Graduate Mathematics

Graduate School
1

<p>I am currently a Mechanical Engineering major with a minor in Mathematics in at Northeastern University, my GPA is a 3.927 and I am a little over halfway through my curriculum.</p>

<p>I enjoy mechanical engineering but I am coming to the conclusion that my real passion is mathematics (its what made me decide to major in engineering in the first place, as a naive high schooler I thought the only available careers for strong math students were engineering). I am a middler at Northeastern (3rd year in the traditional 5 year program), so I have two summer semesters (2 four credit classes per summer semester) and two full semesters remaining (4 four credit classes/labs etc per semester).</p>

<p>As far as math is concerned, so far I have taken Calc I-III, Differential equations & linear algebra (one class, diff equs very comprehensive but linear algebra only basic stuff through eigenvalues/vectors), and Probablity and Statistics (upper level calculus based class).</p>

<p>I am currently considering switching over to a full-time mathematics major, but I am unsure how many more classes I would need to/should take to be an eligible candidate for good graduate mathematics programs.</p>

<p>Also what are some good graduate programs out there, and what steps should I take in order to give myself the best possible shot at admission?</p>

<p>My future goals include possible college level teaching/research, becoming an actuary, and at the very least I would like to be eligible to teach mathematics at the high school level (would an engineering undergraduate degree suffice for this?)</p>

<p>Thanks for any help/advice anyone can offer me, I would like to expand more on this topic but I am at work so I can only type so much at a time</p>

2

<p>To my knowledge, one gotta be very passionate about math to major in it, since the pay is much lower (compare to engineering) and job selection upon graduation is much narrower. And I’ve heard it’s common for a math phd student to graduate without one single paper, is that true?</p>

<p>I am engineering student myself… I’m not saying math is bad, it’s just not for everyone, surely you look very passionate about this field thou which is very good.</p>

3

<p>Well if you keep those type of grades, you will not have much trouble getting into a graduate mathematics program.</p>

<p>As for what you would need for a graduate mathematics program? It would depend on if you plan to go into Pure or Applied Mathematics. Either way, you would need to take:</p>

<ul>
<li>Analysis I and/or Advanced Calculus and/or Real Analysis</li>
<li>Abstract Algebra</li>
<li>Linear Algebra (which you have already taken)</li>
</ul>

<p>The above courses are just about 100% ALWAYS required for a M.A./M.S. program in Pure Mathematics. Those courses will usually be the topic areas of the final comprehensive exam. The Algebra course may/may not be required for an Applied Mathematics program.</p>

<p>If you intend to go into Applied Mathematics or Computational Mathematics (my undergraduate major), you will need additionally:</p>

<ul>
<li>Numerical Analysis</li>
<li>Optimization/Operations Research/Mathematical Programming</li>
<li>Discrete Mathematics (Combinatorics & Graph Theory)</li>
<li>Complex Variables</li>
<li>Computer Science courses like Programming, Algorithm Analysis, Scientific Programming)</li>
</ul>

<p>The requirements for a Masters in Mathematics outside of just courses varies GREATLY from school to school. Some schools have the Thesis requirement, some have project requirement/options, some have non-thesis (usually applied or computational) without ever doing one paper (like Zoo mentioned). At the PHD level, you WILL have quite a few papers…doesn’t matter if the PHD program is ranked 120th.</p>

<p>As for top programs…check the rankings on USNews but pay attention to your specialty. MIT and Princeton are tops for “overall” math programs. NYU is tops for applied math.</p>

<p>As for income, I would go applied math. Areas like Operations Research and Scientific Software Engineering will have much more opportunities than just pure math. Pure math is good if you intend on doing research or academia.</p>

<p>NOTE: Quite a few of the Top-10 graduate Math programs ONLY offer PHD’s not masters.</p>

4

<p>Thanks for your input zoo and traveler.</p>

<p>I would say I am more interested in going the pure mathematics route, as I have found over the years I thoroughly enjoy helping others learn, leaning me towards the academia/research aspect of things. Also I am fairly sure that I am not very interested in the computational/programming aspect of things so much.</p>

<p>Is a pure mathematics major/PhD a good path to becoming an actuary? That would probably be my career choice if teaching at the college level/research didn’t work out. Also in regards to zoo’s comment about salary/job availability it is my understanding that actuaries often make more money than engineers, and are in pretty good demand. </p>

<p>Other options I am exploring are an MBA, and I have done the research and see that an undergraduate degree in mathematics statiscally records some of the best GMAT/GRE scores. This way I could apply to say Harvard’s 2+2 program, and if that doesn’t work out I could go with the back up plan of graduate mathematics at various other programs to which I will have applied to as well.</p>

<p>It is also my understand that in most graduate mathematics programs, you can get your tuition and other expensives paid for by being a teaching assistant, which is something I am extremely interested in learning more about if anyone has undergone this process themself.</p>

<p>Once again I’m at work so I’m going to cut this short, thanks for your input everyone</p>

5

<p>Actuary Science is more of an applied mathematics area. You would still need Analysis, Real Analysis and Algebra (both Abstract and Linear) but you will ALSO need:</p>

<ul>
<li>Probability Courses (strictly probability)</li>
<li>Mathematical Statistics (not those combo Prob/Stat courses for engineers)</li>
<li>Stochastic courses (stochastic modeling, stochastic analysis, etc)</li>
</ul>

<p>If you can get research funding for grad school, by all means do it. As far as a teaching assistant, that depends on the course. Some schools even allow a senior math student to become TA’s for like Calculus I. As a M.A./M.S. student, you may have to actually be the full instructor for certain courses while also doing research.</p>

6

<p>“As far as math is concerned, so far I have taken Calc I-III, Differential equations & linear algebra (one class, diff equs very comprehensive but linear algebra only basic stuff through eigenvalues/vectors), and Probablity and Statistics (upper level calculus based class).”</p>

<p>I’d like to throw out a friendly warning that pursuing graduate studies in pure mathematics is very different than the experience you’ve had in the math courses you’ve taken thus far. The courses you’ve taken are courses that students in several different majors often take, and so they’re not nearly as mathematically rigorous as some of the more traditional math-major-only courses (like abstract algebra and real analysis). In these upper division courses, your homework sets will be proof-based instead of computation-based, which is more of a difference than most students anticipate. The math gets significantly more abstract, and it often moves at a much faster pace because the classes are full of more senior math students who are preparing for graduate school.</p>

<p>Maybe this is something you’re very well prepared for and maybe it’s not, but experiencing these upper division courses is really what distinguishes between someone who has always liked math and someone who might be fit for graduate studies in math. I urge you to take one of these courses (possibly after consulting a mathematics advisor who can provide more insight into your department’s curriculum) before making any grand decisions regarding graduate school.</p>

<p>“At the PHD level, you WILL have quite a few papers…doesn’t matter if the PHD program is ranked 120th.”</p>

<p>This is not necessarily true at all. Having attended a PhD program in pure math (ranked somewhere in the top 20 or 30 - I can’t really remember), I can tell you that I know several people who got their PhDs with no more than one paper published. I didn’t pay too much attention to publications so I can’t say this for certain, but I also believe that we had a couple of people graduate without a single publication under their belts. Granted, these people had decided during grad school that they wanted to be more focused on teaching instead of research (and were interested in teaching at liberal arts colleges), but the assumption that everyone graduates from PhD programs with several publications is just wrong.</p>

<p>“Also what are some good graduate programs out there, and what steps should I take in order to give myself the best possible shot at admission?”</p>

<p>That’s not a question that people on this forum can adequately answer. Admission to the top 10 or so schools is a bit of a crap shoot because there are so many highly qualified candidates with prior research experience that the department can pick whatever they want. At this point, departmental politics may play a role: perhaps one professor got two new students last year and doesn’t have the time to accept new students; if you wanted to work with that particular professor, you’re out of luck.</p>

<p>Honestly, the tip-top schools with the most cutthroat programs might not be the best options for someone who isn’t quite sure of their career goals. If you’re undecided between teaching, research and actuarial sciences, you might be much better served by a lesser-ranked program that has less internal competition. This would provide you a little room to breathe and to figure out what you might want out of the program.</p>

<p>“It is also my understand that in most graduate mathematics programs, you can get your tuition and other expensives paid for by being a teaching assistant, which is something I am extremely interested in learning more about if anyone has undergone this process themself.”</p>

<p>Graduate schools in mathematics generally fund students 100% through fellowships, research assistantships, and teaching assistantships. Each school has a different policy for this; at the school I attended, all first-years were on fellowships, which meant we were paid to focus on our studies. Second-years had teaching assistantships, where we TAed for calculus and linear algebra classes. Third-years and more advanced students had teaching fellowships, where they fully taught their own sections of classes. Some schools only let students TA and not teach full courses, and other schools have smaller sections and don’t have TAs at all. It just depends on the program. Much of this information is available online.</p>

7

<p>math47, I just wanted to address a few things. You’ve mentioned more than one direction here, which is fine, especially given your young age. By the way, your grades are really good.</p>

<p>If you want to be an actuary or teach math in high school you don’t need a graduate degree in math to do that, certainly not a PhD anyway. In fact, I think getting a graduate degree in pure math would take you a little far afield from either of these directions.</p>

<p>I went to graduate school in pure mathematics. I’m not sorry I did but it’s really hard and I bailed with a Masters degree. The focus is very much on training students to do research mathematics. If that is what you want to do then that is fine of course, but that isn’t being an actuary or teaching in high school.</p>

<p>Most of the graduate programs that I know of offer support in the form of being a teaching or research assistant. I thought GLOBALTRAVELER did a great job of talking about the difference between pure and applied mathematics. I think you should go ahead and switch majors now, immerse yourself in the courses GLOBALTRAVELER mentioned, and then see how you feel when the time for applying to graduate schools is closer.</p>

8

<p>The cool thing about where I am right now is I have the next 6 months to figure out what I’m trying to do (I have the luxury of being on co-op at keurig coffee right now, so I’m not missing out on any classtime or anything), which is a good amount of time to get my facts straight before making any huge decisions.</p>

<p>In response to the high school/actuarial sciences stuff, those are just options that don’t require any radical change that I am considering (these are for if I stayed in my engineering major, because I am fairly certain I could accomplish those goals without changing anything I’m doing currently). I would say my primary interest is college professor/research related</p>

<p>In response to globaltraveler and my probability class. This wasn’t a class related to engineering at all, it was a math elective I chose in pursuit of my math minor (covered all the probability necessary for the probability actuarial exam, then the last few weeks as much statistics were covered as possible) </p>

<p>I appreciate your warning emengee, but to be honest most of the stuff that you are warning me about seems exactly like the sort of thing I want to get into. My understanding is that classes like the ones you describe go into the proofs of why all the stuff we’ve been taught actually works, which was always what I wanted to know most about. I was never a student who took the methods given to me for granted, and went out of my way to understand the concepts behind the rules that were always simply given (for example I took it upon myself to research how all the derivation/integration shortcuts came about since none of my teachers ever actually told me why it works that way). I have never demonstrated this sort of self-motivated passion for any other subject, which leads me to believe that my future belongs in mathematics.</p>

<p>Anyways, other than being a math professor/researcher what other opportunities are available for someone with a PhD in pure mathematics?</p>

9

<p>"Honestly, the tip-top schools with the most cutthroat programs might not be the best options for someone who isn’t quite sure of their career goals. If you’re undecided between teaching, research and actuarial sciences, you might be much better served by a lesser-ranked program that has less internal competition. This would provide you a little room to breathe and to figure out what you might want out of the program.
"</p>

<p>I have a question regarding this. I am a pure math major, I have also fulfilled the requirements as an applied math major. (I have taken linear algebra, abstract algebra, analysis, number theory, prob, stat, modeling courses, graduate applied mathematics courses, etc.)</p>

<p>I’m leaning towards applied mathematics for a PhD as I’ve done about 2 years of applied math research (1 pub) and my only pure math research has been on computational algebraic geometry (still somewhat applied). Anyways, I was leaning towards hoping to get into a top 10 program is it really that cut-throat? Are students better off, in say a 10-20 ranked program or such?</p>

10

<p>Math 47,</p>

<p>Ok…I was just throwing that out there about the probability. Many graduate math departments have the same Probability/Statistics course sets: There is that “Prob & Stat” combined course that is for the grad engineering majors and then there is that course just called Probability and you can tell just from it’s description or syllabus that it is for the Math/Stat majors.</p>

<p>I went the M.S. Engineering route after my undergrad Computational Math degree and noticed the course selection. I think I would have to say, thankfully…I only needed the combo Prob/Stat course for graduate engineering…lol. Of course I had to look at the textbook just to see the rigor of the Probability and the Mathematical Stats I & II courses being curious.</p>

11

<p>math47: The warning wasn’t intended to put you off; merely, I wanted to explain that having enjoyed math classes thus far is not necessarily an indicator of how well you would do with more abstract and research-based mathematics. It was more of a suggestion that you (or anyone) should take a few of these upper-division, abstract math classes to see how much you like it.</p>

<p>I was very active with math competitions in middle and high school, and I know a lot of people who had the passion and curiosity that you’ve demonstrated. All of us were crazy interested in math throughout college, but then you see people dropping like flies from PhD programs and academia because it’s different when you’re dealing with mathematical objects that become more and more abstract and less and less relevant.</p>

<p>Once you get into the later parts of grad school, the math that you do ends up being so abstract that only a few dozen people around the world have any idea of what you’re doing, and you might be on a campus where you can’t talk about your problems with any of the other students because everything is so specialized. This is a fantastic working environment for some people and a terrible one for others; the thing is that most people don’t figure out until they’re already into their PhD programs.</p>

<p>I do not mean to deter you from pursuing this; merely pointing out the other side of a coin that most people don’t get exposed to early enough. I’m the sort of person who found that proving laws of calculus in a real analysis class can be a lot of fun, but dealing with metrics on hyperbolic surfaces that only a hundred people have ever heard of before…can really take the fun out of math.</p>

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<p>emengee – I couldn’t have said it better myself. I don’t think anyone is trying to discourage you math47. None of us know you, maybe you are destined to have a career in research mathematics. Even if that isn’t where you end up there are worst things you can do than spend some time in graduate school.</p>

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<p>Pea - We should talk privately sometime. As another person who also bailed with a master’s degree, I think we should start a club ;-)</p>

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<p>I guess it really is hard to tell, could you recommend any specific abstract math classes that would help me with this decision emengee?</p>

<p>I really appreciate all the help everyone is giving me, as I am pretty uneducated on this career path, but still extremely interested.</p>

<p>The only reason I didn’t do it in the first place is because 6-8 years of education was much more than I thought I’d ever be able to afford, and I had never heard of the fellowship programs. For these reasons I picked engineering because it was sort of a safe bet for someone who liked math, and I wanted to make sure I’d be able to take care of my student loans.</p>

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<p>math47: In general, abstract algebra and real analysis (which might be called “advanced calculus”) are probably the two courses (sets of courses, really, as each probably has two semesters) that traditionally give students a view into what graduate school is like. Abstract algebra will give you a view into very abstract mathematical concepts, and real analysis will show you why everything in calculus is as it is. Both courses will give you a chance to develop mathematically rigorous proofs. You’ll need to have gone through both sequences if you want to go to grad school for pure math, so make sure you start taking at least one of them soon.</p>

<p>This still won’t tell you whether or not you’ll thrive in grad school (because grad school is even more rigorous, even more focused, and mostly research-based), but it will give you a better idea of what you can expect than you had from the math classes you’ve already taken.</p>

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<p>emengee: Thanks very much, I really appreciate all the advice you have been giving, just out of curiousity what are you doing now (with regards to mathematics)?</p>

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<p>I’m a math major who is planning on pursuing a double major with physics and i’m wondering if maybe i should put physics aside to concentrate on math. Right now, i have only taken calc 2 honors, and this semester i’m taking linear algebra. In the summer i will be taking Multivariable at my local university, and sophomore year i will most likely take Honors Analysis. From there i don’t know what to take as it all seems very interesting but very spread out into the different, specific branches of math. </p>

<p>As to my own interests, well i have many, there lies the problem. I study math in my spare time, usually problem-solving books from artofproblemsolving or AIME/AMC12 contests at the moment. I’m hoping that by the end of this year, if i keep up my studies, i will be able to work at the USAMO level in my spare time. My goal is to eventually place in the top 200 on the Putnam.</p>

<p>Despite all this i feel disconcerted by the words of caution, i.e, “cut throat”, directed at top school admissions. As i want to go to Harvard, MIT, or Cambridge University, UK, i feel GRE scores and GPA won’t be enough to unlock those doors. I would love to do research, however i don’t have the background necessary, as i’m a freshman. I’m completely confused as to how to even begin research in math.</p>

<p>–sorry if i hijacked the direction of this thread, math47. This thread seems to be teeming with great suggestions and i just wanted to snag a few.</p>

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<p>math47: What am I doing now? That’s an interesting question. I’m teaching math at a branch of The Art Institute, which is an art college. I’m also teaching online for Art of Problem Solving. And I teach flying trapeze (which doesn’t have anything to do with math, but it’s always fun to mention). The eventual goal is a tenure track position at a community college, although I’m considering teaching abroad for a bit beforehand.</p>

<p>ChaoticOrder: Your question of what math classes to take depends on which direction you want to go: pure math or applied math. I believe that question has already been covered in this thread, so I won’t really touch on it here.</p>

<p>As far as the Putnam exam goes, while the mathematical knowledge required to answer questions on the AMC12 and AIME is a bit of a prerequisite to doing well on the Putnam exam, the jump to solving the proof-based problems that the USAMO and Putnam exams offer is quite a bit one. It requires a significant amount of practice with quality feedback. When the answer is a paragraph instead of a number, the difference between right and wrong is much more complicated, and you’ll need someone to work with you to improve your proof-writing skills. If your school offers a course or seminar to prepare students for the Putnam exam, I’d suggest you take it.</p>

<p>And when I was referring to top schools as “cut throat”, I wasn’t actually referring to the admissions standards; I was referring to the programs themselves! I went to Brown, which is a fairly good school, but there’s a big difference between math students at Brown and math students at Princeton, for example. Well, several big differences. But one of them deals with competitiveness. Many of the students who are studying math at Brown didn’t know precisely what they wanted to study when they got there. They’re bright, talented individuals, but many of them didn’t have significant research experience prior to grad school. It was a fantastic environment because everyone felt as though they had time to figure things out.</p>

<p>I’ve never gone to a top 10 school, but from what I’ve heard from friends who do go to these schools, the environments are much more cut-throat, there’s more competition for the best advisors, and students are already living by the publish or perish mantra.</p>

<p>Why do you want to go to Harvard, MIT, or Cambridge? I don’t mean this to sound rude, but most people who want to go to these schools don’t have a good reason. Unless there’s a particular area of math you want to study with a particular advisor at one of these schools, you’ll probably be much happier at any of the other top 50-75 graduate schools that aren’t top 10 schools. You’ll have more room to explore a bit and figure out your interests without as much pressure to compete for an advisor.</p>

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<p>Unfortunately many people dream of going to MIT/Harvard just because of the golden aura around those names. Yes they are fantastic schools, but they are also not for everyone.</p>

<p>Admission office will do their job to filter those people out though.</p>

20

<p>Once again thank you for all the input everyone</p>

<p>Emengee: Thats great, I was thinking about community college work as well, because there is a community college in my hometown that I would like to possibly get involved with in the future</p>

<p>After surfing around a bit I am thinking that the Dartmouth PhD program is what interests me most, does anyone know how difficult it is to get admitted there, or does anyone have any personal experience with the program?</p>

<p>Edit: Also can anyone give some sort of salary insight for mathematics professors? Also what is your yearly schedule like, what sort of time off do you get, how much time researching vs teaching etc? (I’m sure its not like high school where you get full summers and what not but I’m curious to see how similar the schedules actually are)</p>