Ever since I have been in college, whether at HCC or UMD, those who know me have remarked that my classes and activities surrounding mathematics are somehow prodigious or otherwise insane. Perhaps, this is true. I have been doing mathematics for a long time, and I do not deny that I am ahead of mathematics students my age; however, even if you were not a “gifted” or otherwise advanced child, even if you have no USAMO gold medal to boast about, even if you are not the best mathematics student in your class, you can become an outstanding mathematics student provided you are devoted to the subject and you are willing to work not just hard but efficiently as well.
Nota Bene. The hypothetical plans to follow will be based with the requirements and courses of UMD in mind, but they should be fairly similar to something possible at any given university.
While I will not assume any extraordinary background, I will set as a baseline that the student in question has handily passed AP Calculus BC and gotten a 5 on the exam. If this is not the case, then there is a year of catching up to do. Should you fall into this group, do not stop reading now! As you shall soon see, you too can become a mathematics major any department would be proud to call their own.
The most obvious matter to discuss is coursework, so we will begin there. Let us suppose your department requires an introductory computer science course, a numerical analysis course, a probability course, a supporting sequence (e.g., introductory physics, economics, or philosophy courses), and several upper-level mathematics courses. Let us also assume you will take ten general education courses, which do not fulfill any other sort of requirement. For some liberal arts schools, there may be even more required general education courses, but for many others, I suspect this number is higher than the amount required, as frequently mathematics, computer science, and science courses count as general education credit to some degree. Some schools will not require a supporting sequence, numerical analysis (or applied mathematics), probability (and/or statistics), or computer science, but frankly, it is a good idea to take something in these areas. Similarly, not all schools will require the following mathematics courses, but they are the foundation of a solid undergraduate education in pure mathematics (we will get to applied mathematics as well as mathematical statistics in a moment): calculus (3 semesters), differential equations (1 semester of ODEs), linear algebra (1 semester), undergraduate real analysis (1 semesters, covering essentially Baby Rudin), undergraduate abstract algebra (1 semester, covering rigorous group theory, ring theory, and some basic field theory), undergraduate discrete mathematics (1 semester, covering topics such as combinatorics, number theory, and graph theory), and undergraduate topology (1 semester, covering rigorous point-set, or general, topology).
Two of those semesters of calculus are probably out of the way already, and you can may want to add a second course in linear algebra (one which is proof-based), a second course in algebra (covering fields and Galois theory), and a second course in real analysis (covering multivariate analysis in some detail), but the first two can be taken care of with graduate work and the third may be in the one semester course. Similarly, we put complex analysis as a graduate course. That more or less covers the subjects all pure mathematicians ought to know. Also, recall that two semesters of calculus are probably already taken care of, as may very well be the supporting sequence and computer science course (all via AP), though we will assume the latter are not. If you can take an advanced treatment of the introductory courses (e.g., Math 340/341 at UMD), so that you are doing somewhat rigorous mathematics from the beginning, then do so, but it is not essential.
So, let us begin creating a possible schedule.
- Year 1, Fall: Calculus III, Differential Equations, Supporting Sequence I, General Education, General Education
- Year 1, Spring: Linear Algebra, Supporting Sequence II, Computer Programming, General Education, General Education
- Year 2, Fall: Abstract Algebra, Real Analysis, Numerical Analysis, General Education, General Education
- Year 2, Spring: Topology, Discrete Mathematics, Probability & Statistics, General Education, General Education
- Year 3, Fall: Graduate Algebra I, Graduate Real Analysis, Graduate Topology, Easy Class, Easy Class
- Year 3, Spring: Graduate Algebra II, Graduate Complex Analysis, Graduate Differential Geometry, Easy Class, Easy Class
- Year 4, Fall: Graduate Topics or Broadening Course, Graduate Topics or Broadening Course, Graduate Topics or Broadening Course, Easy Class, Easy Class
- Year 4, Spring: Graduate Topics or Broadening Course, Graduate Topics or Broadening Course, Graduate Topics or Broadening Course, Easy Class, Easy Class
The main “problem” here is that one has to work out some details to avoid taking too much math at the end. You see, graduate courses are much more work than undergraduate ones, even ones intended for seniors. A graduate student is usually “full-time” with three courses, and almost nobody takes more than four. They have to teach and worry about research, yes, but you should be worrying about these things, too! Also, it is naive to assume you, a junior mathematics major, is actually as capable as a legitimate Ph.D. student. Even if this does prove true, assuming it from the beginning leaves you liable to failure, and you really do not want to receive a “C” or worse an a graduate course, no matter what year you take it in. (Even a “B-” is pushing it, to be honest.) For this reason, three graduate mathematics courses should be the ceiling, but some exceptional students may be able to take four or take three and a “broadening” course, which I will discuss more later.
I began this semester (my freshmen year, though I have junior standing) taking graduate algebra, analysis, topology, and Lie groups & their representations as well as a Putnam seminar, an Honors seminar, the required Honors colloquium, and computer programming; I am now sitting in on topology and have switched to undergraduate analysis to make my life easier. (I also need an analysis credit, so I am taking it now in case I never take the graduate course for credit.) Do I think I could have gotten good grades wit that schedule? Yes, I am quite confident of that fact, as I have seen all the material before, even if I never sufficiently mastered it then and am not as comfortable with it now. Would I have enjoyed such a hectic schedule? Almost certainly not.
Graduate courses are nice in that they may only have a homework component, but these days the qualifying exam courses tend to give exams, too, and the homework component is probably much more than one might think. Spending twenty hours on a problem set, while not common, would not be surprising. My advisor, Larry Washington, actually graduated from Hopkins undergrad a year early in part because they weighted graduate courses as twice the credit of an undergraduate course. Why all departments do not do this is beyond me.
OK, so I should make several clarifications. First, what is a “broadening course?” Well, this depends on who you are. This is a post primarily intended for a serious undergraduate mathematics major who hopes to go to a good graduate school to obtain a Ph.D. or an M.S. It is therefore a good idea to explore math while you can; it will be much harder to explore partial differential equations in graduate school if your focus is algebraic geometry than it will be in undergrad. Good courses along these lines include: advanced courses in related fields (e.g., quantum mechanics, econometrics, algorithms), dynamical systems, ODEs, PDEs, elementary number theory, etc. For those who really want to explore an area of human thought that is not mathematics, you may also use some of these spots for that. But, beware that if you want to get into the best graduate school possible, you should not have many of these. Graduate admissions is very different from undergraduate admissions. They do not care that you are a well-rounded academic: they care about how you will do at mathematics research. Non-mathematical skills such as perseverance, some writing ability, etc. are good to have, but the primary measures of an application are the math (and, to a lesser degree, related topics) course depth and breadth, the research, the clear love and passion for math, the letters of recommendation, the common professional activities related to math such as expository writing or speaking, etc.
What else should you do? Well, you should essentially try to do those things mentioned above. (And, if this does not sound fun to you, please seriously consider whether graduate school is a good fit for you; graduate school in mathematics, especially at a top university, is not for dilettantes or anyone who doesn’t want to do mathematics for eight, ten, twelve hours a day.) If you can do research in the department or via an REU, do it. Research in applied mathematics is better than nothing, by the way, as is an REU project not worthy of publication. If your department has something like UMD’s Directed Reading Program, do that. Maybe do the Budapest program and study mathematics abroad. We have covered coursework in some depth already. With respect to professional activities, these are basically the mathematics graduate school equivalent of extra-curricular activities. They are less important in graduate school, but you should also want to go to talks and colloquia and workshops, you should want to be a member of the math club, you should want to give talks, you should want to share your course notes online, you should want to talk about mathematics with whoever is willing to (e.g., those online via a blog). Clear love and passion for mathematics essentially follows from these things, and it is perhaps the most important aspect of professional development. Finally, hear a word on letters of recommendation. These are essential. A good letter of recommendation can take you anywhere; you may be a top one hundred math student on paper, but if you have a good friend of Benedict Gross on your side, you may very well get into Harvard, generally reserved for the top twenty students (ten go there, ten to Princeton). Conversely, a bad letter of recommendation can knock you down several “levels,” so ensure those writing letters actually know you and think you are an excellent candidate. A related phenomenon is specialization of application. If you can tailor your application to a particular professor (or contact that professor outside of the application in an analogous manner), they may hand-select you, which is often guaranteed admission. Mentor-ship in general is a good thing. Make a concerted effort to get to know your department and work with or learn from a few of its members.
There is a bit of an elephant in the room that I should address. None of this is possible unless you are willing to work really, really hard. It does not hurt to have natural talent in mathematics or just in general intelligence, but this is not required, in my opinion. More importantly, it is counterproductive to think one is static in his or her ability. Always strive to be able to understand and do more! There is actually a second thing I have forgotten to mention: do not feel the need to rush. I am not saying you need to do the above to be a great math student. Feel free to live a little and enjoy your time at whatever school you attend.
With this plan outlined, I have something to admit: this is not far off from what I did, just two years early. As a kid, I made a few decisions which I consider likely mistakes today, which stopped me from attending college and doing certain other academic things earlier. I still did manage to get to where I am today, though, which is to be on track to have my B.S. at 19. Something I do not really regret doing is generalizing for two years of college (specifically at HCC), even if I do wish I had done it a bit sooner in some way or another (and perhaps changed a few other things). For purely the sake of being a top math student as soon as possible, it would have been optimal to go to a very good mathematics university from the start and get off the ground running, rather than retake the really basic stuff, too. I did not do this. Moreover, I took calculus, linear algebra, differential equations, and discrete mathematics over two years, so many of my peers who call my graduate coursework incredible could actually very well progress more quickly than me in terms of math courses! (To be fair, I did also do special honors courses covering some complex analysis and algebra, but that is the extent of the official math work I did while at HCC.) It is true that I will be in a Ph.D. program two years earlier, and that I have an A.S. in two other areas beside math, but that is largely beside the point. Even without any special early circumstances, a lot of effort and a solid foundation can make a truly wonderful undergraduate student; it is, however, easier said than done. Maybe you are the exception. You are far smarter and more capable than I, and you blow through graduate work with no effort as a mere baby, but this seems unlikely, because even the greatest modern mathematicians such as Terence Tao, Alexander Grothendieck, and so forth have not done anything of the sort. Sooner or later, it becomes hard for everyone, no matter how smart one might think him or her self to be.
Closing Remarks. To end this discussion, let me mention that the schedule featured above can be adjusted to applied mathematics and statistics by appropriately identifying those subjects every good applied mathematician or statistician ought to know and studying them the first three years instead of what the pure math student does, though there will undoubtedly be some overlap. This idea may even extend to related majors such as physics. A student coming in with AP Calculus BC, Computer Science, and Physics C can do wonderful things in four years.