2026-09-20T20:03:03Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/129459 2025-10-20 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-10-19 without embargo terms The student, Omar Musleh, accepted the attached license on 2025-04-28 at 03:26. The student, Omar Musleh, submitted this Dissertation for approval on 2025-04-28 at 19:46. This Dissertation was approved for publication on 2025-04-30 at 12:29. DSpace SAF Submission Ingestion Package generated from Vireo submission #22019 on 2025-10-19 at 18:19:24 A∞-algebras on directed graphs Musleh, Omar 2025-04-30 Pascaleff, James Dunfield, Nathan M Lerman, Eugene M Janda, Felix A∞-algebras Hochschild cohomology Directed graphs Maurer-Cartan equation eng Consider a directed graph Γ and a field k. We construct an associative algebra A(Γ) over k from Γ, whose elements are linear combinations of directed edges, where the multiplication is defined to be zero. The algebra on A(Γ) can be equipped with a graded structure by assigning integer degrees to the edges of Γ. We are particularly interested in the Hochschild cohomology CH∗(A(Γ)) of this graded algebra. Our primary focus is the A∞-algebras on A(Γ), which emerge as solutions to the Maurer-Cartan equation in CH∗(A(Γ)). Under this structure, the product of edges corresponds to their concatenation, though degree constraints on the edges significantly limit the number of such non-trivial structures. The second chapter develops the A∞-algebraic structures on directed graphs. This begins with a discussion of Hochschild cohomology on ungraded algebras, which then extends to graded ones. A∞-algebras are subsequently defined as degree 2 Hochschild cochains that satisfy the Maurer-Cartan equation. Through deformed Hochschild cohomology, an A∞-algebraic structure is then constructed for directed graphs. Later sections introduce tools and notations essential for simplifying work on these structures. The third chapter explores Directed Acyclic Graphs (DAGs), defined as graphs lacking loops. The absence of loops ensures that the Hochschild cochains associated with these graphs are finite-dimensional, which ensures that the MC equation becomes trivial beyond a certain degree. This chapter investigates such graphs, exploring degree assignments that maximize the number of products possible on them, referred to as MCAGs. Additionally, it introduces methods for visualizing and classifying the A∞-algebraic structures, documenting computational results derived using Wolfram Mathematica for these graphs with 5 and 6 vertices. The fourth chapter suggests various possible extensions to this research, revolving around discovering patterns and defining graph operations. The appendix provides the Mathematica file containing the code used to obtain computational results. 2025-05 Thesis https://hdl.handle.net/2142/129459 Copyright 2025 Omar Musleh Mathematics Mathematics University of Illinois Urbana-Champaign Ph.D. Dissertation