Lecturer: Damiano Testa
Term(s): Term 2
Status for Mathematics students:
Commitment: 30 lectures, support classes
Assessment: 3 hour exam 85% and Assignments 15%
Formal registration prerequisites: None
Useful background: Throughout the module, we will work with Abelian groups (finite, finitely generated, free, though also not satisfying any of the previous assumptions), Polynomials and rational functions, extending a field by adding a square root of one of its elements (we will review this in class).
Synergies: Depending on the week, the material that we will cover leads to the following modules:
- MA3A6 Algebraic Number Theory
- MA3G6 Commutative Algebra
- MA257 Introduction to Number Theory
- MA3D5 Galois Theory
- MA3H3 Set Theory though we will introduce everything that we use from these modules.
The module will cover several topics each year. Below is a list of possible topics:
- Sample Topic 1: Fermat's little theorem and RSA Cryptography
- Residue classes modulo primes, Fermat's little theorem, Cryptographic applications. May include Elliptic Curve factorisation
- Sample Topic 2: Hilbert's 10th problem and Undecidability
- Decidability, recursively enumerable set and Diophantine sets, Computing and algorithms
- Sample Topic 3: Hilbert's 3rd problem and Dehn invariants
- Scissor congruence in the plane, Scissor congruence in R^n and Hilbert's 3rd problem, Dehn invariant for R^3
- Sample Topic 4: Four colour theorem
- Graphs, colourings, Five colour theorem, the role of computers
To show how a range of problems both theoretical and applied can be modelled mathematically and solved using tools discussed in core modules from years 1 and 2.
By the end of the module the student should be able to:
- For each of the topics discussed appreciate their importance in the historical context, and why mathematicians at the time were interested in it.
- For each of the topics discussed understand the underlying theory and statement of the result, and where applicable how the proof has been developed (or how a proof has been attempted in the case of unsolved problems).
- For each of the topics discussed understand how to apply the theory to similar problems/situations (where applicable).
- For each of the topics discussed understand the connections between the results/proofs in question and the core mathematics modules that the student has studied.
Depending on the topics, different sources will be used. Most will be available online or with provided lecture notes.