Persistent homology : categorical structural theorem and stability through representations of quivers

Field of study: Mathematics.Dr. Calin Chindris, Dissertation Supervisor; Dr. Jan Segert, Dissertation Supervisor.Includes vita."May 2018."The purpose of this thesis is to advance the study and application of the field of persistent homology through both categorical and quiver theoretic viewpoints. While persistent homology has its roots in these topics, there is a wealth of material that can still be offered up by using these familiar lenses at new angles. There are three chapters of results. Chapter 3 discusses a categorical framework for persistent homology that circumvents quiver theoretic structure, both in practice and in theory, by means of viewing the process as factored through a quotient category. In this chapter, the widely used persistent homology algorithm collectively known as reduction is presented in terms of a matrix factorization result. The remaining results rest on a quiver theoretic approach. Chapter 4 focuses on an algebraic stability theorem for generalized persistence modules for a certain class of finite posets. Both the class of posets and their discretized nature are what make the results unique, while the structure is taken with inspiration from the work of Ulrich Bauer and Michael Lesnick. Chapter 5 deals with taking directed limits of posets and the subsequent expansion of categories to show that the discretized work in the second section recovers classical results over the continuum.Includes bibliographical references (pages 161-166)

Identification of Clinically Relevant Protein Targets in Prostate Cancer with 2D-DIGE Coupled Mass Spectrometry and Systems Biology Network Platform

Prostate cancer (PCa) is the most common type of cancer found in men and among the leading causes of cancer death in the western world. In the present study, we compared the individual protein expression patterns from histologically characterized PCa and the surrounding benign tissue obtained by manual micro dissection using highly sensitive two-dimensional differential gel electrophoresis (2D-DIGE) coupled with mass spectrometry. Proteomic data revealed 118 protein spots to be differentially expressed in cancer (n = 24) compared to benign (n = 21) prostate tissue. These spots were analysed by MALDI-TOF-MS/MS and 79 different proteins were identified. Using principal component analysis we could clearly separate tumor and normal tissue and two distinct tumor groups based on the protein expression pattern. By using a systems biology approach, we could map many of these proteins both into major pathways involved in PCa progression as well as into a group of potential diagnostic and/or prognostic markers. Due to complexity of the highly interconnected shortest pathway network, the functional sub networks revealed some of the potential candidate biomarker proteins for further validation. By using a systems biology approach, our study revealed novel proteins and molecular networks with altered expression in PCa. Further functional validation of individual proteins is ongoing and might provide new insights in PCa progression potentially leading to the design of novel diagnostic and therapeutic strategies