Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132
courses
Courses in Mathematics


(2013-2014)

This document gives a brief description of the various courses in calculus and some of the intermediate level courses in mathematics. It provides advice and pointers for planning your course selections. If you are a Mathematics Concentrator, or are considering entering the Mathematics Concentration, and if you are seeking some overview of the courses and how they fit together, then this document is for you. However, the guidelines presented below are exactly that: guidelines. Keep them in mind when you are deciding how to structure your program, but be sure to talk to your advisor in the Mathematics Department or to the Director of Undergraduate Studies before you turn in your study card each semester.

Calculus

Math 1a/b is the standard first-year calculus sequence. If you are thinking about majoring in math and have not taken calculus before, take Math 1 as soon as possible! If you have had a year of calculus in high school, and if you have passed the Advanced Placement examination in BC Calculus with a score of 4 or better, then you may be advised to begin with Math 21 a/b, the second-year calculus sequence.

If you scored a 5 on the BC Calculus exam and if you are advised to take Math 21 a/b, then you may wish to consider taking Math 23 or Math 25 or 55 instead of Math 21. Be warned: Math 23, 25 and 55 are intense but very rewarding courses, and both 25 and 55 require extensive work outside the classroom. To succeed in the latter two, you must be very committed to mathematics from the start.

Regardless of which calculus course you take, keep in mind that it is important to absorb ideas thoroughly. It's a bad idea to push yourself too far too fast.

For more guidance on choosing your first math course at Harvard please read the pamphlet ``Beyond Math 1: Which math course is for you?'', which you can obtain from Cindy Jimenez, the Undergraduate Program Coordinator (room 334), or from the undergraduate section of the Department's web site.

How to structure a good program

No single program is ideal for all math concentrators. You should design your curriculum based on your background, interests, and future plans. You are strongly urged to consult with your academic advisor or with the Director of Undergraduate Studies in deciding which courses are best suited for you. Do not plan to meet with your advisor on the day study cards are due, since advisors usually don't have more than a few minutes to spend with each student that day. Make an appointment with your advisor well before study cards are due. You should allot about half an hour, so you can discuss your plan of study in depth.

Learning to write proofs

Math 23, 25, 55, 101, 112, and 121 are six courses in which you learn to write proofs, meeting (often for the first time) a style of mathematics in which definitions and proofs become part of the language. Students are generally advised not to take any upper-level math courses before completing (or, at least, taking concurrently) one of these.

  • Math 101 serves three main goals. It lets a student sample the three major areas of mathematics: analysis, algebra, and topology/geometry; it introduces the notions of rigor and proof; and it lets the student have some fun doing mathematics. If you are considering concentrating in Mathematics but are not sure that you are ready to take Math 23, 25 or 55, or if you simply want a glimpse of what ``higher'' math is all about, you are urged to include Math 101 early in your curriculum. Math 101 can be taken concurrently with either Math 21a or 21b. This course is only offered in the fall. If you have had some experience with rigorous proofs and want a different taste of ``higher'' math, you might consider Math 152 in the fall. Neither Math 101 nor Math 152 is appropriate for people from Math 25, Math 55 or (with rare exceptions) Math 23.

  • Math 23, 25 and 55 are the three introductory courses for students with strong math interests. They are geared towards new students. Math 25 and 55 are much more intensive than Math 23, but require much more out of class time. Students who don't wish to make the time commitment will do well to choose Math 23. Meanwhile Math 55 should be taken only by students with extensive college level math backgrounds. Each year several first-year students ask to skip the Math 25/55 level and start with Math 122 or another 100-level course. The Department, based on many years of experience, strongly discourages this. Even if you have taken several years of math at another university, even if you have seen every topic to be covered in Math 25 or 55, you will not be bored in these accelerated courses. The topics covered in Math 25 and 55 are not as important as the level and the depth of mathematical maturity at which they are taught. Taking Math 25 or 55 is the most intense mathematical experience you are going to have in any Harvard course, shared with the most talented of your peers. You may learn more advanced material in other 100- and 200-level courses, but never with the same speed and depth as in Math 25 or 55. These courses are not taught in any other university because no other university has the same caliber of first-year mathematicians. And the courses are simply a lot of fun. Many students who have skipped 25 and 55 have been dissatisfied with their decision. In any event, you must speak with the Director of Undergraduate Studies if you plan to skip the Math 21-55 level.

  • Math 112 and Math 121 are courses suitable for students from Math 21, and they provide an alternative entry-point for the department's more advanced courses in Analysis and Algebra respectively. They should not be normally be taken by students who have been through Math 23 or 25. If you are a sophomore and have taken Math 21 but are not yet comfortable with writing proofs, then consider including these courses in your plan of study.

Key courses at the 100 level

If you have taken Math 23, 25 or 55, or if you have taken Math 21 and gained some experience in writing proofs through courses such as Math 101, 112 and 121, then you are ready to take some of the courses at the 100-level that form the core of the Mathematics curriculum. Most of the courses at this level can be classified as belonging to one of the three main streams of mathematics: ``Analysis'', ``Algebra'' and ``Geometry and Topology''. Courses belonging to these areas are numbered in the ranges 110-119, 120-129 and 130-139 respectively. In each stream, there are two courses which are regarded as ``core'' courses, making a total of six central courses. These are:

  • Math 113. Analysis I: Complex Function Theory
    Math 114. Analysis II: Measure, Integration and Banach Spaces
  • Math 122. Algebra I: Theory of Groups and Vector Spaces
    Math 123. Algebra II: Theory of Rings and Fields
  • Math 131. Topology I: Topological Spaces and the Fundamental Group
    Math 132. Topology II: Smooth manifolds

It is not necessary to include all six of these courses in your plan of study, but here are some points to bear in mind.

  • Students from Math 55 will have covered in 55 the material of Math 122 and Math 113. If you have taken 55, you should look first at Math 114, Math 123 and the Math 131-132 sequence.

  • With the exception just noted, you should consider including Math 122 early on in your curriculum. Algebra is a basic language of modern mathematics, and it is hard to comprehend advanced material without some familiarity with groups and related topics in algebra. The same remark applies to Math 123, to a lesser degree. By the same token, Math 113 should also be taken early on as Complex Analysis is used in many other fields of mathematics. You will also find the topology you learn in Math 131 useful in many other areas: amongst other things, it provides the mathematical language with which to discuss continuity and limits in wide generality.

  • Math 123 cannot be taken before Math 122; but in the other two streams, the courses can be taken in either order. Thus, Math 114 can be taken before or after Math 113, and the same applies to Math 131 and 132.

  • You should try to fulfill the distribution requirement (i.e., the requirement to take at least one course in analysis, algebra, and geometry) early in your academic career. By your junior or senior year, you should be exposed to the main branches of mathematics; then you can choose the department's advanced courses. In any case, most 200-level courses assume (at least informally) familiarity with the basic tools of analysis, algebra, and topology.

Other courses at the 100 level

At this level, there are many other courses to choose from: Number theory in Math 124 or Math 129, Differential Geometry in Math 136, Probability in Math 154, Logic and Set Theory in Math 141 and Math 143, amongst others.

  • It is a good idea to take a tutorial (Math 99r) during the sophomore or junior year. Many students found the tutorial to be one of the best courses they took at Harvard. Tutorials generally satisfy the Math Expository requirement and often lead to senior thesis topics. More about tutorials appears below.

  • Students wishing to take a rigorous course in mathematical logic in years when Math 141, 142, 143, or 144 are not offered at Harvard should consider taking logic courses at M.I.T. In any event, the Harvard courses offer a good introduction to model theory, set theory and recursion theory -- the three main branches of Mathematical Logic. Students interested in the more philosophical aspects of logic and/or in proof or set theory may want to take Philosophy 143, and those interested in mathematics of computation should look into Computer Science 121 and some of the other theoretical CS courses.

  • Students interested in Combinatorics should look at Math 155, and may also want to look up M.I.T.'s listings in that area. If you want M.I.T. courses to count for the concentration credit, you must get permission in advance from the Director of Undergraduate Studies, Prof. Peter Kronheimer (kronheim@math).

  • Students are encouraged to take courses from a variety of professors in the department and not just to ``follow'' one teacher. It is advisable to be exposed to different views and styles of doing mathematics.

200-level courses

100, 200 - What's the Difference?

The difference between 100-level and 200-level courses is fairly easy to summarize: 100-level courses are designed for undergraduates, whereas the 200-level courses are generally designed for graduate students. As far as course material goes, the 100-level courses are designed to offer a comprehensive view of all the major fields in pure mathematics. They emphasize the classical examples and problems that started each field going and they all lead to one of the fundamental results that motivates the further development of the field. In contrast, a 200-level course will assume you understand the basic ideas of a field. A 200-level course will set out the systematic, abstract foundations for a field and develop tools needed to get to the present frontiers.

The 100-level courses give you a good overview of mathematics, they foster intellectual growth, and they prepare you for your chosen career. This is not true of 200-level courses. These courses assume that you are interested in the subject, and that you are already fairly certain of becoming an academic mathematician. The amount you learn in such a course is often also entirely up to you. Your prerequisites, though correct according to the course catalog, may be entirely inadequate. Many courses are paired into 100-level and 200-level sequences:

Corresponding 100-level, 200-level Courses
Math 114