MA3J9 Historical Challenges in Mathematics

Lecturer: Damiano Testa

Term(s): 2

Status for Mathematics students:

Commitment: 30 lectures, support classes

Assessment: Two hour exam 85% and Assignments 15%

Prerequisites: Mathematics core: MA131 Analysis, MA106 Linear Algebra, MA251 Algebra 1, MA249 Algebra 2, MA259 Multivariate Calculus (or equivalents)

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, 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.