An Undergraduate Curriculum in Mathematics

by

Dr. Alexander Coward

Senior Director of Studies in Mathematics


Introduction

This curriculum is intended for students who want to learn mathematics to the standard of someone graduating as a math major from a top university. 

Topics range from classical geometry dating back to Ancient Greece, through pioneering work done in the 17th, 18th and 19th centuries, to modern mathematics that forms the foundations of current research. 

The course of study balances pure mathematical topics like Galois Theory and Geometry with elements of applied mathematics like Fluid Mechanics and Statistical Mechanics. It also covers important topics in mathematical physics such as Special Relativity and Quantum Theory, as well as topics in theoretical computer science and the mathematical foundations of Machine Learning and Blockchain. 

How to use this Curriculum

At a high level, this curriculum divides mathematics into to halves: pure mathematics and applied mathematics. Under each heading, there are a broad topics such as “Algebra”, “Analysis” and so forth. These broad topics are not sorted in any particular way.

Within each broad topic however, the individual courses are generally speaking in the order that they should be studied. Despite the fact that so many notes are available, with accompanying problem sets, it would be very difficult indeed to work through all these unaided, perhaps even impossible. If you have a family member or friend who can help you they would be a good place to start.

Once you’ve figured out a course to start with, the next thing is to work through it. It’s a much better idea to pick one course and work all the way to the end rather than picking about at different courses here and there. Also, it’s absolutely essential to do all the exercises and get feedback on them, and address little points you don’t understand as you go.

Good luck with your studies!

Base Curriculum

Pure Mathematics

Foundations

General Overview: Review of High-School Calculus, writing Mathematics at University level, elementary Logic and Set Theory.

Algebra

General Overview: Introduction to Complex Numbers, Group Theory, Linear Algebra, Rings and Modules, Representation Theory, Galois Theory.

Analysis

General Overview: Real analysis, Integration (including Lebesgue Integration), Measure Theory, Metric Spaces, Topological Spaces, Complex Analysis, Functional Analysis (i.e. Banach Spaces and Hilbert Spaces). 

Number Theory and Combinatorics

General Overview: Elementary Number Theory, Introductory Combinatorics, Graph Theory, Combinatorial Geometry, Probabilistic Combinatorics. 

Geometry and Topology

General Overview: Classical Geometry of the Plane. Elementary geometry in ℝ^3. Metric and Topological Spaces. Projective Geometry. Differential Forms. Algebraic Topology. Differentiable Manifolds and Riemannian Metrics.


Applied Mathematics

Mathematical Methods and Mathematical Physics

General Overview: Multivariable Calculus, Ordinary Differential Equations, Introductory Dynamics, Numerical Analysis, Introductory Special Relativity, Calculus of Variations, Electromagnetism, Introductory Quantum Theory

Probability Theory

General Overview: Combinatorial Probability, Measure Theory, Applied Probability, Stochastic Analysis, Stochastic Calculus. 

Engineering Mathematics

General Overview: Fourier Analysis, Fluid Dynamics, Statistical Mechanics.

Mathematical Biology

General Overview: Population Modeling. Mathematical Physiology. 

Computational Mathematics, Statistics, Data Analysis, Optimization and Machine Learning

General Overview: Using Python for Mathematics, Numerical Analysis, Introduction to Statistics, Statistical Inference, Optimization, Foundations of Machine Learning.

Complexity, Cryptography, Game Theory and Computer Science

General Overview: Complexity Theory, Cryptography, Game Theory, Data Structures and Algorithms, Theoretical Computer Science, Foundations of Blockchain.