Core Math Courses Beyond Freshman Year

The math department offers a wide variety of options beyond the introductory math courses. Courses listed in this section are considered “core” because the material covered in them forms the foundation of modern math, but it is by no means necessary to take all or most of them in order to become a mathematician; you should pick the ones that interest you the most. To help guide you in selecting your math courses, here are some guidelines:

  • Remember to consider the professor while choosing courses. Course material and difficulty can vary dramatically from year to year depending on who is teaching the course. We strongly recommend reading the Q guide for instructors’ previous courses and reading syllabi (Syllabus explorer) for specific course material.
  • Prerequisites are not strictly enforced and may vary depending on the instructor. If you’re not sure about your preparation for a particular course, ask the professor what background they expect. It is also acceptable to learn prerequisite material on your own via reading a textbook or something similar. 
  • Courses in this section are not ordered by difficulty; they are grouped by subject and numerically ordered. See expected workload, prerequisites, and comments for an estimate of the difficulty of any particular course.
  • See this section for other topics (which are not considered “core” for other courses).

The information here is taken from the <a href="http://my.harvard.edu/">course catalogcourse catalog and supplemented with information from students. The following courses are organized by Spring/Fall and in numerical order following the math department’s organization:

110s, Analysis Courses: Analysis is the branch of math focused on how real and complex functions change, by rigorizing calculus and extending it to more unifying applications.

120s, Algebra Courses: Algebra is the branch of math focused on generalizations of number systems such as the integers and the reals, as well as other applications.

130s, Geometry and Topology Courses: Geometry and topology are branches of math focused on the study of space, generalizing fundamentals of geometry to more general conceptions of space.

Spring

Math 101: Sets, Groups, and Topology (both semesters)

Course Content: Math 101 is primarily an introduction to rigorous math, including axioms and proofs, essential for anyone who is interested in math and has not seen this material before. It also covers material including set theory, basic group theory, and low-dimensional topology. This course is the preferred introduction to proofs and it is strongly recommended that you take this course as soon as you are able to.

Anti-requisite: Math 23a/b, 25a/b or 55a/b (and some intro to proofs overlap with 22a/b, but not a strict anti-requisite)

Prerequisites: Basic differential and integral calculus (Math Ma/b, 1a, or 5 on AP Calculus AB). You can take this course simultaneously with Math 1b or 21a/b. 

Expected Workload: 7 – 10 hrs/wk (average ~9 hrs/wk)

Note: Previous students in Math 101 have had a wide range of experiences, depending on the year, their background, and how much time they have to devote to the course. Here is a sample of student testimonials (all from the Q guide): 

  • “The most interesting math class I’ve ever taken. One of the few that actually changed the way I think about and solve problems.”
  • “I took 101 as prep for real analysis, with 21a/b as background. I had no idea what a proof was before this class, but now I feel confident going into upper-level classes.”
  • “The quality of this class heavily depends on the instructor, as the class has a new instructor each semester or year, so you should learn about your instructor before deciding to take the class.”
  • “It’s a great introduction to proof–based mathematics. I came in not knowing what it really meant to write a mathematical proof and now I’m a lot more knowledgeable about constructing a proof and broad topics in advanced mathematics.”

110s: Analysis Courses

Math 112: Introductory Real Analysis (spring)

Course Content: Math 112 is an introduction to real analysis–the rigor behind calculus–covering continuity and limits in metric spaces, uniform continuity and convergence, and the Riemann integral. This course provides an introduction to proofs for students from Math 21. The material in this course is essential for modern mathematics, so it is recommended that you take it as soon as you are ready.

Anti-requisite: Math 23a/b, 25a/b, or 55a/b

Prerequisites: Multivariable calculus and computational linear algebra (Math 19a/b,21a/b, 22a/b)

Useful Knowledge: Students may consider taking Math 101 or Math 22a/b before or concurrently with this course to gain further exposure to proofs.

Expected Workload: 7 – 10 hrs/wk (average ~11 hrs/wk)

Note: Previous students in Math 112 have had a wide range of experiences, depending on the year, their background, and how much time they have to devote to the course. Here is a sample of student testimonials (all from the Q guide): 

  • “This course is absolutely fantastic. […] The material is very interesting and you prove a lot of the theorems and methods you have already studied in calculus. I would make sure to have some background with mathematical proofs before taking this course, like Math 101.”
  • “This is a strong class, and the content is really important. It is challenging but not crushing like other math classes. You get an interesting mix of people because it is required for a lot of subjects besides math.”
  • “This was my first proofs-based course coming out of the 21 track, and I highly recommend taking it if you’re in a similar situation. While it might seem a little intimidating at first, the proofs element quickly becomes easier, and the material is cool. Taking this course got me excited about pure math, and I’m definitely planning on taking more math courses in the future.”

Math 113: Complex Analysis (spring)

Course Content: Analytic functions of one complex variable: power series expansions, contour integrals, Cauchy’s theorem, Laurent series and the residue theorem. Some applications to real analysis, including the evaluation of indefinite integrals. An introduction to some special functions.

Anti-requisite: Math 55b

Prerequisites: Multivariable calculus (Math 19a/b,21a, 22a/b or equivalent) and a background in proofs (22a/b, 23a/b, 25a/b, Math 101) 

Useful Knowledge: Depending on the instructor, it may have been helpful to have taken other 100-level math courses.

Expected Workload: 7-10 hrs/wk (average varies widely based on who is teaching the course)

Note: Previous students in Math 113 have had a wide range of experiences, depending on the year, their background, and how much time they have to devote to the course. Here is a sample of student testimonials (all from the Q guide): 

  • “Take it! Complex analysis is an engaging subject with connections to many other topics, including algebra, number theory, topology, and physics. The course varies year to year, depending on who’s teaching it. Under the direction of Professor Siu, we covered more than I was expecting. The workload was heavy, but not unreasonable. Overall, a fantastic experience.” (Spring 2019, Siu)
  • “Complex analysis is an interesting area. In particular, it helped me understand some results that I had seen referenced in other courses. The book used and topics covered in this course vary significantly with the course head, I would recommend shopping the course to see if the current version is right for you.” (Spring 2018, Boney)

120s: Algebra Courses

Math 123: Algebra II: Theory of Rings and Fields (spring)

Course Content: This course is often split into two parts: i) Fields and Galois theory ii) Rings and Modules. Material of the former includes field extensions, the fundamental theorem of Galois theory, classification of finite fields, solvability of polynomials. Material of the latter includes the structure theorems for modules. Sometimes this class covers some elements of algebraic number theory and representation theory, among other things.

Prerequisites: Mathematics 122 or Mathematics 55a

Expected Workload: 7-10 hrs/wk (average ~9 hrs/wk)

Note: Previous students in Math 123 have had a wide range of experiences, depending on the year, their background, and how much time they have to devote to the course.Here is a sample of student testimonials (all from the Q guide): 

  • “This is a solid continuation of Math 122. The material is important and you need to learn it.”
  • “It is probably a good idea to take this class right after Math 122. Being fresh out of introductory abstract algebra will make the material here easier to grasp. Galois theory is, in my opinion, very interesting but somewhat hard to follow, at least initially.”
  • “Taking math 122 and not this class is like setting up dominoes and then putting them away without knocking them over. Although rings and modules are not particularly exciting material, Galois theory is without a doubt the coolest thing I have learned in math in college so far.”

130s: Geometry and Topology Courses

Math 130: Classical Geometry (spring)

Course Content: Math 130 covers traditional geometry, including affine, projective, spherical, hyperbolic, and Euclidean geometry. This class provides an introduction to proofs for students from Math 21.

Prerequisites: Math 19a/b, 21a/b, 23a, 25a, or 55a (can be taken concurrently)

Useful Knowledge: Students from Math 19/21 may consider taking Math 101 for further exposure to proofs, as the class is heavily proof-based from the get-go.

Expected Workload: 7 – 10 hrs/wk (average ~8 hrs/wk)

Note: Math 130 depends extremely heavily on the course head for that year. Previous students have also had a wide range of experiences depending on the year, their background, and how much time they have to devote to the course. Here are a sample of student testimonials (all from the Q guide):

  • “This class was very easy: low workload, no tests, only a midterm and final paper. Take this class only if you are genuinely interested in learning something about geometry. […] I took as my first post-21 series math class and it was a great hands-on introduction to proofs and higher-level mathematics.”
  • “If you’re a math concentrator who’s afraid of topology (like I am), this is an easy way to fulfill your ‘one course in the 130s’ requirement.”
  • “MATH 130 is a great introduction to affine/projective geometry and its major theorems.”

Math 132: Differential Topology (spring)

Course Content: Generalizing the techniques and results of calculus to spaces beyond R^n. An introduction to smooth manifolds and their morphisms, transversality, differential forms, Stokes’ theorem, and de Rham cohomology.

Prerequisites: Math 22ab, 23ab, 25ab, 55ab, or an equivalent background in proofs, linear algebra, and real analysis. Math 131 is recommended but not required.

Expected Workload: Spring 2019, 11-14 hrs/wk; historically, 8 hrs/wk

Note: Previous students in Math 132 have had a wide range of experiences, depending on the year, their background, and how much time they have to devote to the course. Here is a sample of student testimonials (all from the Q guide): 

  • “This class covers some material that is very interesting and very modern, but has the potential to get out of hand very quickly. There seemed to be an emphasis on covering a huge volume of material rather than dwelling on more important aspects. The plus was that we saw some interesting Morse theory and went deep into differential forms.”
  • “The material is quite challenging. Be prepared to work hard and be ok with not understanding much.”
  • “Definitely take it! Great class. The later material particularly interesting.”

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