PhD in mathematics... worried... need advice
PhD in mathematics... worried... need advice
I really want to pursue a PhD in mathematics. Some background info, I have my B.S. in mathematics with minors in chemistry, biology, and computer science. I graduated with a 3.8 GPA and had a very good scholarship for my undergrad. This semester I am graduating with my M.S. in computer science. I have two papers under review with journals in fields not related to mathematics (virology and CS). My GPA is good, but sometimes I truly feel like I didn't truly deserve it, maybe will make sense below
Now my worries. Everyone tells me PhD programs for math are impossible to get into. I am really worried. I also wonder if I am competent to be successful in a PhD program. The reason why I wonder is because initially I struggled a lot with pure mathematics. Abstract algebra was hell for me (I got a B in the class), and then real analysis was so difficult for me (Though I got an A but I truly believe it's because the prof took some mercy/pity on me because I just worked so damn hard in the class). I really gave up on math after those courses. But then suddenly, I took topology this semester (this is my last semester of M.S. and I begged them to let me take more math courses) and it all "clicked". Real analysis, algebra, topology, these classes all became very easy for me suddenly. Topology is like cake for me, and I can't for the life of me understand why the other classes were so difficult for me before. The reason I think it is easy for me now is just because I put such a huge amount of time and effort into these classes. I can't stress enough how none of it came naturally to me, but only after a lot of dedication and hard work. But now honestly in topology, I breeze through the course although I find it completely fascinating so I just study so much and love every moment of it. I just don't find it difficult at all (note, I don't say any of these thing to brag, just trying to give a complete picture of my current situation. Again, I really sucked so much at all of this originally which is why I'm confused)
Anyway, is it even worth it to pursue a PhD in mathematics or do you need to be one of these people who just don't have to work and it's all cake from the first moment?
Also, I have had an "interesting" academic journey. I come from a low income area and dropped out of high school and left home very young. When I began college, I started in intermediate algebra. So I have come a very long way and because of this sometimes I feel I am very behind because I have had to teach myself a lot. My peers in college had a huge head up on me, when I started I had literally never taken a single course in chemistry/physics/calculus/biology, anything beyond basic elementary algebra. I have always had a difficult time learning in lectures, I don't know why, and so during my B.S. and M.S. I pretty much never went to lectures and opted to teach myself from text books. The classes that were exceptions were abstract algebra and real analysis, I really depended on bugging my profs day and night with questions over that material. Anyway, I don't know if this is at all relevant just that, I really feel I am behind even still today and just... I don't have much confidence in mathematics even though I really love it more than I can say. Pure mathematics gives me such a thrill and I am so curious in many areas.
I'll also add that the reason I went for my M.S. in CS rather than math was because I just felt like I would never be able to make it in math after struggling with real analysis so much. I really learned how much I don't enjoy CS and just have no desire to stay in this field. But I feel like this M.S. is going to make me look bad for abandoning math I guess
EDIT: The math dept. here is small and not many classes are offered. The only pure math courses I've taken: some intro to logic course, abstract algebra, real analysis, functional analysis, topology. I am self studying complex analysis because I really want to learn it but it's never offered here. I'd be happy to self study any other subjects that are fascinating like this but I don't know what to go after next! Also, I really *really* hate statistics. Not sure if that is relevant.
Now my worries. Everyone tells me PhD programs for math are impossible to get into. I am really worried. I also wonder if I am competent to be successful in a PhD program. The reason why I wonder is because initially I struggled a lot with pure mathematics. Abstract algebra was hell for me (I got a B in the class), and then real analysis was so difficult for me (Though I got an A but I truly believe it's because the prof took some mercy/pity on me because I just worked so damn hard in the class). I really gave up on math after those courses. But then suddenly, I took topology this semester (this is my last semester of M.S. and I begged them to let me take more math courses) and it all "clicked". Real analysis, algebra, topology, these classes all became very easy for me suddenly. Topology is like cake for me, and I can't for the life of me understand why the other classes were so difficult for me before. The reason I think it is easy for me now is just because I put such a huge amount of time and effort into these classes. I can't stress enough how none of it came naturally to me, but only after a lot of dedication and hard work. But now honestly in topology, I breeze through the course although I find it completely fascinating so I just study so much and love every moment of it. I just don't find it difficult at all (note, I don't say any of these thing to brag, just trying to give a complete picture of my current situation. Again, I really sucked so much at all of this originally which is why I'm confused)
Anyway, is it even worth it to pursue a PhD in mathematics or do you need to be one of these people who just don't have to work and it's all cake from the first moment?
Also, I have had an "interesting" academic journey. I come from a low income area and dropped out of high school and left home very young. When I began college, I started in intermediate algebra. So I have come a very long way and because of this sometimes I feel I am very behind because I have had to teach myself a lot. My peers in college had a huge head up on me, when I started I had literally never taken a single course in chemistry/physics/calculus/biology, anything beyond basic elementary algebra. I have always had a difficult time learning in lectures, I don't know why, and so during my B.S. and M.S. I pretty much never went to lectures and opted to teach myself from text books. The classes that were exceptions were abstract algebra and real analysis, I really depended on bugging my profs day and night with questions over that material. Anyway, I don't know if this is at all relevant just that, I really feel I am behind even still today and just... I don't have much confidence in mathematics even though I really love it more than I can say. Pure mathematics gives me such a thrill and I am so curious in many areas.
I'll also add that the reason I went for my M.S. in CS rather than math was because I just felt like I would never be able to make it in math after struggling with real analysis so much. I really learned how much I don't enjoy CS and just have no desire to stay in this field. But I feel like this M.S. is going to make me look bad for abandoning math I guess
EDIT: The math dept. here is small and not many classes are offered. The only pure math courses I've taken: some intro to logic course, abstract algebra, real analysis, functional analysis, topology. I am self studying complex analysis because I really want to learn it but it's never offered here. I'd be happy to self study any other subjects that are fascinating like this but I don't know what to go after next! Also, I really *really* hate statistics. Not sure if that is relevant.

 Posts: 7
 Joined: Wed Dec 01, 2010 1:28 pm
Re: PhD in mathematics... worried... need advice
There is an excellent piece on the topic written by Terence Tao: http://terrytao.wordpress.com/careerad ... domaths/
The bottom line is no, you don't have to be "one of those people who just doesn't have to work" and in a sense being that person can actually be detrimental to one's success.
Research in mathematics is very different from learning mathematics. Maths textbooks essentially rewrite history in a logical order and do not give any sense as to what you'll encounter in a PhD. I would suggest that you try get some feel for the sort of work people do daily and the sort of problems that really interest you. This information can then be used to strengthen your grad school application when the time comes.
There is a lot more invaluable advice on Terry's blog: http://terrytao.wordpress.com/ and http://terrytao.wordpress.com/careeradvice/ and I would highly suggest reading it. You can even read about the problems he's working on right now.
The bottom line is no, you don't have to be "one of those people who just doesn't have to work" and in a sense being that person can actually be detrimental to one's success.
Research in mathematics is very different from learning mathematics. Maths textbooks essentially rewrite history in a logical order and do not give any sense as to what you'll encounter in a PhD. I would suggest that you try get some feel for the sort of work people do daily and the sort of problems that really interest you. This information can then be used to strengthen your grad school application when the time comes.
There is a lot more invaluable advice on Terry's blog: http://terrytao.wordpress.com/ and http://terrytao.wordpress.com/careeradvice/ and I would highly suggest reading it. You can even read about the problems he's working on right now.
Re: PhD in mathematics... worried... need advice
This is very natural, and it was (and still is) my story. I had to go through hell selfstudying Analysis and Algebra before they made sense to me. In fact, even in high school I thought I was awful in mathematics until I got to calculus (when I realized how wellprepared I had been by my precalculus teachers.) Also, I've gotten into a few solid math PhD programs without ever having taking a topology course, though sometimes I do feel like a bit of a fraud because of thisfleazo wrote:But then suddenly, I took topology this semester...and it all "clicked". Real analysis, algebra, topology, these classes all became very easy for me suddenly. Topology is like cake for me, and I can't for the life of me understand why the other classes were so difficult for me before. The reason I think it is easy for me now is just because I put such a huge amount of time and effort into these classes. I can't stress enough how none of it came naturally to me, but only after a lot of dedication and hard work.
You're going to be fine. Try taking some practice subject GRE exams (since, after all, that is the topic of this forum), and read Terrence Tao's excellent blog posts like irreducible said. Terry Tao, of course, had the enviable journey of a child prodigy, but his wisdom for the rest of us is still very relevant.
If you don't mind, who is "everyone," what exactly are they telling you, and which programs are they saying are "impossible" to get into? MIT et. al. might be a bit out of your reach if you don't do some killer research first, but that doesn't mean you don't have a future in mathematics.Everyone tells me PhD programs for math are impossible to get into. I am really worried.
Re: PhD in mathematics... worried... need advice
thanks a lot for the input! I went and talked to a few of my profs after reading these responses, and they encouraged me to go for the PhD and loaned me some books, etc. They spent quite a few hours talking with me and just giving me some ideas and encouragement. Even the prof who I taught me abstract algebra and analysis (who I was for sure he would tell me to give up this dream because I sucked) he was just so encouraging! Thanks for all the encouragement on this thread because it really changed my perspective!
Yesterday was talking with my roommate and told her of my plans to apply, she just kind of stopped awe struck and said "are you sure? you have to have a really innate talent to do your PhD in math if it's not applied math." I guess this is the kind of discouragement I'm talking about. If I ever say anything like "well, I've worked hard and it's really catching on now." they just say "well everything you're studying is so basic and elementary that you don't really know yet". I don't get it, these are graduate level math theory courses, when will I know? This is the response I get from almost every student I mention it to! Then they just keep going on and on about how it's so damned impossible so I should reconsider. Imagine my shock when I spoke to my profs and they encouraged me! For the first time I feel I have some confidence here. Well, maybe I won't be able to make it through a PhD program but I at least want to try!
Yesterday was talking with my roommate and told her of my plans to apply, she just kind of stopped awe struck and said "are you sure? you have to have a really innate talent to do your PhD in math if it's not applied math." I guess this is the kind of discouragement I'm talking about. If I ever say anything like "well, I've worked hard and it's really catching on now." they just say "well everything you're studying is so basic and elementary that you don't really know yet". I don't get it, these are graduate level math theory courses, when will I know? This is the response I get from almost every student I mention it to! Then they just keep going on and on about how it's so damned impossible so I should reconsider. Imagine my shock when I spoke to my profs and they encouraged me! For the first time I feel I have some confidence here. Well, maybe I won't be able to make it through a PhD program but I at least want to try!
Sorry, yeah everyone just means the kind of grad students in my current department. I say anything about a PhD in math and they just raise an eyebrow and go on about how impossible it is. If not that then it's about how I don't know if I'm capable of it yet so why try because it's impossible. If not that then it's how you can't get a job when you're done. All this talk had me down but the thing is I am not really concerned with the name of the school. I just want to find good professors that I can learn a lot from!If you don't mind, who is "everyone," what exactly are they telling you, and which programs are they saying are "impossible" to get into? MIT et. al. might be a bit out of your reach if you don't do some killer research first, but that doesn't mean you don't have a future in mathematics.
Re: PhD in mathematics... worried... need advice
It sounds like you have the right (realistic) idea of primarily wanting to find good professors to work with, regardless of name, and your being in the habit of absorbing mathematics by selfstudy is another good sign in my opinion.fleazo wrote:Yesterday was talking with my roommate and told her of my plans to apply, she just kind of stopped awe struck and said "are you sure? you have to have a really innate talent to do your PhD in math if it's not applied math." I guess this is the kind of discouragement I'm talking about.
Your roommate and these grad students are likely jealous and bitter and attend a second or thirdtier school. Haters gonna hate. I remember one guy in my last institution who dropped out of grad math, claiming resentfully that it would take anyone "one hundred years" to get through Royden. Not surprisingly, his general malcontent was not confined to his studies of Lebesgue theory.
Best of luck to you, and let us know how it goes. Not that I intend to be on this forum next year...so at least pay it forward by helping out younger people when you are a little older and wiser

 Posts: 15
 Joined: Sat May 14, 2011 6:24 am
Re: PhD in mathematics... worried... need advice
Right..Ergodic..for me too the same opinion. He got the idea quite clearly.
Re: PhD in mathematics... worried... need advice
What are your goals in doing a PhD in mathematics? I'm sorry, there was a lot of text flying around, and I read it a bit quickly.If I ever say anything like "well, I've worked hard and it's really catching on now." they just say "well everything you're studying is so basic and elementary that you don't really know yet". I don't get it, these are graduate level math theory courses, when will I know? This is the response I get from almost every student I mention it to! Then they just keep going on and on about how it's so damned impossible so I should reconsider.
The truth is that math research tends to involve knowing quite a lot, because really, you are contributing an original something or the other to the massive body of math. If you want to get a job doing math, then what you produce will have to be deep  this involves both going deep into a subject and producing something people care about. The classes you take, regardless of how challenging, are still classes, intended to introduce you to the subject.
I suggest you try emailing someone who has taught you, and get advice on a simple miniproject or so that you can do to understand some topic in mathematics not well represented in the literature.
My feeling is that if you really want to do it, you can read Royden, etc, etc, and make it through the PhD program fine with your background; you have to take some classes at the start anyway, which gives catchup time. Some people may be better prepared at the beginning, but that's the beauty of research, right? Everyone has to go on his/her own path.
What IS true is that what you have done right now is not representative of what you will do in the program, because it will be much less structured there. And getting a good faculty position is hard even for strong people at good programs. If your 5 years of study would be worth it to you regardless, go for it, but make sure you know what you are getting yourself into.
Re: PhD in mathematics... worried... need advice
I wanted to also comment on this. People give very conflicting answers, and I wanted to reconcile the different ones.But now honestly in topology, I breeze through the course although I find it completely fascinating so I just study so much and love every moment of it. I just don't find it difficult at all (note, I don't say any of these thing to brag, just trying to give a complete picture of my current situation. Again, I really sucked so much at all of this originally which is why I'm confused)
Anyway, is it even worth it to pursue a PhD in mathematics or do you need to be one of these people who just don't have to work and it's all cake from the first moment?
First off, it is GOOD that you are finding it easier to follow topology. That is more abstract material, and it is not easy. But do not expect your next class in mathematics to be easy  you can have good and bad moments.
What is going for you is that you appreciate the value of bashing your head against something (for the love of doing it) until it makes sense. That will help in the PhD.
Be careful, however, or you will get discouraged  expect it to be tough in the program. Expect yourself to have moments when you feel it isn't worth it, and you won't make it. Remember to never forget that it's meant to be enjoyable. If you get to the point where you're learning it like swallowing medicine, you will 100% burn out.
It is most important to have a consistent climb.
All said though, you can't be someone who enters the program finding all the basic material hard. With some significant work, you should find it doable.
And remember that some day, if you strive for faculty positions, given they are even harder to come by than PhD spots, you will end up competing with people who did find it all cake when they started. The beauty is nobody finds it cake when they get far enough. And if you want to prevail, the important thing is to prevail at that particular point  when everything is insanely hard for everyone, not at the point when stuff is easy for some people and not for others.