it is a good idea to apply to programs of varying selectivity. A few REUs regularly publish in standard professional journals, but most don't publish at all or "only" in undergraduate research journals or some obscure journal that no one ever reads. If you are concerned about a publication, apply to programs that work in areas you have never heard of before rather than doing standard math. The reason is that all of the easy problems in standard math have been solved, and the remaining ones are too hard for undergraduates to tackle. Many REUs try to admit applicants with a similar level of mathematical maturity because otherwise the students will have a hard time working together. Surprisingly I ended up not getting into the less selective programs I applied to, but typically it's the other way round. It's hard to estimate beforehand how competitive of an applicant you are, so it is safest to apply to a variety of programs.

The following report contains a lot of info about individual REU programs that may not be published on each program's website. Some of them specify exactly how selective they are, how they select applicants, how they approach undergraduate research, etc. http://www.ams.org/employment/PURMproceedings.pdf There is also a version from 1999 that has some reports from REUs that are not in the 2007 proceedings. You can probably find it on google. When reading the latter, keep in mind that most programs have become a lot more selective since 1999 because REUs only became popular very recently.I am just curious, why would you need more math courses than currently possible at your institution to take algebraic topology and differential geometry in your senior year? At my college the first graduate courses in those fields realistically only require linear algebra, multivariable calculus, abstract algebra and point-set topology (and officially also real analysis but one would be fine w/o). That doesn't sound all that impossible. I took both classes as a sophomore.

 

[zpconn]

Thanks for so much detailed information, owlpride!

I am just curious, why would you need more math courses than currently possible at your institution to take algebraic topology and differential geometry in your senior year?

There's only a single course in algebraic topology offered here, and it's a very high-level graduate course. It requires as a prerequisite a year-long graduate sequence in topology plus undergraduate topology plus multivariable analysis.

Differential geometry is similar.

I also want to take graduate courses in complex analysis (since, at least right now, my strongest interest essentially involves the intersection of algebraic topology, differential geometry, and complex manifolds). I'm just not sure how to fit it all in!

That said, I just discovered, unknown to me until this point, that there's apparently a lower-level graduate course in differential geometry here which I'm qualified for. I'd see if there would be a way for me to get into that next semester, but I think abstract algebra, analysis, and another class (topology or number theory) will probably be enough. Based on your advice, it sounds like I should definitely take topology as soon as possible. Unfortunately the class is full right now. I'll see if I can request permission to get in anyway; I don't think professors really care whether there's one extra person in the class, especially when classrooms always have empty seats anyways.

Just out of curiosity, what courses did you take your first and second year in order to be able to complete algebraic topology/differential geometry so early?

Thanks again!

 

[zpconn]

Upon further investigation, it appears that topology, neither undergraduate nor graduate, is being offered here next semester. Now that's definitely bizarre.