My First Conference

摘要

It began when I received an email from the maths department describing a conference for undergrad mathematicians. They encouraged people to attend, and possibly present. I had some ideas I had been playing with, and I thought it would be a good idea to get feedback from mathematicians outside my small circle of friends. So I wrote up an abstract and applied to speak. My talk was about drawing graphs without self-intersections in 3-dimensional space. The analogous problem for two dimensions is well known and completely solved by Kuratowski's theorem, a theorem of graph theory. In three dimensions, the flavour of the problem completely changes. The extra dimension gives us so much more room that it's much easier to draw a graph without self-intersections, and the natural tools to use become those of topology. In fact, it turns out that the extra dimension gives us so much more room that any graph can be drawn without self-intersections by arbitrarily small perturbations of its vertices! (More generally, any subset of R~n can be put in a general linear position by an arbitrarily small perturbation.)