Introductory description

The module will cover several topics each year, usually inspired by questions asked by David Hilbert in his address to the International Congress of Mathematicians in 1900.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Sample Topic 1: 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 2: Hilbert's 17th problem and Sums of squares
Polynomials having only non-negative values. Polynomials and rational functions that are sums of squares. Real closed fields.

Sample Topic 3: Hilbert's 10th problem and Undecidability
Decidability, recursively enumerable set and Diophantine sets. Computing and algorithms.

Sample Topic 4: Four colour theorem
Graphs, colourings. Five colour theorem. The role of computers.

Sample Topic 5: Fermat's little theorem and RSA Cryptography
Residue classes modulo primes. Fermat's little theorem. Cryptographic applications. May include Elliptic Curve factorisation.

Learning outcomes

By the end of the module, students 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.

Subject specific skills

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.
Develop tools and techniques to approach and solve problems from a variety of sources.

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.