Abstracts
for the Seminar

Fall 2014

**Speaker: ** Raazesh Sainudiin, University of Canterbury and Cornell

**Title: **Some Arithmetic, Algebraic and Combinatorial Aspects of Plane Binary Trees

**Time:** 2:30 PM, Monday, October 27, 2014

**Place:** Malott 206

**Abstract:**
Plane binary trees underpinning Thompson's group are closed under union operations.
The first part of this talk will exploit this property to implement arithmetic operations recursively over such trees. Depending on interest, I'll give some insights into the algebra and arithmetic of plane binary trees for purposes of (1) statistical operations in non-parametric regression and density estimation (2) rigorous operations over inclusion algebras in computer-aided proofs in analysis.
The second part will describe a family of 'split-exchangeable' distributions on these trees using 'Catalan coefficients', which give
the number of distinct ways in which you can obtain any plane
binary tree by sequentially splitting the leaves from the root node.
This is analogous to how the binomial distribution is obtained
from the binomial coefficients. I will try to motivate why these
distributions on tree spaces arise naturally in (i) some consistent
density estimation rules, (ii) biodiversity models of speciation and
extinction in macro-evolution, (iii) infection trees in epidemics on
various hidden contact graphs and (iv) causal trees underlying
self-exciting point processes, among others.