Which Colleges Have Strong Undergraduate Math Programs?

[d_leet]

Well I'm going to be a senior in high school this coming school year, and since it's starting to get close to the time to apply to colleges I've been doing a lot of research into colleges. I've been taking math classes at a local university since the summer after my sophmore year of high school and I'm pretty set on being a math major in college so I'm trying to find schools that have good undergraduate math programs besides the ivy league schools, and the top public universitys like The Universities of Wisconsin and Michigan, as well as the University of Chicago which is where I would really like to go, but I've had a lot of trouble finding similar schools. If anyone could recommend any other schools that I should take a look at I would be very grateful.

Thanks.

 

[mathwonk]

i think chicago is outstanding, and because it is in a big city that many people find scary, it is not nearly as hard to get into as the ivies are. it may have changed but the acceptance rate at chicago used to be 3 or 4 times as high as at harvard and columbia for instance or brown.


I don't know hoe strong ior advanced you are but you probbaly have no idea how many good programs are out there. even at average state schools like UGA there are small programs suited to good students still offering spivak calculus and smalkl classes in differential topology and so on for undergrads.

the upside to higher profile school is more top students to learn from, but there are less prestigious ones with good subprograms for their best students.


read the catalogs and see if they lump everyone into the same big classes where yopu will be held back by slow students or if they have a good honors prgram for the well prepared and interested.

actually in my opinion some of the ivies are less suited to this good instruiction these days, since e.g. i do not believe harvard or stanford still have a spivak calc course. they did not have them 15 years ago when my son was applying.

small outstanding math schools include haverford and swarthmore and rice, or do you mean less prestigious than those?

 

[mathwonk]

its helpful pages like this from chicago that convince me it is still a superb place for students:

A Guide for the Perplexed On the College Calculus Sequences

This guide is meant to dispel a widely held misconception about calculus at the U. of C. The misconception is that the mathematical theory of calculus is taught only in the 16000's. In a related matter, some people think that the 13000's sequence is for those who have "never been any good at mathematics." Neither of these views is correct (as hinted by the use of 'misconception'). Here are the facts (or some facts--a kind of distinction you should get used to at the U. of C.).

The sequence Mathematics 16100-16200-16300 is called Honors Calculus; it might be better referred to as Introduction to Mathematical Analysis. The course starts with the mathematical definition of the real numbers, and concentrates on theory all year. The mechanics of differentiation and integration and all the standard applications are covered, but the emphasis throughout the year is on rigorous proof. Students in the 16000's are expected to prove things in their homework assignments and on tests. You don't have to be a mathematics major to take the 16000's, and you don't have to take the 16000's to be a mathematics major.

Mathematics 15100-15200-15300 and Mathematics 13100-13200-13300 focus less on theory, in the sense that students don't prove as many things for themselves. However, it is not the case that these courses only provide students with recipes and formulas (plug and chug). Students are expected to understand the definitions of key concepts (limit, derivative, integral) and to be able to apply definitions and theorems to solve problems. In particular, ALL calculus courses require students to do epsilon-delta limit proofs.

The major difference between the 13000's and 15000's is the amount of mathematics students have seen and mastered before coming to the U. of C. The 13000's sequence is designed for those who are less familiar with pre-calculus mathematics, specifically trigonometry, logarithms, and exponential functions, and to a lesser extent, high school algebra. Therefore, these subjects are covered more thoroughly in the 130's than in the 15000's. The 13000's and 15000's sequences end up covering almost the same material, but 15000's classes usually see more advanced applications.

Placement in these courses depends on tests you took during Orientation Week. Many students entering the U. of C. took calculus in high school; this fact alone will not determine your placement.

Despite our best efforts, some students will find themselves placed in the wrong level of calculus. It's easy to switch from the 16000's to the 15000's. Switching from the 15000's to the 13000's requires a little more persuasion. Going the other way is unusual but can be done. If you are convinced that you are in the wrong class, talk to your instructor and to Diane Herrmann.

The College only requires two quarters of calculus. Many departments (including the physical sciences, HiPSS, and economics) require concentrators to take three quarters of calculus. So, even if you don't like calculus, it may be worth taking all three quarters your first year so that you don't have to worry about taking the third quarter later on when you find out that your concentration requires it.

 

[courtrigrad]

how is Columbia's math department (particularly applied math)?

 

[d_leet]

Mathwonk thank you very much for your response I greatly appreciate your help.

i think chicago is outstanding, and because it is in a big city that many people find scary, it is not nearly as hard to get into as the ivies are. it may have changed but the acceptance rate at chicago used to be 3 or 4 times as high as at harvard and columbia for instance or brown.

I've done a lot of research on the university of chicago's website and from what I've seen it is definitely my top choice in schools and it seems like a place where I would truly enjoy learning and would get a great well rounded education. I think chicago's acceptance rate is around 40% compared to harvard's 9% so it hasn't changed very much from what you remember.

I don't know hoe strong ior advanced you are but you probbaly have no idea how many good programs are out there. even at average state schools like UGA there are small programs suited to good students still offering spivak calculus and smalkl classes in differential topology and so on for undergrads.

This is pretty much exactly what I'm looking for in a school. I've seen you mention Spivak's Calculus many times in other posts and I've been trying to find schools that use this book for calculus, and your exactly right that I don't really have any idea about how many good programs there are.

read the catalogs and see if they lump everyone into the same big classes where yopu will be held back by slow students or if they have a good honors prgram for the well prepared and interested.

Again this is what I've been looking for and not having a very easy time doing so. I definitely enjoy thje idea of a good honors program because I've become rather afraid of the idea of huge lecture classes where there will be little or no possibility for personal contact with the professor.

small outstanding math schools include haverford and swarthmore and rice, or do you mean less prestigious than those?

I took a look at haverford and it seems like a very nice school, but if you could list some of the less prestigious schools that would be great because my list of schools that I may apply to seems to consist of more places where I don't stand as great a chance of getting into as I would like.

Again thank you very much for your responses.

 

[Plastic Photon]

What is the significance of this spivak textbook? My first two calculus courses did not have a requirement for a textbook and the professors taught from their notes.

Consider that you will have to pay out of state tuition or high tuition costs for private schools. A full year at univeristy of texas austin is costing my friend $17k in tuition, fees, books, rent and the essentials; Rice adds up to $17k in tuition alone, and both of these schools are considered fairly cheap.

 

[d_leet]

Consider that you will have to pay out of state tuition or high tuition costs for private schools. A full year at univeristy of texas austin is costing my friend $17k in tuition, fees, books, rent and the essentials; Rice adds up to $17k in tuition alone, and both of these schools are considered fairly cheap.

I'm going to be out of state practically everywhere so I don't think tuition costs are going to be vastly different for me regardless of if I attend a public or private university.

 

[trinitron]

Michigan also offers an honors sequence using Spivak in the first course. They go on (typically) to use Spivak and Hoffman (for linear algebra) in the second semester, Spivak or Munkres (analysis on manifolds) in the third semester and Spivak for differential geometry in the fourth semester.

From the course guide (these are a bit dated, pace is usually a little faster now and content a bit different):

295: Real functions, limits, elementary topology of the real line, continuous functions, derivatives, indefinite and definite integrals.

296: Infinite series, power series, vector spaces, structure of linear maps, duality, eigenvalues, normed vector spaces, higher-dimensional derivatives (Chain rule, inverse/implicit function theorems).

395: Structure of bilinear forms, tensor products, metric spaces, function spaces, topology of vector spaces, higher-dimensional integrals, change of variables formula, partitions of unity.

396: Geometry on manifolds, differential forms, vector fields, Stokes' theorem, deRham cohomology.

Most people would also take a Algebra (512) in their first or second year, as well as a diff eq course if they haven't had one (unfortunately there aren't many good options, even the honors diff eq class here is somewhat 'plug and chug'). Then many people start taking graduate courses in their third year (590s).

See the course guide here-
http://www.math.lsa.umich.edu/courses/undergrad.shtml
http://www.math.lsa.umich.edu/courses/grad.shtml

 

[d_leet]

Michigan also offers an honors sequence using Spivak in the first course. They go on (typically) to use Spivak and Hoffman (for linear algebra) in the second semester, Spivak or Munkres (analysis on manifolds) in the third semester and Spivak for differential geometry in the fourth semester.

I've looked at Michigan's curriculusm quite extensively and it is one of the schools that I plan on applying to. Are you a student at Michigan? If so how do you like it there?

 

[mathwonk]

Michigan is a top school on a par with many of the famous private schools mentioned here. Professors there include Lazarsfeld, Fulton, Dolgachev, Hochster, and many other outstanding people.

University of georgia is on a lower level, with many weaker students, but keeps a small math program for majors with small classes and good teachers and good books. The undergrad program includes a diferential topology course from Guillemin and Pollack, and a differential geometry course from Shifrins notes or other good book like doCarmo. The 3rd option calc course is from Spivak, but some courses are on a lower plane like ode. advanced undergrads are also allowed to take grad coureses.

one thing abut the top schools, like harvard, etc... is the profesors really are the experts in the field and the courses will be morelikely to be as authoritative as possible. in 1964 at harvard i bought the niotes for richard brauer's course on galois theory and still have them. these are not available anywhere, were never published, and he was one of the world eladers in group theory. these notes are more detailed than usual, more elementary, and also more ambitious, with many topics seldom found such as galois theory in the theory of riemann surfaces. this meant little to me at the time, but now i can appreciate them. and have done for many years.

 

[trinitron]

"Are you a student at Michigan? If so how do you like it there?"

Yes, I'm a current student going into my second year. I like the school academically but it's not really my cup of tea socially. Of course, most people here love the social life, so you should really try and visit the schools you're choosing from to get an idea where you'd be happy. There are some quality of life issues (read food) I suppose. The math department is excellent, and is very welcoming, especially to top students. I believe they give out several scholarships every year. With the exception of one course, I have really liked their curiculum, and the professors are very friendly and approchable. Some of the physics courses aren't as good, because their is a wider range of people in the course and so they teach to the center. Also I personally don't think that their undergrad humanities courses are on the same level as Chicago.

I think the main advantages here are that it's easy to stand out and to do well. Prerequisites are hardly enforced, for example, so you can decide for yourself what you can handle. Also there is not an entrance exam into the honors sequence like at Chicago. Instead they let those who are interested sign up and then about half the class drops out in the first few weeks. I like that philosohpy. It's also easy to get involved in reasearch, although I'm sure this is true at most places.

Mathwonk, I am actually taking a course from Prof. Dolgachev next term, so that's good to hear.

 

[mathwonk]

Igor is terrific. youll like his course i think. you might also enjoying searching on "dolgachev surfaces", to see one of his famous constructions.