Termcard for Hilary 2012
Unless otherwise stated, events take place on a Tuesday in the Maths Institute at around 8:15pm
Wednesday Week 2, 25th Jan - Complex Numbers, Quaternions and Beyond
Sam Evington
In 1835 Hamilton gave a rigorous definition of a complex number as simply an order pair of real numbers (a,b). He defined addition coordinatewise and multiplication by (a,b)(c,d) = (ac-bd,ad+bc). He then went on to show that the expected properties follow from these definitions and nothing else. There’s no need to pluck a square root of -1 out of thin air!
This leads one naturally to ask if there is a similar rule for multiplying triples of reals (a,b,c). Given the many applications of complex numbers to 2-dimensional geometry and physics a generalisation to the 3 dimensions of space would be immensely useful.
Alas we shall see that no such system exists. All is not lost however. If one ventures into a fourth dimension (and gives up the notion that A times B should equal B times A) then a generalisation of complex numbers does exist. These are the Quaternions, and we shall establish their basic theory then look at their applications in 3-dimensional geometry, the invention of vector and modern physics.
If time permits we shall look beyond Quaternions and discover why (with perhaps 1 exception) there’s nothing else worthy of being called a number system.
Wednesday Week 8, 7th Mar - Graph Theory: an Introduction to Cycle Spectra
Lauren Kutler
One attribute of graphs that can be investigated is cycle length. Roughly, a graph has a cycle of length n if you can pick some vertex, then travel along n edges back to that initial vertex without going through the same vertex twice (excepting, of course, the initial vertex). One 'interesting' kind of graph has all possible different cycle lengths. We call such a graph pancyclic. Graph Theorists have been studying these graphs for the past 40 years or so, asking questions such as what conditions guarantee pancyclicity in an arbitrary graph. Recently, there has been some interest in a modification of this kind of question, asking instead what conditions guarantee that a graph has at least p different cycles lengths, where |p| has been dubbed the 'cycle spectrum' of the graph. I had the privilege of investigating this question during a summer research project. I will discuss our findings along with some of the background preceding our research, as well as other current work. The talk should be very accessible to mathematicians in any year (and hopefully interesting too).
