Date of Award

Spring 2019

Project Type

Dissertation

Program or Major

Mathematics

Degree Name

Doctor of Philosophy

First Advisor

David V Feldman

Second Advisor

John F Gibson

Third Advisor

Don Hadwin

Abstract

Every set X, finite of cardinality n say, carries a set M(X) of all possible pseudometrics. It is well known that M(X) forms a convex polyhedral cone whose faces correspond to triangle inequalities. Every point in a convex cone can be expressed as a conical sum of its extreme rays, hence the interest around discovering and classifying such rays. We shall give examples of extreme rays for M(X) exhibiting all integral edge lengths up to half the cardinality of X. By intersecting the cone with the unit cube we obtain the convex polytope of bounded-by-one pseudometrics BM(X). Analogous to extreme rays, every point in a convex polytope arises as a convex combination of extreme points. Extreme rays of BM(X) give rise to very special extreme points of ̄BM(X) as we may normalize a nonzero pseudometric to make its largest distance 1. We shall give a simple and complete characterization of extremeness for metrics with only edge lengths equal to 1/2 and 1. Then we shall use this characterization to give a decomposition result for the upper half of BM(X). BM(X) contains the set of bounded-by-1 pseudoultrametrics, U(X). Ultrametrics satisfy a stronger version of the triangle inequality, and have an interesting structure expressed in terms of partition chains. We will describe the topology of U(X) and its subset of scaled ultrametrics, SU(X), up to homotopy equivalence. Every permutation on a set X can be written as a product of disjoint cycles that cover X. In this way, a permutation generalizes a partition. An iterated cycle structure (ICS) will then be the associated generalization of a partition chain. Analogously, we will compute the “Euler-characteristic” of the set of iterated cycle structures.

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