How competitive are admissions to math phd programs ?

[inknit]

So I understand admissions to the top PhD programs are close to impossible (being a math prodigy is a necessary requirement), but how competitive are admissions to programs ranked in 15 - 20 (Northwestern, Brown, UPenn, Wisconsin - Madison to name a few)?

Also, just how important is a solid research background for someone who wants to do pure mathematics? I've been told that simply doing well in courses is good enough, since doing math homework is closer to math research, then say doing chemistry homework is to chemistry research.

Also, I don't see how it's possible for someone who's just scratching the surface of mathematics to do 'pure math' research as an undergrad.

 

[fss]

PhD programs are competitive in general. "Ranks" 15-20 are still quite competitive. You really should not be looking at places based on ranking, in my opinion.

 

[holomorphic]

Also, I don't see how it's possible for someone who's just scratching the surface of mathematics to do 'pure math' research as an undergrad.

Right there, you've got an issue. There are REUs galore on abstract mathematics. They wouldn't exist if you couldn't do some kind of "pure" math research as an undergrad. It isn't all about prerequisites--just as you can read in some well-known math texts, often what is called for is "maturity" or "sophistication."

That said, in math, it's much harder to find someone to introduce you to a topic and help you see what you can do with it than it is in, say, chemistry.

Try not to see it as impossible. You might have to ask a lot of faculty members a lot of naive-sounding questions, but it will be worth it if you can find something interesting to do as a directed independent study, or senior project, or some such format. Also, there are bound to be undergraduate research conferences of a very general type, for which math submissions are often competitive. Even if you can't do a DIS or senior project at your school, you might go to a faculty member with the goal of presenting at such a conference a year later.

edit: Also, I wouldn't rule out the possibility of doing research in applied math, physics, or computer science.

 

[mathwonk]

of course they are competitive, but good students are always at a premium. And never forget the value of being a good listener. A student who hears what he is told and tries to follow the advice, will sometimes succeed more than a brilliant one who does not.

 

[zpconn]

A successful research record is not required even for the graduate programs at Harvard or Princeton or MIT, but what is more or less required at a place like those is a solid record of repeatedly trying your hand at research. Even the best students will struggle with getting publications in mathematics at an undergraduate, and many of the best REUs in the country also struggle despite having the strongest students. Most REUs in mathematics are about struggling, failing, and maybe having some glimpses of success; there are exceptions (see Duluth).

So, basically, try to do research as much as you can. Don't fret if you don't succeed, because simply trying is success at this point.

 

[doodle_sack]

I second zpconn's views on this.

 

[inknit]

Thanks for the replies. The problem is that I haven't taken any "real" math courses yet.
Right now, as a freshman, I'm taking Calc III and linear algebra. Next semester, I plan on taking real analysis, advanced linear algebra, and modern algebra. Could I do something then?

And what exactly will I do? Create a theorem? Prove the Riemann hypothesis..?

I feel naive.

 

[Cider]

Check out some of the sites for various math REUs that are being held this summer. Most of them tend to have descriptions of what type of projects will be done and what necessary courses there are for it. There are a few that don't have prerequisites past linear algebra (though there are ones that are recommended and useful), so that can give you a better idea of what type of things you'd be able to do. It's too late to really apply to any now, but with the courses you'll be taking next year, there will be an increase the number of projects you satisfy the prereqs by a good amount.

 

[deRham]

Right there, you've got an issue. There are REUs galore on abstract mathematics. They wouldn't exist if you couldn't do some kind of "pure" math research as an undergrad.

Some kind is, in my opinion, more what zpconn says than anything else - a glimpse of success at research, not much more.

So I understand admissions to the top PhD programs are close to impossible (being a math prodigy is a necessary requirement)

I strongly suspect you have understood a falsehood. Yes, given Harvard, MIT, etc are some of the most insane schools to get into period, you will face people who were doing PhD level mathematics (mind you, actual creative research, not just taking hard classes) and had won tons of international competitions in high school.

That said, you will meet superstars at various schools, including ones below the so-called top 5. Being at Stanford for math doesn't mean you are one, and on the other hand, you'll find some crazy international students at so-called 'lower-ranked' schools.

The trouble is that when your schools are tiny in size, some of them with entering classes of roughly 8 students, and people are applying from all over the world, you end up with a mix of superstars and more 'normal' really smart, motivated students.

What can be more important than being a prodigy is having clear direction, a connection with a professor who really thinks you have potential, etc. People like this can and do get into schools like Princeton. Remember, not every child prodigy wants to be a mathematician, and not every mathematician was a child prodigy. But you can bet that there's something special about anyone who does become a successful mathematician.

 

[holomorphic]

Some kind is, in my opinion, more what zpconn says than anything else - a glimpse of success at research, not much more.

Yes--that is certainly the spirit in which I meant to say it.

And what exactly will I do? Create a theorem? Prove the Riemann hypothesis..?
I feel naive.

Like the above said, most undergraduate math research is about getting a glimpse. It's okay to be naive to what kind of problems are tractable for an undergraduate--no one's ever mentioned any to you, probably.

The way I did it, the first time, was to send emails to a lot of professors in the math department. I didn't know any of them because I was new to the school and all my classes had been taught by graduate students. I think I was respectful and courteous, but about half of them didn't even reply. One of them said he didn't know anyone in math who had undergrad students doing such things and told me to talk to a chemistry professor. Another offered me a job at a biology lab. ONE of them set up a meeting with me to discuss possibilities, and I worked with him and his students for four semesters.

Later on, when I'd had some actual faculty members as instructors, I politely asked a professor whose class I enjoyed whether he knew any problems I could maybe look at in a directed independent study. He had an idea right away. I was lucky here, because he was enthusiastic about having an undergrad work a little on this problem related to his research. It ended up being useful (a little, tiny bit) to him. Mostly, though, he introduced me to a rich subject in a way that I would understand, and gave me an idea of how "pure" math research is done. I also got a good idea of how papers are written.

Bottom line, no blockbuster theorems can be attributed to me.

I also felt very naive, but getting answers about these things is worth being a tiny bit humiliated. I thought undergrad education was a little frustrating without the extra component of doing to go with the classes--I got the sense of "all this paper with homework problems on it, and nothing new."