Non-Commutative Probability for the Spectral Analysis of Simplicial Complexes

Free probability theory, invented by Voiculescu, and greatly expanded by Speicher, is a young and active area of research with numerous applications in pure and applied mathematics. This Master thesis is a comprehensive study of a specific result in the recent preprint by C. Vargas, in which Vargas presents a survey of applications of non-commutative and free probability to topological data analysis. The relevant result from the preprint reveals a new interpretation of Betti numbers for simplicial complexes in terms of distributions in an operator-valued probability space. This thesis is mostly an exposition of the areas of free probability and algebraic topology; here, we do not present cutting-edge research in either free probability or algebraic topology. The author did a literature review for both fields and presents here the results in a comprehensive way along with detailed proofs and motivating examples that one may not find in a research paper. We believe that this thesis would help researchers to quickly grasp the main ideas and tools in both fields, and we hope it will help to advance the research in both areas and to develop applications in related areas