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.
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.
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.
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.
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.
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.
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: