# MA3J9 Historical Challenges in Mathematics

**Lecturer: **Damiano Testa

**Term(s): **Term 1

**Status for Mathematics students:**

**Commitment: **30 lectures, support classes

**Assessment: **3 hour exam 85% and Assignments 15%

**Formal registration prerequisites: **None

**Assumed knowledge:**** **Core Maths modules in Year 2 especially MA249 Algebra II: Groups and Rings and MA251 Algebra 1: Advanced Linear Algebra.

**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.

**Content:**

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

** ****Aims: **

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.

**Objectives:**

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.

**Books:
**

Depending on the topics, different sources will be used. Most will be available online or with provided lecture notes.