Math programs of different schools

[oab729]

This is to compare the quality of different math programs. I'll start. I go to dartmouth. To find course content you can peruse most course webpages by going to math.dartmouth.edu. I'm comparing the courses I know to courses I've seen at other universities.

UGrad courses

Math 54 (topology with munkres) probably at the level of most intro topology courses. Ended on urysohn metrization theorem.

Math 63 (honors real analysis with baby rudin) is probably at the level of most introductory analysis courses

Math 71 (honors abstract algebra with gallian) is probably closer to the non honors version of algebra at harvard

Grad Courses
Math 101 (Abstract Algebra with Dummit and Foote) probably at the level of math 55, though the content changes.

Math 103 (Measure Theory and Lebesgue integral) is probably at the level of Harvard math 214a and the chicago sophomore program.

Overall, I'd say Dartmouth has great teaching (lectures are very clear), but courses lack as much depth, breadth, and creative problem solving as those at tier 1 schools (based on problem sets). A lot of the time, you don't really get to see the bigger picture of the material, since classes simply don't cover enough material or problems. Of course, this is only based on the more advanced courses after the introductory ones in algebra, analysis, and topology.

[JRav]

I go to Hopkins. I'll leave out numbers because I don't think that's particularly relevant:

Undergraduate:
Complex Analysis: We use the book that MIT's complex analysis course uses (I forgot the number), and I know Terry Tao has taught out of it in his course at UCLA. Standard stuff of complex analysis.

Algebra I & II: Prior to this year, this course used Dummit and Foote but the professor changed the book to Nicholson's book (it's easier, but I think it's good). Depending on which book is used, might be closer to a graduate course.

Honors Analysis I & II: We use Carothers' book, which is the book used in Caltech's Classical Analysis sequence. I think the course is modeled after UChicago's Honors Analysis sequence, and the difficulty is probably in between Math 25 and 55.

Topology: Probably similar to every other undergrad topology course that uses Munkres.

Differential Geometry, Fourier Analysis, PDE, Other Stuff: I don't really know any basis for comparison.

Graduate Courses:
Algebra: Uses Lang with D&F as a reference.
Real Variables: Measure Theory with Wheeden & Zygmund's book.
I'm not really familiar with too many others...yet.

I'm happy with the teaching but I wish there were opportunities for research, which there aren't. Hopkins is obviously really big on biology/medicine so a lot of the research money goes there.

[calc rulz]

I'm just starting at Princeton, so more experienced others can add to this but I'll write about the sense I've gotten so far.

The math department is of course one of the best, although it's fairly small, so many standard courses aren't offered all the time. We do have intro analysis/linear algebra every semester, and intro algebra, topology, differential geometry, analysis courses happen once every year. For more advanced courses, however, we only have advanced topics in algebra/analysis/geometry, so you can only take whatever advanced topic they're offering that semester. E.g., there's no standard undergraduate algebraic number theory or algebraic geometry course; you only get such a course every 2-3 years when it's taught in one of the topics classes. You can also take a graduate (500-level) course, though they tend to be quite difficult (and the graduate courses here are more intended as introductions to research-level math than to graduate-level math, since the graduate students here are expected to learn the basics on their own). On the other hand, many professors are willing to hold small independent studies, so if you can find 4-5 who want to study a particular book, you can probably find a professor to run that as an independent study. For example, I heard that a group of students studied Miranda's Algebraic Curves and Riemann Surfaces with Professor Gunning, and another group learned about modular forms through Apostol's Modular Functions and Dirichlet Series in Number Theory.

As for professors, I get the sense that the professors are generally good teachers and nice people, but of course this can vary (and this is where others can chime in, since I've only seen three professors so far). Many people tell me that professors are very willing to talk to you during office hours. The grading tends not to be too harsh in the more advanced classes. However, I've heard that in the lower level classes (e.g. intro calculus and multivariable calculus), the professors aren't as good, and the grading can be extremely harsh.

Another very nice thing about the math department here is that they pretty much let you take whatever you want to take. I had no trouble getting into algebraic number theory (MAT 453) and complex analysis as a freshman.

Now I'll do a quick run-through of the specific course options. First, I start with the beginning courses that most math majors take.

MAT 214 (Numbers, Equations, and Proofs): An introduction to number theory, which few people take, but is probably a really good idea if you haven't seen number theory and/or haven't seen much beauty in mathematics.
MAT 215 (Analysis in a Single Variable): This is the introductory analysis course, which almost all math majors take as freshman. It covers the first 8 chapters or so of Rudin.
MAT 217 (Linear Algebra): This is the theoretical introduction to linear algebra, which loosely follows Hoffman and Kunze.
MAT 218 (Analysis in Several Variables): This is the rigorous introduction to multivariable analysis, which culminates in proving Generalized Stokes's Theorem and deriving the differential form version of Maxwell's equations. Unlike some multivariable analysis classes, it's not required to already know some multivariable calculus, and there are problem sessions designed to give the student an intuitive feel for the computational aspects of multivariable calculus.

Next come the 300-level courses.

MAT 322 (Algebra with Galois Theory): This is Princeton's introduction to algebra course, which, unlike most other places, is only one semester (and does Galois theory up from no abstract algebra background within that semester!). I've heard that this class is quite hard, and the professor may discuss advanced topics such as Galois cohomology or infinite Galois theory at the end. After this, if you want to learn more algebra, you can take MAT 424, which gives advanced topics varying from year to year. If you want to learn some of the algebra taught in most universities but skipped in MAT 323, you could learn it on your own (which many do), or you could take:
MAT 323 (Algebra): This is an introduction to abstract algebra eventually covering Sylow theory, modules over a PID, and applications of group theory to coding.

MAT 325 (Topology): This is a standard topology course, covering all the chapters of Munkres's topology book.
MAT 326 (Algebraic Topology): This is the undergraduate introduction to algebraic topology, which (possibly surprisingly) doesn't require 325, but does require knowledge of abstract algebra and MAT 218. In recent years, this course has mainly used the book From Calculus to Cohomology.

MAT 327+328 are the introductory courses in differential geometry, the first talking intuitively about curves and surfaces, and the second developing the machinery of manifolds and vector bundles.

MAT 330-332: This is the analysis sequence famous at Princeton, from which three textbooks have been written. The courses are Fourier analysis, complex analysis, and Hilbert spaces/measure theory, though neither is a prerequisite for any other. These courses are somewhat difficult and teach students to think analytically (in the sense of analysis), with lots of difficult problems requiring all sorts of bounding and other analytic arguments. The classes also touch on relations to other areas of math, especially number theory. You'll have seen proofs of Dirichlet's Theorem and the Prime Number Theorem by the end of 330 and 331. I'm in the complex analysis course right now, and we proved Cauchy-Goursat in the third lecture and have been moving on quickly from there; it's quite intense.

There are 400-level topics classes in analysis, algebra, and geometry, respectively (usually 1-2 of each a semester) numbered MAT 451-456. The algebra one usually covers algebraic or analytic number theory, or algebraic geometry. There's also MAT 424, which is a another topics class in algebra that happens in the spring, and this may cover topics such as advanced Galois theory, algebraic number theory, algebraic geometry, representation theory, or Wedderburn theory. There's also an extra topics in analysis class which happens in the fall and covers distribution theory (MAT 433).

That's the Princeton math department. Looking over this, I feel like the department here has a somewhat unconventional and creative tone, which I especially feel about classes like MAT 322 and 326, the willingness of professors to supervise independent studies, and the lack of a set curriculum. In some sense, the department doesn't teach you a specific set of mathematical facts that an undergraduate should know, but it teaches you how to think mathematically.

[oab729]

Wow mat 325 sounds ridiculous... All of Munkres?! that's insane, simply because Munkres' text covers so much material. He goes into algebraic topology and such later on.

[calc rulz]

I don't know if it covers every little detail in Munkres, but it does deal with basic topology, function spaces/paracompactness, and the fundamental group.

[JRav]

I would think that's about right for most topology courses. Chapter 1 is a lot of set theory stuff so if you omit that, you could probably do a lot more. We did the first 4 chapters completely and chapter 9 completely-but we probably could have done more if we skipped the set theory stuff.

[Kileel]

I go to Cambridge and am quite interested in how the tripos's level compares to that of top US programs. (I'm Canadian.) The question is nontrivial since most Cambridge courses do not follow a textbook. So when I name some below it means they are drawn from a list of 5 or 6 texts suggested for the corresponding class -- hopefully a meaningful approximation. Also, due to the exam system (arcane to most North Americans), nth-year students very usually do nth-year courses. Exceptions typically see a student just sitting in on higher-year lectures. The keen can locate current, very detailed syllabuses or past example sheets and tripos exams online to get a fuller picture of the course's quality.

Here's IA, the first year.

Numbers & Sets: once called discrete maths. Basic number theory, set theory/logic, intro to series and sequences, countability.
Differential Equations: standard ODE stuff through to a bit on PDEs.
Vectors & Matrices: an applied/practical approach to linear algebra and vector algebra. Easy.
Groups: usual theorems (Lagrange, isomorphism thm, Orbit-Stabilizer). Mobius maps and the group. Matrix groups. S_n and A_n.
Analysis I: in a single variable. Up thru Riemann integration.
Vector Calculus: the various integrals, differential operators and integral thms. Applications to Maxwell's eqns and fluids. Laplace's and Poisson's eqn. Then an applied treatment of tensors.
Probability: discrete and continuous random variables. Inequalities, CLT (with heuristic mgf proof). Loosely Grimmett and Welsh. Lectured by Timothy Gowers last year.
Dynamics & Special Relativity: note maths equals math union theoretical physics. Newtonian stuff, orbits, rotating frames, rigid bodies, SR.

Now a second year, I'm presently taking

Analysis II: all things uniform, normed spaces, metric spaces, contraction mapping, differentiation from R^m to R^n. Rudin or Korner's Companion appropriate.
Markov Chains: discrete time ones; quite like Norris's first chapter.
Linear Algebra: theoretical treatment. Dual spaces, JNF, mainly (not completely) restricted to finite-dimensional vector spaces. Halmos.
Methods: principally for PDEs, though started off with Fourier Series and Sturm-Liouville. 'Applied'.
Quantum Mechanics: Gasiorowicz's book at a similar level. Ummm... apparently?

Next term I'll do Complex Analysis (I imagine standard), Geometry (based on Wilson's Curved Spaces), Groups, Rings & Modules (affectionately read GR(i)M; Artin or Herstein), Metric & Topological Spaces (read Sutherland) and Variational Principles. Possibly, I'll turn up to the non-examinable Algorithms lectures too. If I wanted, I could be doing Fluid Dynamics, EM, Statistics or Numerical Analysis. Throughout both terms there are computing projects.

The third year concludes the undergraduate degree and leads into Part III. There are several courses at an advanced level. My class list might be: Algebraic Geometry, Riemann Surfaces, Galois Theory, Graph Theory, PDEs, Algebraic Topology, Linear Analysis, Differential Geometry and Representation Theory. Part III is an intense 1-year Masters equivalent course which looks awfully fun. Around ninety courses are offered; pick six.

Overall, I'm really happy with Cambridge. There are no general requirements and, to my knowledge, the supervision system has no North American analog. calc rulz, eg, might have had issues diving into higher-level stuff but commonly (in the pure subjects at least) much harder and more interesting problems are included at the end of assignments. Lecturers are concise, necessarily since a "long" class consists of only 24 lectures. The once-per-annum evaluation setup demands you to continually synthesize material from different areas. Unfortunately (correctly?), there seems to exist here the attitude that undergraduate maths research is too trivial to warrant support. Finally, while there are 250+ math students per year university-wide and loads of choice especially in parts II and III, Cambridge's collegiate system gives rise to a nice "small department" feel.

[JRav]

It seems that Cambridge places a lot more emphasis on applied math than most programs in the US, which makes sense. It also seems like Cambridge gives more of a survey of many different areas than a detailed look at anything particular. For example, I didn't see anything in your post that seems like an algebra II course (Galois theory) or anything to do with measure theory, which I think many programs here offer. Correct me if I'm getting the wrong impression.

[Kileel]

Notice I said I will be taking Galois Theory next year. Measure theory is also taught, in Probability & Measure. (Logic & Set Theory is too... so is Number Fields and Number Theory.) However my tentative list of 9 courses for next year is lengthy at present. But Measures is something I'll consider or least try to pick up on my own, especially if it's standard at most US places.

So I think your impression is half-correct. You definitely can take a detailed look at something; maybe my personal tastes obscured this. I would argue, and most people who have actually done the tripos would, that the strength is precisely depth. Different classes by design blend into each and the expectation is, in IB for example, everything from IA you've more or less mastered and can prove on the spot. An older student from my college took in her fourth year strictly algebra-related classes. The breadth of options makes such depth feasible. Also bear in mind we just do math. It is interesting to note 3rd-year MIT students on exchange tend to complete their math requirements (so that's two more years) here in just one year. Most also do our second year. (To be fair, some students at MIT go well-beyond the requirements, diving into grad classes off the bat.)

You're right though Cambridge emphasizes the applied. In the first two years, you are "forced" into some applied courses and if you in fact prefer this madness, then you can get a math degree having done GR, QM, Fluids, EM etc and not much, beyond the first 1.5 years, in pure math.

[JRav]

A lot of people here double in math and physics or some engineering field so they see a lot of applied stuff (I'm actually in economics as well, which is not quite physics ), but Cambridge seems to "force" it on people more (that's not meant to be a bad thing).

Most universities here will teach parts of measure theory as part of an advanced probability course, but that will usually be in an applied math department. Typically, there is a separate course devoted to measure theory (perhaps called Measure Theory, Real Variables, Integration,...) which is taken by upper level undergrads and grad students. They typically use a book like Stein's "Real Analysis," Wheeden/Zygmund's "Measure and Integral," "Papa Rudin," or Folland's book, or Royden's book, books which are theory-based instead of probability-based.

You said you were taking Galois theory but I didn't see anything. I saw "Groups, Rings, & Modules" but I didn't think that Galois theory would be taught in that class. Most US universities have a two-part algebra curriculum-groups, rings, and fields in the first half and Galois theory in the second.

[Kileel]

[JRav]

Looking through that list (briefly), I noticed a few things. First, indeed it looks like "Probability and Measure" is the closest thing to what I am referring to, although "Linear Analysis" certainly has elements. However, I noticed an absence of Differentiation, Maximal Functions, Approximations to the Identity and so forth (or maybe I just didn't read closely enough), which is stuff you're guaranteed to see in a Measure Theory/Real Variables course. The other thing is the analysis sequence. Your Analysis I class is what would typically be called "Advanced Calculus" here, or "Honors Calculus" or something. Spivak's book is just calculus. Your "Analysis II" is what would typically be done in the first semester of analysis here (say, from baby Rudin or some other book). The second semester for us would be integration, although how far depends on the course. For example, my analysis II course was devoted mostly to measure theory, but I don't think that's necessarily true of every analysis II course.

Your course catalog is quite interesting.

[oab729]

Yes, Cambridge seems surprisingly applied, which is not at all a bad thing! My measure theory course is far too theoretical sometimes, and, especially as we delve into functional analysis and banach spaces and especially dual spaces (I basically blanked out when my professor talked about isomorphisms from X to X**), I have no sense of the motivations behind these theorems and definitions. Here, there's some amount of overlap between undergrad studies and graduate studies, i.e. Graduate Algebra goes over undergrad Algebra, but with more depth and breadth such as group actions etc. I know that MIT is also much more "applied" in that my friend's Analysis II course has psets requiring linear algebra, differentials, measures, etc, in what must be a really fun experience.

I'll also admit that the problems I do are typically pretty easy. They're typically just quick problems that require only an understanding of the theorems. I typically finish a week of homework in 1 or 2 hours?

Honestly though, as someone who wants to do more applied stuff, I'm quite lost about what to do in the future and what courses to take. I'm quite sick of theory and would like to see some applications of functional analysis soon!

[JRav]