# MA3J9 Historical Challenges in Mathematics

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

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:

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.