From: "Daniel Ueltschi" Sender: ueltschid@googlemail.com To: "John Kerl" Subject: Topics for exam Cc: "Tom Kennedy" Date: Fri, 13 Jun 2008 19:42:07 +0100 Hello again, After reflection, I could suggest the following topics for your comprehensive exam. Most can be found in my article with Betz, or in the proceeding. Some of the background needs to come from elsewhere. 1. Model of spatial permutations. General definition. The case of the lattice. Description of main result about occurrence of infinite cycles. 2. Description of the steps for proving the existence of infinite cycles. It involves certain lemmas that could be stated without proof. 3. Relation with quantum Bose gas. Again, you should know the results, without necessarily understanding all the steps. But it would be great if you can precisely state which steps you are still missing. 4. The simple model with 2-cycle interactions. Conjecture for the critical density (recall that it can be computed, but it is not proved that it corresponds to infinite cycles). This model may be the first one that you will investigate numerically. Let me know if you have questions! Daniel -- Daniel Ueltschi Mathematics, Warwick University http://www.ueltschi.org