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, difficulty, and quality can vary dramatically from year to year depending on who is teaching the course. Thus, we strongly recommend shopping a variety of courses with different professors if possible and considering the professor while selecting courses, in addition to prerequisites and course material. Professors who have received Q scores over 4.5 over the past 3 years include Professors Dennis Gaitsgory, Joe Harris, Peter Kronheimer, Jacob Lurie, Curtis McMullen, Martin Nowak, and Wilfried Schmid. Note that this section only represents average student opinion over the past few years, and your experience with any particular professor may vary depending on your learning style and the course.
  • Prerequisites listed here are not strictly enforced and may vary from year to year. If you’re not sure whether you’re qualified to take a particular course, talk to the professor and ask what material he/she is assuming you know. It is also perfectly acceptable to learn the material taught in any particular course on your own via reading a textbook or something similar. Prerequisites are material/knowledge that all students surveyed have agreed was necessary for the course. Useful Knowledge is material/knowledge that only some students surveyed agreed was necessary for the course.
  • Courses in this section are not ordered by difficulty. They are grouped by subject and numerically ordered. See expected workload and prerequisites for an estimate of difficulty of any particular course.
  • Many interesting topics in math (such as number theory, probability/combinatorics, logic/set theory, mathematical biology, and others) are not covered in this section because they are not considered core for other courses. If you are interested in these topics, do not let this discourage you; we strongly recommend you look at this section.

Note: Some courses listed here have anti-requisites; these are courses that you cannot have taken previously to gain credit for a particular class due to overlapping content.

The information here is taken from the course catalog and supplemented with information from students.

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
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: 5 – 10 hours/week
Notes: 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 are a sample of quotes (all from the Q guide):

  • “Fairly easy, painless intro-math course. If you’re looking to get your feet wet, then take it. But if you want something super in-depth and rigorous, I would look somewhere else.”
  • “This class is a moderately interesting and fun. It may seem a bit overwhelming at first, and the material may become much more difficult during the middle of the semester, but the homework problems are not as difficult to understand as the material taught.”
  • “This class is a very good introduction to higher level math classes. It gave me confidence in my ability to handle the content of difficult math classes and my ability to think mathematically. It can be a little bit all over the place in terms of material but you do get a good sense of everything there is in higher math.”
  • “This is the course that you should definitely if you have never had exposure to any proof based mathematics. I recommend this course with enthusiasm to anyone who would like to learn to think mathematically!”

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.

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 or 21a/b)
Useful Knowledge: Students may consider taking Math 101 before this course to gain further exposure to proofs.
Expected Workload: 7 – 12 hours/week
Notes: 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 are a sample of quotes (all from the Q guide):

  • “Math 112 is a difficult course, especially if you haven’t taken a real mathematics class before. However, I throughly enjoyed the rigor of the class, and it was a great introduction to rigorous mathematics. I took Math 101 before taking this, but this class required a new level of mathematical reasoning.”
  • “This is a good intro material wise to proofs. Definitely can go straight from the 21s.”
  • “This class is pretty hard and will require a lot of time and effort, but it is very rewarding at the end when you finally feel like you understand all of the math you have ever learned prior. It is possible to take this class as an introduction to the 100s, but it will require you to reach out for help often.”
  • “The course was taught incredibly well and it is a fantastic platform for becoming more comfortable with proof based mathematics.”

Math 113: Analysis I: Complex Function Theory (spring)
Course Content: Math 113 covers introductory complex analysis, including the Cauchy integral formula, contour integrals, Laurent series, power series expansion, and the evaluation of some indefinite integrals.
Anti-requisite: Math 55b
Prerequisites: Basic multivariable calculus (Math 21a, 23b, or 25b) and exposure to proofs (Math 23, 25, 101, 112, 121, or 130).
Useful Knowledge: Students who took Math 21a may consider taking Math 112 for real analysis background.
Expected Workload: 5 – 12 hours/week
Notes: 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 are a sample of quotes (all from the Q guide):

  • “Complex analysis is a lot of fun–there are many very interesting, counterintuitive results about how complex functions behave, and you will come across a lot of clever arguments in the proofs.”
  • “For some reason, calculus over functions in the complex numbers is incredibly convenient. I am also now able to do integrals I thought were previously beyond mortal reach, which is pretty great.”
  • “The course is primarily designed to teach methods to solve new problems (read: integrals) and the rationale behind those methods. It’s not overwhelmingly focused on proof, but it does provide a good look at complex analysis in general.”
  • “It was fairly easy compared to some of the other 100-level math classes, but you do get a very solid grounding in complex analysis.”

Math 114: Analysis II: Measure, Integration, and Banach spaces (fall)
Course Content: Math 114 covers introductory functional analysis, including Lebesgue theory, Banach and Hilbert spaces, and an introduction to $L^p$ spaces, among other topics depending on the year.
Prerequisites: Real analysis (Math 23a/b, 25b, 55b, or 112).
Useful Knowledge: This course is often considered one of the more difficult undergraduate math courses, so greater mathematical maturity is helpful. See the quotes below for more details.
Expected Workload: 7 – 15 hours/week
Notes: Previous students in Math 114 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 are a sample of quotes (all from the Q guide):

  • “Math 114 gives a good introduction to several topics in real analysis. The one thing to note about this class is that it is fairly difficult, especially compared to other undergraduate math courses.”
  • “The course is not too hard. Very manageable.”
  • “This class, though tough, covers a lot of interesting material (especially near the end). […] The problem sets were often difficult, and rarely could you do them without help from the CA’s or other students. Personally, I don’t think a problem is instructive it is so hard that you have to get hints every step of the way. This class also assumes a good deal of familiarity with basic topology and analysis (presumably taught in Math 23/25).”
  • “The course material is very interesting […] but the class was a bit too difficult. The pace was very quick and falling behind by a little proved to be fatal.”

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.

Math 121: Linear Algebra and Applications (fall)
Course Content: Math 121 covers introductory abstract linear algebra, including the notions of vector spaces, linear maps, inner products, and eigenvalues, along with some applications to other fields. This course provides an introduction to proofs for students from Math 21.
Anti-requisite: Math 23a/b, 25a/b, or 55a/b
Prerequisites: Computational linear algebra (Math 19b or 21b).
Useful Knowledge: Students may consider taking Math 101 before this course to gain further exposure to proofs.
Expected Workload: 3 – 10 hours/week
Notes: Previous students in Math 121 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 are a sample of quotes (all from the Q guide):

  • “The class is great for learning more on linear algebra after taking 21b while learning how to write proofs.”
  • “This course is very well designed and it trains you well in proof writing and logic thinking. You will gain a deeper understanding of linear algebra by learning how it is related and can be applied to other areas of math, e.g. topology, group theory. However, a trade-off is that the course cannot go into details of those topics and sometimes you need to do some extra reading in order to truly appreciate the beauty of the application of linear algebra in those fields.”
  • “This is a great course for learning proof-based mathematics for those who have never taken any math class with proofs. Also, the course reviews several different interesting applications of linear algebra.”

Math 122: Algebra I: Theory of Groups and Vector Spaces (fall)
Course Content: Math 122 covers the foundation of modern algebra, including group theory, abstract linear algebra, and representation theory. In some years, it has also included introductions to topics such as ring theory and module theory. The material taught in this course is essential to understand most of modern mathematics, so it is strongly recommended that you take this course as soon as you are ready.
Anti-requisite: Math 55a
Prerequisites: Experience with computational linear algebra (Math 21b, 23a, or 25a) and exposure to proofs (Math 23, 25, 101, 112, 121, or 130)
Useful Knowledge: Students who took Math 21b but not Math 101 may consider taking Math 121 for further linear algebra exposure.
Expected Workload: 5 – 12 hours/week
Notes: Previous students in Math 122 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 are a sample of quotes (all from the Q guide):

  • “It’s a great course. Foundational material for mathematics and physics. If you’ve never written a proof before, it’s going to require extra work from you at the start — you basically have some catching up to do.”
  • “This course is challenging, time consuming, and arduous. But in the end, you have finally become part of a world of mathematics. A new language. This is the doorway to higher math. And the work it takes to get there feels as stimulating and challenging as the years of math to come will be.”
  • “I would recommend taking this course if you’re not a math concentrator and want to appreciate abstract math more (if you’re AM, CS, physics, chem, etc). I had lots of fun with this course as an AM concentrator not coming from the 23/25 track (I took the 21s and then Math 152 for intro to proofs).”
  • “This class was wonderful! I came out of Math 23 and had a lot less proof experience than most of my classmates, but I still felt like I was able to keep up and learn a lot without falling behind. […] There’s no real knowledge prerequisite for the class; you just need some comfort with mathematical notation/lingo and with writing/understanding proofs. If you have those, you’ll be fine in the class.”
  • “The material in 122 is foundational mathematical knowledge and so you should take this course if you want or need that knowledge, or if you’re looking for an accessible introduction to the sort of abstract things mathematicians study beyond linear algebra. It’s for the most part very elegant, and some of the results have astonishing structure (e.g. Sylow theorems).”

Math 123: Algebra II: Theory of Rings and Fields (spring)
Course Content: Math 123 covers basic ring theory, the basic structure theorems of modules over rings, and Galois theory. Depending on the year, it can also include material like representation theory and the tensor product.
Prerequisites: Familiarity with group theory and linear algebra (Math 55a or 122).
Useful Knowledge: Depending on the instructor and the year, additional familiarity with rings and greater mathematical maturity may be helpful (Math 55a or 122).
Expected Workload: 4 – 12 hours/week
Notes: 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 are a sample of quotes (all from the Q guide):

  • “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.”
  • “The class covers a lot of very interesting material and will force you to think deeply in a way that is often rewarding. However it did require a much greater time commitment than Math 122, for instance, so people in concentrations related to but outside of math might want to keep that in mind while considering this as an elective.”
  • “I found this to be one of the most rigorous and difficult math classes I’ve taken. I think it was great in giving me a deeper understanding of algebra, but do be prepared to put in quite a bit of work.”
  • “This course was a great introduction to fields, Galois theory, and representation theory. Background in groups (e.g. Math 122) probably useful but definitely not necessary (I came from 25, and didn’t take 122).”

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.

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: Computational linear algebra and multivariable calculus (Math 19a/b, 21a/b, 23a, 25a, or 55a), which can be taken concurrently.
Useful Knowledge: Students from Math 21 may consider taking Math 101 for further exposure to proofs.
Expected Workload: 1 – 5 hours/week
Notes: Previous students in Math 130 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 are a sample of quotes (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. If you’re not in that category however, the course material isn’t very interesting (the first 2 months are things that you learned in 10th grade geometry and the last month is ‘projective’ geometry stuff, which is just kind of weird), so I would avoid it.”
  • “Mad easy. If you like math but aren’t serious about it (i.e., you’re concentrating in some non-STEM where it isn’t an essential part of what you’ll be doing for the next few decades after college) it might be a good fit for you. Otherwise, you might want to hold off.”

Math 131: Topology I: Topological Spaces and the Fundamental Group (fall)
Course Content: Math 131 begins with an introduction to the basic properties of topological spaces and point-set topology. It then covers the fundamentals of algebraic topology, including the fundamental group and covering spaces.
Prerequisites: Familiarity with real analysis (Math 23a/b, 25b, 55b, or 112) and with groups (Math 55a, 101, or 122).
Useful Knowledge: Students who took Math 101 may consider taking Math 122 for a stronger background in group theory.
Expected Workload: 4 – 10 hours/week
Notes: Previous students in Math 131 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 are a sample of quotes (all from the Q guide):

  • “I think everyone in math and physics should take an intro to topology class, because it is hugely important as a branch of mathematics and has really surprising applications for physicists.”
  • “Topology is an intuitive, beautiful subject! The difficulty of the course and topics covered will vary greatly depending on who is teaching it. But this is an elementary building block for a lot of math so it’s a great class to take junior year.”
  • “This is a really cool course. Topology is fun: visual and mind-blowing, especially algebraic topology towards the end.”
  • “This class is very good, but very difficult conceptually and has an extremely heavy workload.”

Math 132: Topology II: Smooth Manifolds (spring)
Course Content: Math 132 covers the properties of differential manifolds and vector fields, as well as the proof of Stokes’ theorem, differential forms, and some homology.
Prerequisites: Familiarity with computational linear algebra, multivariable calculus, and real analysis (Math 23a/b, 25a/b, 55a/b, or 112).
Useful Knowledge: Math 131 may be a good idea for students interested in rigorizing all proofs about basic topology.
Expected Workload: 4 – 10 hours/week
Notes: 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 are a sample of quotes (all from the Q guide):

  • “It is a challenging topic, but very interesting that only through studying very rigorously can you. If you only have the time to glaze over things, you probably won’t appreciate it much.”
  • “This class is a great introduction to many interesting topics. It is more concrete than 131, but it still presents many beautiful results. Definitely take this if you enjoyed the analysis portions of Math 23/25/55.”
  • “This is a hidden gem of the math department. This course is full of powerful techniques and really cool results and serves as good preparation to study tons of neat topics in physics from a rigorous perspective (symplectic manifolds, Lie groups, etc.).”
  • “This really fascinating material that’s very hard to get a handle on. Expect to spend a […] ton of time confused.”

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