Mathematics Course Outline
This course is designed to give students an introduction to the areas of maths that they are likely to study as part of a mathematics degree. The topics cover a broad range of areas in pure and applied mathematics, formal proof, and metamathematics.
The following sections are a useful background to what we will be doing:
- Chapter 1 (all) Natural Numbers and Induction
- 1 Sets: Definitions and Notation
- 1 Addressing quantifiers. Analysing Proofs
Day 1: Algebra I – Polynomials
- What are polynomials?
- The solutions of polynomial equations in a single unknown through history. Students will see solutions to cubic equations, learn the impossibility of solutions by radicals of higher order polynomials and discover some of the history surrounding the topic.
- Diophantine equations. Students will discover the idea of a Diophantine equation and consider possible techniques that could be used to solve them.
- Fermat’s Last Theorem.
Day 2: Algebra II – Groups
- What are groups?
- Why are they important?
- Symmetry groups and their applications to shapes and permutations of strings.
Day 3: Algebra III - Linear algebra
- How can we do things in n-dimensions and how can we generalise what know from linear simultaneous equations?
- What are matrices? Students will be able to learn how to use matrices to encode simple and then more complicated sets of simultaneous equations. They will also learn about abstract concepts to do with linear maps of vector spaces.
- How can we use matrices to solve problems in algebra? Students will learn about determinants and inverses of matrices and how they can be applied.
- Cayley-Hamilton Theorem.
Day 4: Logic
- How can you formalise a rule set into symbols?
- How do propositional and predicate logics work and how can you produce ‘true’ statements using rules.
- What is the difference between truth and being provable?
Day 5: Dynamics
- How can we use mathematics to study motion?
- Newtonian models for dynamics and motion.
- Differential equations and tangent fields
- How could we use computers to solve problems that are otherwise difficult or impossible?
Day 6: Analysis I: Limits
- Why is it useful to formalise the language we use to prove things in maths?
- What are sequences and series?
- What does it mean to say a sequence or series converges to a limit? Students will consider epsilon definitions of limits and learn how to construct definitions which then allow you to analyse problems which at first seem a bit vague in a concrete and precise way.
Day 7: Analysis II: Differentiation
- What does it mean to say a function is continuous or differentiable?
- What are indeterminate forms?
- Students will learn about L’Hopital’s rule and how it can be used to evaluate things which might at first appear to be indeterminate.
Day 8: Probability and statistics
- Why is probability useful?
- How can we use probability to predict future events?
- Students will learn about the Poisson distribution and how it can be used to model random events.
Day 9: Geometry
- How did Euclid axiomatize geometry?
- What are axioms?
- What would happen if we lived in a world without parallel lines?
- Students will learn about Euclidean and non-Euclidean geometry and how they foundations laid by the Greeks set a standard which is now followed all over the mathematical world.
Day 10: Set theory
- What are numbers? Students will learn about set theoretic definitions of counting numbers.
- What is infinity?
- Is it possible to have something bigger than infinity? Students will learn about different cardinalities and how we can and cannot compare some ‘infinitely big’ sets using techniques such as Cantor’s diagonal slice.