Student Profile: Tim Banova
Tim is a student at the School of Mathematical Sciences at Monash University. Having just finished his Bachelor of Science, he is keen to begin his Honours year. In the future, he is interested in continuing studying graph theory, combinatorial geometry and other forms of combinatorial problems. Outside of mathematics, Tim enjoys theatre, playing guitar and playing video games.
Exploring Combinatorial Geometry
Combinatorial geometry is the study of the combinatorial properties of arrangements of geometric objects and is a rich source of simply stated but difficult open problems. It combines elements of combinatorics, especially graph theory, with ideas from linear algebra, convexity theory, topology and algebraic geometry. Together, we will select an open problem, study its history and existing partial solutions in the literature, and look for new ways to attack it.
Here is just one example: Given any drawing of the complete graph in the plane with straight edges, how many colours are needed to colour the edges so that no two edges of the same colour cross or share an endpoint?
Here is a second example: Consider a set of great circles arranged on the sphere so that no three cross at a common point. This arrangement can be viewed as the drawing of a planar graph whose vertices are the crossing points, and whose edges are the arcs between crossing points. Since it is planar, it is 4-colourable. But can such graphs also be 3-coloured?