Mathematics is one of the oldest academic subjects. Ideas relating to number and magnitude date back to pre-historic times, and all the major ancient civilizations have left a legacy that still can be seen in the modern study of mathematics.

Modern mathematics can be broadly divided into two main types, pure mathematics and applied mathematics. Pure mathematics considers abstract questions that may or may not be related to the world we live in, and seeks to find precise answers in the form of proofs or counterexamples. Because of the precision required, it is wise to start with a grounding in deductive logic, and gradually build on that to form a precise understanding of what we mean by common ideas such as set, number, and function.

Applied mathematics has a different emphasis. Instead of being concerned with precise proofs, applied mathematicians use ideas from mathematics to shed insight on real world phenomena ranging from economics, to biology, to physics.

Duration: Three to four years (full-time study); six to eight years (part-time study)

Difficulty: Advanced

Entry Requirements: Undergraduate degree or equivalent general education

The core topics introduce the field of mathematics and give students the skills and understanding to choose and pursue their interests in the specialized topics. The Director of Studies will make adaptations based on student ability and schedule. On completing the core, students will have the grounding necessary to explore advanced mathematics. The core topics are:

- Logic for Mathematicians
- Real Analysis
- Groups, Rings and Fields
- Linear Algebra
- Complex Analysis
- Elementary Number Theory
- Elementary Geometry
- Point-Set Topology

- Multivariate Calculus
- Differential Equations and Fourier Analysis
- Mechanics
- Elementary Probability
- Calculus of Variations
- Introductory Statistics
- Complexity and Cryptography
- Special Relativity and Elementary Quantum Theory
- Scientific Programming

Specializations should be discussed with the Director of Studies. On completing specializations, students will be able to independently read advanced literature and conduct a final project in that topic. Some suggested specializations are:

- Galois Theory
- Functional Analysis
- Combinatorics
- Differential and Riemannian Geometry
- Advanced Mathematical Physics
- Stochastic Analysis
- Mathematical Biology

A final project in mathematics consists in one or more papers written on an area of specialization. Sometimes advanced students may prove a new theorem and submit the resulting paper for publication. More commonly, students will write an expository dissertation on a topic of interest. All final projects should be discussed with your Director of Studies, who will assist you in choosing an appropriate area to focus on.

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