Cellular Sheaves of Hilbert Spaces Julian Joseph Gould

Format:

Book

Thesis/Dissertation

Author/Creator:

Gould, Julian Joseph, author.

Contributor:

University of Pennsylvania. Mathematics., degree granting institution.

Language:

English

Subjects (All):

0364.

0405.

0753.

0984.

Local Subjects:

0364.

0405.

0753.

0984.

Physical Description:

1 electronic resource (190 pages)

Contained In:

Dissertations Abstracts International 87-07B

Place of Publication:

Ann Arbor : ProQuest Dissertations and Theses, 2025

Language Note:

English

Summary:

This dissertation extends the theory of cellular sheaves from finite-dimensional to infinite-dimensional Hilbert spaces, thereby broadening the scope of cellular sheaf theory through the incorporation of functional and non-smooth analytic techniques. While classical cellular sheaves, particularly weighted cellular sheaves valued in finite-dimensional Hilbert spaces, have found applications in network analysis, opinion dynamics, and neural networks, some applications naturally require sheaves valued in infinite-dimensional spaces. The passage from finite to infinite dimensions introduces fundamental complications that necessitate careful theoretical development. When restriction maps are unbounded operators with partial domains, the composition of morphisms requires precise domain considerations, cochain complexes may fail to satisfy the standard hypotheses for cohomology theory, and even elementary sheaf operations become problematic. This work systematically addresses these challenges through a trio of technical tools: the restriction categories of Cockett and Lack, the formalism of Hilbert complexes as developed by Brüning and Lesch, and the analysis of block operators between direct sums of Hilbert spaces. The central construction of this thesis is the Hilbert sheaf. Pre-Hilbert sheaves are introduced as functors from combinatorially well-behaved acyclic categories to the category of Hilbert spaces and unbounded operators. While these objects generalize weighted cellular sheaves directly, they may exhibit pathological behavior. The dissertation therefore identifies necessary conditions for well-behaved objects, leading to the definition of Hilbert sheaves proper. A Hilbert sheaf is a pre-Hilbert sheaf whose associated coboundary operators are closable, ensuring the formation of genuine Hilbert complexes. The theoretical framework encompasses several key developments. First, it establishes conditions under which Hilbert sheaves admit meaningful cohomology groups and spectral theory. Second, it identifies distinguished classes including bounded Hilbert sheaves (where all restriction maps are bounded) and closed Hilbert sheaves (where coboundary operators have closed range), each possessing favorable computational and theoretical properties. Third, it develops dynamical systems on these sheaves, including heat flows, wave propagation, and nonlinear diffusion processes, which serve as tools for the study of consensus problems. This work establishes cellular sheaves of Hilbert spaces as a rigorous mathematical framework in the intersection of algebraic topology, functional analysis, and applied mathematics, opening new avenues for the analysis of complex systems with infinite-dimensional local structure

Notes:

Advisors: Ghrist, Robert Committee members: Merling, Mona; North, Paige Randall

Source: Dissertations Abstracts International, Volume: 87-07, Section: B.

Ph.D. University of Pennsylvania 2025

Vendor supplied data

Local Notes:

School code: 0175

ISBN:

9798276001463

Access Restriction:

Restricted for use by site license