## DEPARTMENTAL PHD THESIS EXAM – Abraham Isgur

Everyone welcome.  Refreshments will be served in the Math Lounge before the exam.

The thesis can be viewed at

https://sites.google.com/site/adisgur/thesis

Monday, February 27, 2012
10:10 a.m., in BA 2195, 40 St. George Street

PhD Candidate:  Abraham Isgur

Thesis Title:  Solving Nested Recursions With Trees

Thesis Abstract:
This thesis concerns the use of labelled infinite trees to solve families of nested recursions of the form
$$R(n)=\sum_{i=1}^kR(n-a_i-\sum_{j=1}^pR(n-b_{ij}))+\nu ,$$
where $a_i$ is a nonnegative integer, $\nu$ is any integer, and $b_{ij},k,$ and $p$ are natural numbers. We show that the solutions to many families of such nested recursions have an intriguing combinatorial interpretation, namely, they count nodes on the bottom level of labelled infinite trees that correspond to the recursion. Furthermore, we show how the parameters defining these recursion families relate in a natural way to specific structural properties of the corresponding tree families. We introduce a general tree “pruning” methodology that we use to establish all the required tree-sequence correspondences.