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**“**core” topics**.

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. The following courses are organized by Spring/Fall and in numerical order following the math department’s organization:

**Tutorials and Reading Courses**

**Math 60r: Reading Course for Senior Honors Candidates**

*Course Content:* Math 60r is designed for students writing senior theses, in order to provide more time to work on the senior thesis. It is not mandatory to sign up for Math 60r if you are writing a senior thesis; the only real purpose of this course is to free up time to work on the thesis. It can be taken senior fall or spring or both.

**Math 91r: Supervised Reading and Research**

*Course Content:* Math 91r is an opportunity for students who engage in a directed reading course about a topic not typically covered in another math course, under the supervision of a faculty member, Benjamin Peirce fellow, or postdoc. This is a great opportunity to learn about a unique math topic in a different style from the typical problem set/exam cycle of a normal math class. Open to math concentrators in any year. The student needs to take the initiative to ask the faculty member; a group of students can also ask a faculty member to do a reading course together.

**Math 99r: Tutorial**

*Course Content:* Math 99r is a supervised small group (less than 8 students) tutorial that results in a final paper or presentation. Tutorial topics are typically posted to the math department website at the start of the academic year. Previous topics include Morse Theory, Etale Cohomology, Category Theory, and Elliptic Curves. It may be repeated for course credit with permission from the Director of Undergraduate Studies, but only one tutorial may count for concentration credit. Tutorials usually automatically satisfy the concentration expository requirement.

**110s: Analysis Courses**

**Math 110: Vector Space Methods for Differential Equations** (spring)

*Course Content:* Math 110 covers the theory of finite and infinite-dimensional inner product spaces and then applies it to solve differential equations, discussing existence and uniqueness theorems, Sturm-Liouville systems, Fourier series and transforms, and eigenvalue problems.

*Prerequisites:* Computational linear algebra and multivariable calculus (Math 19a/b, 21a/b, 22a/b, 23a/b, 25a/b, or 55a/b) and exposure to proofs (Math 22a/b, 23a/b, 25a/b, 55a/b, 101, 112, 121, or 130). Real analysis (Math 23a/b, 25b, 55b, or 112) may be helpful but is not necessary.

*Expected Workload:* 5 – 9 hrs/wk (average ~7 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):

- “I enjoyed the class very much and learned a lot about differential equations and Hilbert spaces. Math 23a was very good background for this course.”

**Math 118r: Dynamical Systems (spring)**

*Course Content:* Introduction to dynamical systems theory with a view toward applications. Topics include existence and uniqueness theorems for flows, qualitative study of equilibria and attractors, iterated maps, and bifurcation theory.

*Prerequisites:* Mathematics 19a,b or 21a,b or Math 22a,b,or Math 23a,b or Math 25a,b or Math 55a,b; or an equivalent background in mathematics.

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

*Note: *Previous students in Math 118r 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 (both from the Q guide):

- “A lot of the stuff we did had cool real world applications or geometric interpretations. It uses tools in linear algebra and analysis and thus is the perfect way to continue to build skills from those classes […]”
- “This is important subject material which I think is useful and interesting for students in many different areas of study. This course does a good job of blending theory and application and giving everyone the means of studying what is of interest for them. This is also a much better way of learning ODEs without the typical ‘recipe-style’ introduction.”

**120s: Algebra Courses**

**Math 129: Number Fields (spring)**

*Course Content:* Math 129 covers introductory algebraic number theory, including number fields, unique factorization of ideals, finiteness of class group, structure of unit group, Frobenius elements, local fields, ramification, weak approximation, adeles, and ideles.

*Prerequisites: *Familiarity with Galois theory (Math 123) and elementary number theory (Math 124). Familiarity with complex analysis (Math 113 or 55b) can help but is not essential.

*Expected Workload: *4-8 hrs/wk (average ~8 hrs/wk)

*Note: *Previous students in Math 129 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 (both from the Q guide):

- “Algebraic Number Theory has seemed to me to be a hard subject to get a feel for on one’s own. In light of that, taking Math 129 with professor Mazur is perhaps the most wonderful way of getting a familiarity with the subject. In particular, he lectures with great clarity, and makes deep, often difficult things to prove seem effortless and natural. If Mazur teaches this course again, it is a must take.”
- “Number theory is a difficult subject matter. It can be dense and hard to understand the greater picture of what is going on. It can also be very interesting and fun. To get anything out of this course you need to seriously devote the time to making sure you are understanding at every stage (for me this was doing all the reading carefully and attending all sections and office hours). Professor Mazur is an absolute treasure, so I would definitely seize the opportunity to take it if he is teaching it again.”

**Math 137: Algebraic Geometry (spring)**

*Course Content:*** **“This class is an introduction to algebraic geometry. Some topics we will cover include Hilbert’s Nullstellensatz, affine and projective varieties, plane curves, Bézout’s Theorem, morphisms of varieties, divisors and linear systems on curves, Riemann-Roch Theorem.”

*Prerequisites: *Knowledge of the material in Math 123. Algebraic geometry tends to assume a great deal of background from a variety of fields, so greater mathematical maturity may be helpful.

*Expected workload:*** **7-10 hrs/week (average ~8 hrs/wk)

*Note:* Previous students in Math 137 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 (both from the Q guide):

“Fairly standard course in the department. Taught by a postdoc who works in the field, which was nice. No exams – grade based entirely on problem sets and a final presentation – this is in keeping with the very low stress nature of the class, as the problem sets were always very reasonable. Lecture notes made available online. Overall a very good class, on interesting material, with a good professor. Highly recommended.”

“Before taking the class, you might want to ask yourself: do you really care about these things? The commutative algebra is interesting and useful, but the Algebraic Geometry is very old, mostly the kind people did before 1900. You’ll get to some cool results at the end of the class, but if you just want to do Algebraic Geometry, you might want to pick up the Commutative Algebra from elsewhere and do 232/233 (also, the problem sets are a lot of work, so it is a significant time commitment).”

#### Math 152: Discrete Mathematics (spring)

*Course Content: *An introduction to finite groups, finite fields, finite geometry, finite topology, combinatorics, graph theory, and (for section 2 only) elementary algebraic topology. A recurring theme of the course is the symmetry group of the regular icosahedron. Elementary category theory will be introduced as a unifying principle. Taught in a seminar format: students will gain experience in presenting proofs at the blackboard. Covers material used in Computer Science 121 and Computer Science 124.

*Anti-requisites: *Not to be taken in addition to Computer Science 20, Mathematics 55a/b or Mathematics 122.

*Prerequisites: *For section 1: Mathematics 19b or 21b. Previous experience with proofs is not required. For section 2: Mathematics 23a or 25a or an equivalent background in mathematics that includes experience with proofs.

*Expected Workload:* 4-7 hrs/wk (average ~6 hrs/wk)

- “Math 152 is a fun, approachable seminar. I had only taken Math 21a and 21b, so it was a good first step into more abstract concepts and proofs. I initially took this because it fulfilled an Applied Math requirement and prepared me for Math 112, but I ended up enjoying it much more than expected. If you’re looking for a relatively chill math course and want to try out abstract math, this is a great option. It’s very approachable and the flipped classroom approach works pretty well. But know that while it’s relatively chill, it’s not a walk in the park.”
- “This is a fantastic course and is what I would consider to be the gem of the math department. This probably isn’t the most “useful” course in the world, as I don’t see it really preparing you for many other courses. However, if you’re genuinely curious about mathematics, you learn a lot of really interesting concepts and how different parts of math connect together. You also get the opportunity to be in a seminar format with Bamberg, who is one of the best and funniest professors I’ve had.”

**Math 154: Probability Theory**

*Course Content: *An introduction to probability theory. Discrete and continuous random variables; distribution and density functions for one and two random variables; conditional probability. Generating functions, weak and strong laws of large numbers, and the central limit theorem. Geometrical probability, random walks, and Markov processes.

*Prerequisites: *A previous mathematics course at the level of Mathematics 19a/b, 21a/b, or a higher number. For students from 19ab or 21ab, previous or concurrent enrollment in Math 101 or 102 or 112 may be helpful. Freshmen from Math 22a, 23a, 25a or 55a fall term can also take the course.

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

*Note:* Previous students in Math 154 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 (both from the Q guide):

- “This class is a nice introduction to probability from a proof-based math perspective. It is very broad and therefore at times feels scattered, but it is good exposure to various topics in probability that will make you more comfortable with the field.”
- “This is a very good elective math course which gives a relatively clear introduction to probability, and covers many interesting topics. It is not a difficult course but gives a good basis to learn from. I would recommend this for anyone who would like something with more of a mathematical/theoretical understanding than Stat 110.”

**Math 157: Mathematics in the World** (spring)

*Course Content: *Math 157 teaches problem-solving techniques in a wide variety of areas such as logic, number theory, probability, combinatorics, and algorithms, with each class meeting being unified by one common topic.

*Prerequisites:*** **None

*Useful Knowledge: *Math 19/21/22/23 is recommended as background, prior knowledge of coding/Python is useful but not required since there are workshops at the beginning to help

*Expected Workload:*** **3-6 hrs/wk (average ~7 hrs/wk)

*Note:*** **Each class meeting begins with a short introduction on the day’s topic, followed by time to work through the day’s set of problems collaboratively with classmates and finally time to review answers as shared by other groups or by the course head. Note that while Joe Harris is officially listed as course head, it is the Head TF that runs and teaches the course. Problems closely resemble high school competition math or quantitative interview questions. Problem sets usually contain some coding component.

- “A good class for someone looking for a nontraditional math course!”
- “The set-up of this class is so great: no tests, very interesting final paper, fun collaborative work in class.”
- “Amazing course. Not as easy as everyone makes it seems. The psets are not effortless. But it is really easy to get a good grade if you just do the homeworks and the final project.”

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