# An Undergraduate Curriculum in Mathematics

*by *

### Dr. Alexander Coward

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.

# Base Curriculum

## Pure Mathematics

### Foundations

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

- (Open-access book with Exercises) Single Variable Calculus: Late Transcendentals by Prof. David Guichard of Whitman College. (Chapters 1–12.)
- (Lecture Notes with Problem Sets) Introduction to University Mathematics by Prof. Alan Lauder of the University of Oxford.
- (Lecture Notes with Exercises) Introduction to Logic and Set Theory by Dr. Ahuva C. Shkop of Ben-Gurion University of the Negev.

### Algebra

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

- (Lecture Notes with Problem Sets) Introduction to Complex Numbers by Dr. Peter Neumann of the University of Oxford.
- (Lecture Notes with Problem Sets) Introduction to Group Theory by Prof. Jeremy Rickard of the University of Bristol.
- (Lecture Notes with Problem Sets) Linear Algebra I by Dr. Peter Neumann of the University of Oxford.
- (Lecture Notes with Problem Sets) Group Theory by Dr. Richard Earl of the University of Oxford.
- (Lecture Notes with Problem Sets) IB Linear Algebra by Dr. Simon Wadsley of the University of Cambridge.
- (Lecture Notes with Problem Sets) A3: Rings and Modules by Dr. Richard Earl of the University of Oxford.
- (Lecture Notes with Exercises) Introduction to Representation theory by Prof. Pavel Etingof of Massachusetts Institute of Technology.
- (Lecture Notes with Exercises) Fields and Galois Theory by Prof. James Milne of the University of Michigan.

### 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).

- (Lecture Notes with Problem Sets) M2: Analysis - Sequences and Series by Prof. Hilary Priestley of the University of Oxford.
- (Lecture Notes with Problem Sets) M2: Analysis - Continuity and Differentiability by Prof. Hilary Priestley of the University of Oxford.
- (Lecture Notes with Problem Sets) M2: Analysis - Integration by Prof. Ben Green of the University of Oxford.
- (Lecture Notes with Exercises) Metric and Topological Spaces by Prof. Tom Körner of the University of Cambridge. (Co-listed with Geometry and Topology.)
- (Lecture Notes with Problem Sets) Math 206: Measure Theory by Prof. John Hunter of the University of California at Davis.
- (Lecture Notes with Exercises) Complex Analysis by Prof. Christian Berg of the University of CopenHagen.
- (Lecture Notes with Exercises) Lectures in Functional Analysis by Prof. Roman Vershynin of the University of California, Irvine.

### Number Theory and Combinatorics

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

- (Lecture Notes with Problem Sets) Number Theory by Prof. Ben Green of the University of Oxford.
- (Lecture Notes with Problem Sets) Math 155: Combinatorics by Prof. Jacob Lurie of Harvard University.
- (Lecture Notes with Exercises) Math 485: Graph Theory by Prof. Christopher Griffin of Penn State University.
- (Problem Sets and Accompanying Reading List) Combinatorial geometry by Dr. Bartosz Walczak of Jagiellonian University.
- (Lecture Notes with Problem Sets) C8.4 Probabilistic Combinatorics by Dr. Michal Przykucki of the University of Oxford.

### 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.

- (Lecture Notes with Exercises) MA2219: An Introduction to Geometry by Dr. Yan Loi Wong of the National University of Singapore.
- (Lecture Notes with Problem Sets) Math 433/533: Differential Geometry by Prof. Richard Koch of the University of Oregon. (Problem sets here.)
- (Lecture Notes with Exercises) Metric and Topological Spaces by Prof. Tom Körner of the University of Cambridge. (Co-listed with Analysis.)
- (Lecture Notes with Exercises) Math 477: Projective Geometry by Dr. William D. Gillam of Boğaziçi University. (Additional problem sets here.)
- (Lecture Notes with Exercises) Introduction to Differential Forms by Prof. Donu Arapura of Purdue University.
- (Open-access book with Exercises) Algebraic Topology by Prof. Allen Hatcher of Cornell University. (Chapters 0, 1 and 2.)
- (Lecture Notes with Exercises and Solutions) Introduction to Differentiable Manifolds by Prof. Robertus Vandervorst of VU University Amsterdam.
- (Lecture Notes with Exercises) An Introduction to Riemannian Geometry by Prof. Sigmundur Gudmundsson of Lund University.

## 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

- (Lecture Notes with Exercises) MA1104: Multivariable Calculus by Dr. Yan Loi Wong of the National University of Singapore.
- (Lecture Notes with Exercises) MA3220: Ordinary Differential Equations by Dr. Yan Loi Wong of the National University of Singapore.
- (Lecture Notes with Problem Sets) Dynamics and Special Relativity by Dr. Stephen Siklos of the University of Cambridge. (Alternative Lecture Notes and problem sets by Prof. David Tong for the same course available here.)
- (Lecture Notes with Exercises and Solutions) Calculus of Variations and Optimal Control by Dr. George Halikias of City University. (Exercises and Solutions available here.)
- (Lecture Notes with Exercises) Part IB and Part II Electromagnetism by Prof. David Tong of the University of Cambridge.
- (Lecture Notes with Problem Sets) A11: Quantum Theory by Prof. Andrew Hodges of the University of Oxford.

### Probability Theory

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

- (Open-access book with Exercises) Introduction to Probability by Prof. Charles Grinstead of Swarthmore College and Prof. J. Laurie Snell of Dartmouth College.
- (Lecture Notes with Problem Sets) MAT/STA 235A: Probability Theory by Prof. Dan Romik of UC Davis. (Open-access book to accompany the course available here.)
- (Lecture Notes with Problem Sets) MAT/STA 235B: Probability Theory by Prof. Dan Romik of UC Davis. (Open-access book to accompany the course available here.)
- (Lecture Notes with Problem Sets) ACM 217: Stochastic Calculus and Stochastic Control by Dr. Ramon van Handel of Princeton University.

### Engineering Mathematics

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

- (Lecture Notes with Exercises) Lecture Notes in Fourier Analysis by Prof. Mohammad Asadzadeh of Chalmers University of Technology.
- (Lecture Notes with Problem Sets.) A10: Fluids and Waves by Prof. Irene Moroz of the University of Oxford.
- (Online Course with Video Lectures, Lecture Notes and Problem Sets) Statistical Mechanics I: Statistical Mechanics of Particles by Prof. Mehran Kardar of Massachusetts Institute of Technology.

### Mathematical Biology

General Overview: Population Modeling. Mathematical Physiology.

- (Lecture Notes with Problem Sets) ASO: Mathematical Modeling in Biology by the University of Oxford. Lecture Notes by Ruth Baker.
- (Lecture Notes with Problem Sets) An Introduction to Mathematical Physiology by Prof. Andrew Fowler the University of Limerick.

### 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.

- (Open-access Course with Exercises) Scipy Lecture Notes edited by Dr. Gaël Varoquaux of INRIA, Dr. Emmanuelle Gouillart of CNRS and Prof. Olav Vahtras of KTH Royal Institute of Technology, Stockholm.
- (Open-access Book with Exercises) Introduction to Python for Econometrics, Statistics and Numerical Analysis by Dr. Kevin Sheppard of the University of Oxford.
- (Lecture Notes with Problem Sets) M3: Statistics and Data Analysis by the University of Oxford. Lecture Notes by Neil Laws.
- (Lecture Notes with Exercises) Introduction to Statistical Thinking by Prof. Benjamin Yakir of The Hebrew University of Jerusalem.
- (Lecture Notes with Problem Sets) Part IB Optimization by Dr. Mike Tehranchi of the University of Cambridge.
- (Lecture Notes with Problem Sets and Solutions) MATH20602 - Numerical Analysis 1 by Dr. Martin Lotz of the University of Manchester. (Problem sets and solutions are available in the week-by-week summary pages.)
- (Lecture Notes with Problem Sets and Solutions) MATH36022 - Numerical Analysis 2 by Dr. Martin Lotz of the University of Manchester.
- (Lecture Notes with Exercises) Mathematics of Machine Learning by Dr. Christopher Griffin of Penn State University.

### Complexity, Cryptography, Game Theory and Computer Science

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

- (Lecture Notes with Problem Sets) Codes and Cryptograpy by Dr. Keith Carne of the University of Cambridge.
- (Lecture Notes with Exercises) Math 486: Game Theory by Dr. Christopher Griffin of Penn State University.
- (Lecture Notes with Exercises and Solutions) Concise Notes on Data Structures and Algorithms by Prof. Christopher Fox of James Madison University.
- (Lecture Notes with Exercises) CS-121: Introduction to Theoretical Computer Science by Prof. Boaz Barak of Harvard University.
- (Open-access Course with Open-access Book, Problem Sets and Projects) CS 251: Bitcoin and Cryptocurrency Technologies by Prof. Edward Felten, Dr. Arvind Narayanan and Steven Goldfeder of Princeton University, Dr. Joseph Bonneau of New York University and Dr. Andrew Miller of the University of Illinois, Urbana-Champaign.