###Continued fractions and total positivity in enumerative combinatorics
Project ID: 2228bd1094 (You will need this ID for your application)
Research Theme: Mathematical Sciences
UCL Lead department: Mathematics
Lead Supervisor: Alan Sokal
Project Summary:
The expression of combinatorial generating functions as continued fractions goes back to Euler (1746), but it has gained impetus in recent years, especially as a tool for proving total positivity. It connects with many other areas of mathematics and mathematical physics. The aim of this project is to find and apply new continued fractions, including the branched continued fractions recently introduced by Petreolle, Sokal and Zhu. The student will be trained in all aspects of enumerative combinatorics, including both algebraic and bijective techniques, and in symbolic computation. The goal will be to discover new continued fractions, in part by computer experimentation, and then to prove them by bijective or algebraic methods. Where possible we will also apply these continued fractions to prove total positivity. We will be working with Professor Sokal’s collaborators around the world, notably Jiang Zeng (Lyon, France) and various colleagues in China, France and Spain.
The student should have a good background in undergraduate mathematics, computer science or mathematical physics.