# Math Major Roadmaps

This document outlines “roadmaps” of course options for undergraduates interested in particular fields and applications of mathematics. Each roadmap consists of three stages:

**Stage 1:**Introductory courses with few prerequisites, accessible to a typical sophomore.**Stage 2:**More advanced classes for students who have mastered several stage-1 classes.**Stage 3:**The most advanced classes, often beginning graduate-level subjects, for students who have mastered many stage-1 and stage-2 classes. (Many math majors will never take a stage-3 course, and that’s okay!)

The **stages also reflect priorities**—for example, students interested in pursuing analysis probably want to take 18.100 as early as possible, whereas a student pursuing mathematical finance might take it later to deepen their understanding.

In each specialized field or application of mathematics below, we list courses that could be **relevant for a career in that field**—not just courses narrowly in that specialization, but also useful courses in broadly related areas. On the other hand, these specific courses are **not requirements** to pursue that field.

Note that a math degree requires 18.06/18.C06/18.700/701 (or approved substitutions thereof), but these are not necessarily listed in every roadmap below, nor do we list GIRs like 18.02.

## Pure Mathematics

General comments: Below is a list of research areas. **As an undergraduate, however, you should not conceive of yourself as specializing in one or another of these areas**. A much better idea is to gain experience in several of them. You may find yourself taking more courses in one or another area, but all of these fields of study reinforce each other.

Stage 2, but not specific to any particular area: 18.821 (Project Lab in Mathematics) is a great introduction to some essential skills (research, teamwork, and communication).

### Algebra

- Stage 1: 18.090, 18.700+18.703, 18.700+18.701+18.702 or 18.701+18.702
- Stage 2: 18.704, 18.715, 18.721, 18.782
- Stage 3: 18.705, 18.706, 18.725, 18.745, 18.755, 18.783

### Analysis & Geometry

- Stage 1: 18.090, 18.100, 18.700/18.701
- Stage 2: 18.101, 18.102, 18.103, 18.104, 18.112, 18.152, 18.950, 18.994
- Stage 3: 18.125, 18.155, 18.156, 18.952, 18.965, 18.966

### Logic

- Stage 1: 18.090, 18.510, 18.701 or 18.700+18.703
- Stage 2: 18.400, 18.404, 18.504
- Stage 3: 18.515 (not offered regularly)

The MIT Philosophy department also offers subjects in logic: 24.241–24.245 and 24.711. Also consider logic classes at Harvard.

### Number Theory

- Stage 1: 18.090, 18.700+18.703 or 18.701+18.702, 18.781
- Stage 2: 18.704, 18.721, 18.782, 18.784
- Stage 3: 18.705, 18.725, 18.783, 18.785

### Probability & Statistics

- Stage 1: 18.090, 18.065, 18.100, 18.600, 18.700/18.701
- Stage 2: 18.102, 18.103, 18.112, 18.200, 18.211, 18.615, 18.642, 18.650
- Stage 3: 18.125, 18.675, 18.676, 18.677, 18.338, 18.416, 18.424, 18.655, 18.657

### Topology & Geometry

- Stage 1: 18.090, 18.100, 18.112, 18.701+18.702 or 18.700+18.703, 18.900, 18.950
- Stage 2: 18.101, 18.102, 18.901, 18.904, 18.952, 18.994
- Stage 3: 18.116, 18.155, 18.721, 18.755, 18.782, 18.905, 18.906, 18.965, 18.966

## Applied Mathematics

Stage 2, but not specific to any particular area: 18.821 (Project Lab in Mathematics) is a great introduction to some essential skills (research, teamwork, and communication).

### Combinatorics

- Stage 1: 18.090, 18.100, 18.200, 18.400, 18.600, 18.701+18.702 or 18.700+18.703, 18.900
- Stage 2: 18.204, 18.211, 18.212, 18.112, 18.404, 18.410, 18.453, 18.721, 18.781, 18.901, 18.950
- Stage 3: 18.217, 18.218, 18.225, 18.226, 18.338, 18.455, 18.615, 18.705, 18.715, 18.725, 18.745

### Computer Science

- Stage 1: 18.06/18.C06/18.700, 18.062, 18.090, 18.200, 18.330, 18.600, 18.650, 18.701
- Stage 2: 18.204, 18.211, 18.400, 18.404, 18.410, 18.424, 18.434, 18.453
- Stage 3: 18.337, 18.415, 18.416, 18.425, 18.435, 18.437, 18.455, 18.783

Students in this area should strongly consider supplementing their math courses with several courses in computer science; see the 18c major requirements for typical choices.

### Economics and Finance

- Stage 1: 18.03, 18.05, 18.06/18.C06/18.700, 18.065, 18.090, 18.300, 18.330, 18.600, 18.650
- Stage 2: 18.100, 18.152, 18.303, 18.453, 18.642
- Stage 3: 18.103, 18.125, 18.675, 18.676, 18.677, 18.335, 18.337, 18.338, 18.352, 18.353, 18.355, 18.615, 18.655

Students interested in economics and finance should strongly consider supplementing their math courses with several classes in course 14 and 15, if not a minor or double major.

### Computational Science and Engineering

- Stage 1: 18.03, 18.06/18.C06/18.700, 18.04, 18.05, 18.062, 18.065, 18.090, 18.200, 18.303, 18.330, 18.600, 18.650, 18.S190, 18.S191
- Stage 2: 18.100, 18.404, 18.434, 18.410, 18.701
- Stage 3: 18.335, 18.336, 18.337, 18.338, 18.367, 18.415, 18.437

Students in this area should consider supplementing their math courses with several courses in computer science, both for software engineering (e.g. 6.100A, 6.1010) and numerical methods or optimization (e.g. 6.7201, 6.7330). For students interested in computational modeling of physical systems, see also the Physical Applied Math classes below.

### Physical Applied Math

- Stage 1: 18.03, 18.04, 18.06/18.C06/18.700, 18.090, 18.300, 18.303, 18.330, 18.352, 18.353, 18.354, 18.600, 18.650
- Stage 2: 18.100, 18.112, 18.152, 18.384, 18.417
- Stage 3: 18.102, 18.103, 18.305, 18.306, 18.335, 18.355, 18.357, 18.367, 18.369, 18.376, 18.377

Students interested in physical applied mathematics should strongly consider supplementing their math courses with at least 2–3 non-GIR courses in physics and/or engineering, depending on their field of interest, if not a minor or double major.

### Statistics and Data Science

- Stage 1: 18.05, 18.06/18.C06/18.700, 18.062, 18.065, 18.090, 18.100, 18.200, 18.330, 18.600, 18.650, 18.701
- Stage 2: 18.102, 18.112, 18.204, 18.642
- Stage 3: 18.125, 18.675, 18.676, 18.335, 18.338, 18.615, 18.657

Students in this area should consider supplementing their math courses with courses in computer science on machine learning and courses in economics on econometrics.