Firdan Rifaldi
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Pure Mathematics Undergraduate Curriculum: A Comparison and Open Resource Guide

Disclaimer: This guide is a reference for self-study and supplementary learning. It does not replace a formal degree, accredited coursework, or guidance from qualified instructors. Completing these materials on your own does not grant you the same qualifications, depth of understanding, or credentials as an actual undergraduate programme. If you are serious about pursuing mathematics professionally, formal education with proper assessment, mentorship, and peer interaction is strongly recommended.

Executive Summary

This report compares the undergraduate pure mathematics curriculum requirements at several leading universities (Stanford, MIT, UC Berkeley, Harvard), outlining core courses, electives, credits, capstone/thesis requirements, degree tracks, and honours criteria. It then presents a list of freely available learning resources (courses, lecture notes, textbooks, videos) for each major subject. The report also offers a study plan spanning four years, with weekly milestones and a resource-to-course mapping table. All sources reference official university materials or open platforms.

Pure Mathematics BSc Requirements at Various Universities

Stanford University (BS Mathematics)

Stanford’s curriculum requires a minimum of 57 units in Mathematics courses taken for a letter grade, covering at least eight upper-division courses (beyond Math 63CM/DM). In addition to calculus and linear algebra, students must take core courses such as Real Analysis, Abstract Algebra, and Topology, along with electives. A capstone requirement (thesis or final project) can be fulfilled through an honours thesis, a collaborative project (MATH 195), or an advanced course.

Students who qualify for the Honors program must complete additional coursework (e.g. Math 120, 171, 197) and an honours thesis, with a minimum of 7 graduate-level courses covering Algebra, Analysis, and Geometry.

The total unit requirement post-GIR (General Institute Requirements) is 180. A typical path: Calculus I-II, Linear Algebra, and Discrete Math in the first year; Real Analysis and Advanced Algebra in the second year; Topology, Geometry, and additional electives in the third year; and capstone and advanced courses in the fourth year.

MIT (Course 18, BS Mathematics - Pure Option)

MIT requires the following core courses: 18.03 (Differential Equations), 18.100 (Real Analysis), 18.701-18.702 (Algebra I-II), and 18.901 (Topology). Additionally, students choose one course from Advanced Analysis options (18.101, 18.102, 18.103) and one special seminar (e.g. Analysis Seminar, Logic, etc.). Two additional free upper-division (advanced) courses are also required. The total units in the major are 108 (total SB degree: 180).

MIT does not have a formal undergraduate qualifying exam, but students are encouraged to participate in the Putnam competition and research. For honours, MIT students can write a BS thesis by enrolling in Math 199.

Typical path: Year 1 covers Calculus (18.01-18.02), Linear Algebra (18.06/18.700), and ODEs; Year 2 begins Real Analysis (18.100) and Abstract Algebra; Year 3 enters pure elective courses; Year 4 completes the seminar/final project.

UC Berkeley (BA or BS Mathematics)

Berkeley requires 5 lower-division courses: Math 51-54 (Calculus I-IV or equivalent) and Math 55 (Discrete Mathematics). Required upper-division courses include Math 104 (Real Analysis I), 110 (Linear Algebra), 113 (Abstract Algebra I), and 185 (Complex Analysis). Additionally, students choose two semi-electives (covering Computation, Geometry, or Logic topics) and two other mathematics electives, for a total of eight upper-division courses.

There is no formal qualifying exam; high-achieving students may pursue an honours thesis with faculty supervision.

Typical path: Year 1 focuses on Calculus and Discrete Math; Year 2 moves to Real Analysis and Algebra; Year 3 covers Topology/Geometry/Advanced Analysis; Year 4 focuses on the thesis or advanced coursework.

Harvard University (AB Mathematics)

Harvard applies a mathematics concentration requiring a minimum of 12 letter-graded courses, of which at least 8 must carry a Mathematics label and the remainder in a related accredited field. Of the mathematics courses, there must be at least one Analysis course, one Algebra course, and one Geometry/Topology course.

Students must also submit an expository paper (~5 pages) in the third year as a requirement, and are advised to take a senior thesis in the fourth year if targeting High Honors. Without a thesis, students can earn Straight Honors by taking four additional mathematics courses.

Harvard also offers a Concurrent AB-AM option (MMath degree) over 4 years. The curriculum is flexible: Year 1 generally covers Calculus, Linear Algebra, and Discrete Structures; Year 2 continues to Advanced Analysis; Year 3 includes Geometry, Mathematical Statistics, or advanced topics; Year 4 is for the thesis and advanced electives.

Free Learning Resources per Core Subject

Below are freely available resources (Creative Commons or Public Domain) for subjects commonly required in pure mathematics programmes:

Calculus (Single Variable & Multivariable)

Linear Algebra

Differential Equations (ODE/PDE)

Real Analysis

Abstract Algebra

Topology

Complex Analysis

Discrete Mathematics / Logic

Statistics & Probability

All OCW sources listed above are freely accessible and carry Creative Commons licensing. Other open sources (Paul’s Notes, Khan Academy, OpenStax, etc.) provide text and video materials at no cost.

Four-Year Study Plan

The following study plan outlines a self-paced approach to undergraduate mathematics over four years (8 semesters). It assumes the student begins at a standard high school level, with 4-6 hours of study (lectures, practice, review) per day as a starting range.

Year 1, First Semester (Introductory Phase)

Typically covers introductory Calculus and Discrete Mathematics.

Year 1, Second Semester (Deepening)

Continue to Calculus II and III.

Year 2 (Advanced Theory)

Focus on upper-division core courses.

Year 3 (Specialization)

Choose advanced electives (complex analysis, algebraic geometry, computation, etc.).

Year 4 (Integration & Capstone)

Complete remaining requirements and the capstone.

Contingency: Adjust daily/weekly workload as needed. If this week’s material isn’t mastered, add a remedial session on the weekend. If already comfortable, use extra time for advanced problems or a mini research project.

Downloadable Study Roadmaps

Below are CSV files you can open in any spreadsheet app (Excel, Google Sheets, LibreOffice Calc). Each file contains a week-by-week checklist for one year: topics to cover, resources, and a checkbox column for each day.

Course to Resource and Weekly Study Hours Mapping

Below is an example mapping of common courses to primary free resources and estimated weekly study hours (average):

CoursePrimary Free ResourcesEst. Hours/Week (incl. lectures)
Calculus I-IIMIT OCW 18.01; Paul’s Online Notes; Khan Academy10-15 hrs
Linear Algebra IMIT OCW 18.06 (Strang); Khan Academy Linear Algebra8-12 hrs
Differential EquationsMIT OCW 18.03; Paul’s Notes ODE; Khan Academy ODE6-10 hrs
Real AnalysisMIT OCW 18.100; Understanding Analysis (Abbott)10-15 hrs
Abstract AlgebraAbstract Algebra: T&A (Judson, free); Khan Academy8-12 hrs
TopologyMIT OCW 18.901; Munkres Topology (older edition)6-10 hrs
Complex AnalysisMIT OCW 18.04; Brown & Churchill (older edition)6-10 hrs

Study hours above are total estimates (lectures + self-study). Workload may increase around exam periods.

Note: All MIT OCW sources listed are freely accessible with Creative Commons licensing. Other open sources (Paul’s Notes, Khan Academy, OpenStax, etc.) provide text and video materials at no cost.

Sources: Curriculum information taken from official university faculty/department websites (Stanford Math Major, MIT Course 18, UC Berkeley Math Major, Harvard Math Concentration), as well as MIT OCW course materials and open textbooks referenced above.